diff --git a/LeanPool/MetricCodes/Binary.lean b/LeanPool/MetricCodes/Binary.lean
index 79a3e72ca..4b81ea01d 100644
--- a/LeanPool/MetricCodes/Binary.lean
+++ b/LeanPool/MetricCodes/Binary.lean
@@ -424,7 +424,7 @@ theorem tendsto_logb_supportHarmonicDimension
   · have hb : 0 < b := lt_of_le_of_ne h.support_nonneg
       (Ne.symm hbzero)
     have hba : b < a := by
-      nlinarith [h.support_lt_half, h.weight_pos]
+      linarith [h.support_lt_half, h.weight_pos]
     have hhalf :
         ∀ᶠ n : ℕ in atTop,
           2 * supportDegree b n ≤ shellWeight a n :=
@@ -459,7 +459,7 @@ theorem tendsto_logb_complementHarmonicDimension
   · have hg : 0 < g := lt_of_le_of_ne h.complement_nonneg
       (Ne.symm hgzero)
     have hga : g < 1 - a := by
-      nlinarith [h.complement_lt_half, h.weight_complement_pos]
+      linarith [h.complement_lt_half, h.weight_complement_pos]
     have hhalf :
         ∀ᶠ n : ℕ in atTop,
           2 * complementDegree g n ≤ n - shellWeight a n :=
@@ -1241,7 +1241,7 @@ theorem splitTensor_pairing {n w p q : ℕ}
       rw [Finset.mul_sum]
       apply Finset.sum_congr rfl
       intro B _
-      ring
+      ac_rfl
 
 theorem splitTensor_orthonormal {n w p q r s : ℕ}
     (x : JohnsonSphere n w)
@@ -1405,7 +1405,7 @@ theorem dot_fintype_weighted_sum
       rw [Finset.mul_sum]
       apply Finset.sum_congr rfl
       intro S _
-      ring
+      ac_rfl
 
 theorem coupledTensor_dot {n w p q t : ℕ}
     (x : JohnsonSphere n w)
@@ -2308,9 +2308,7 @@ def johnsonUpperScale (n j : ℕ) : ℝ :=
 theorem johnsonMiddleScale_pos {n j : ℕ}
     (hj : 0 < j) (hhalf : 2 * j < n) :
     0 < johnsonMiddleScale n j := by
-  have hj' : (0 : ℝ) < (j : ℝ) := by exact_mod_cast hj
   have hn : 0 < n := by omega
-  have hn' : (0 : ℝ) < (n : ℝ) := by exact_mod_cast hn
   have hgap := johnsonHarmonicGap_pos hhalf
   have hlast : 0 < (n : ℝ) - (j : ℝ) + 1 := by
     have hcast : (j : ℝ) < (n : ℝ) := by
@@ -3147,7 +3145,7 @@ theorem johnsonMiddleChannel_isometry {n j : ℕ}
             Finset.mul_sum]
           apply Finset.sum_congr rfl
           intro a _
-          ring
+          ac_rfl
     _ = ((Real.sqrt (johnsonMiddleScale n j))⁻¹ *
           (Real.sqrt (johnsonMiddleScale n j))⁻¹) *
         (johnsonMiddleScale n j * MetricCodes.Boolean.dot f g) := by
@@ -3156,7 +3154,7 @@ theorem johnsonMiddleChannel_isometry {n j : ℕ}
           (Real.sqrt (johnsonMiddleScale n j))⁻¹ *
           johnsonMiddleScale n j) *
         MetricCodes.Boolean.dot f g := by
-          ring
+          ac_rfl
     _ = MetricCodes.Boolean.dot f g := by
           rw [hscalar, one_mul]
 
@@ -3516,7 +3514,7 @@ theorem johnsonUpperChannel_isometry {n j : ℕ}
             Finset.mul_sum]
           apply Finset.sum_congr rfl
           intro a _
-          ring
+          ac_rfl
     _ = ((Real.sqrt (johnsonUpperScale n j))⁻¹ *
           (Real.sqrt (johnsonUpperScale n j))⁻¹) *
         (johnsonUpperScale n j * MetricCodes.Boolean.dot f g) := by
@@ -3525,7 +3523,7 @@ theorem johnsonUpperChannel_isometry {n j : ℕ}
           (Real.sqrt (johnsonUpperScale n j))⁻¹ *
           johnsonUpperScale n j) *
         MetricCodes.Boolean.dot f g := by
-          ring
+          ac_rfl
     _ = MetricCodes.Boolean.dot f g := by
           rw [hscalar, one_mul]
 
@@ -3783,7 +3781,7 @@ theorem johnsonCoordinateDot_smul {n : ℕ}
     Finset.mul_sum]
   apply Finset.sum_congr rfl
   intro a _
-  ring
+  ac_rfl
 
 theorem johnsonLowerChannel_orthogonal_middleChannel {n j : ℕ}
     (f g : MetricCodes.Boolean.Function n)
@@ -4345,7 +4343,7 @@ theorem johnsonDot_axisRaise {n w : ℕ}
   intro a _
   apply Finset.sum_congr rfl
   intro S _
-  ring
+  ac_rfl
 
 theorem johnsonDot_axisLower {n w : ℕ}
     (x : JohnsonSphere n w)
@@ -4362,7 +4360,7 @@ theorem johnsonDot_axisLower {n w : ℕ}
   intro a _
   apply Finset.sum_congr rfl
   intro S _
-  ring
+  ac_rfl
 
 theorem johnsonDot_axisMembership {n w : ℕ}
     (x : JohnsonSphere n w)
@@ -4380,7 +4378,7 @@ theorem johnsonDot_axisMembership {n w : ℕ}
   intro a _
   apply Finset.sum_congr rfl
   intro S _
-  ring
+  ac_rfl
 
 theorem johnsonLowerChannel_axis_inner {n w j : ℕ}
     (x : JohnsonSphere n w)
@@ -4398,7 +4396,7 @@ theorem johnsonLowerChannel_axis_inner {n w j : ℕ}
   rw [johnsonDot_axisRaise, Finset.mul_sum]
   apply Finset.sum_congr rfl
   intro a _
-  ring
+  ac_rfl
 
 theorem johnsonUpperRaw_dot_harmonic_right {n j k : ℕ}
     (f g : MetricCodes.Boolean.Function n)
@@ -4434,7 +4432,7 @@ theorem johnsonUpperChannel_axis_inner {n w j k : ℕ}
   rw [johnsonDot_axisLower, Finset.mul_sum]
   apply Finset.sum_congr rfl
   intro a _
-  ring
+  ac_rfl
 
 theorem johnsonMiddleRaw_dot_harmonic_right {n j k : ℕ}
     (f g : MetricCodes.Boolean.Function n)
@@ -4592,7 +4590,7 @@ theorem sum_geometricAxis_on_subset {n w : ℕ}
     _ = _ := by
       rw [sum_coordinateIndicator_on_subset x S,
         MetricCodes.Boolean.sum_mem_indicator]
-      ring
+      ac_rfl
 
 theorem johnsonAxisMembership_apply {n w : ℕ}
     (x : JohnsonSphere n w)
@@ -4680,7 +4678,7 @@ theorem johnsonDot_fintype_weighted_sum_right
   intro i _
   apply Finset.sum_congr rfl
   intro S _
-  ring
+  ac_rfl
 
 theorem johnsonAxisMembership_coupledTensor {n w p q : ℕ}
     (x : JohnsonSphere n w)
@@ -4758,7 +4756,7 @@ theorem johnsonAxisMembership_coupledHarmonic_dot
   simp only [Finset.mul_sum]
   apply Finset.sum_congr rfl
   intro r _
-  ring
+  ac_rfl
 
 theorem coordinateSplitEquiv_erase_support {n w : ℕ}
     (x : JohnsonSphere n w) (i : SupportCoordinates x)
@@ -4989,7 +4987,7 @@ theorem raise_splitTensor_eq {n w p q r s : ℕ}
     complementRaisedFunction_raise x hq hcomplement]
   simp only [Pi.add_apply, Pi.smul_apply, smul_eq_mul,
     splitTensor]
-  ring
+  ac_rfl
 
 theorem johnsonHarmonic_dot_splitTensor_succ_mul
     {n w p q t r : ℕ}
@@ -5117,7 +5115,7 @@ theorem johnsonHarmonic_dot_coupledHarmonic
         clebschNormSq w (n - w) p q t =
       Real.sqrt (clebschNormSq w (n - w) p q t) := by
     field_simp [hsqrt]
-    nlinarith [hsq]
+    linarith [hsq]
   calc
     (Real.sqrt (clebschNormSq w (n - w) p q t))⁻¹ *
         (∑ r : Fin (t + 1),
@@ -6011,7 +6009,7 @@ theorem clebschFirstMoment_eq
       (((w : ℝ) - 2 * (p : ℝ)) +
         ((N : ℝ) - 2 * (q : ℝ)) - 2 * (t : ℝ) + 2) *
         clebschFirstMoment w N p q t := by
-          ring
+          ac_rfl
     _ = (t : ℝ) *
         ((N : ℝ) - 2 * (q : ℝ) - (t : ℝ) + 1) *
           clebschNormSq w N p q t :=
@@ -6490,8 +6488,6 @@ theorem clebschCenteredExpectation_eq_johnsonDiagonal
   have hn : 0 < n := by omega
   have hnreal : 0 < (n : ℝ) := by exact_mod_cast hn
   have hwreal : 0 < (w : ℝ) := by exact_mod_cast h.weight_pos
-  have hNreal : 0 < ((n - w : ℕ) : ℝ) := by
-    exact_mod_cast (Nat.sub_pos_of_lt h.weight_lt)
   have hjbound := h.window_degree_le_weight source
   have hjreal :
       2 * ((p + q + source.val : ℕ) : ℝ) < (n : ℝ) := by
@@ -6502,10 +6498,6 @@ theorem clebschCenteredExpectation_eq_johnsonDiagonal
       exact_mod_cast hstrict
     push_cast at hs
     linarith
-  have hjgap :
-      0 < (n : ℝ) -
-        2 * ((p + q + source.val : ℕ) : ℝ) := by
-    linarith
   have hJ :
       0 < MetricCodes.johnsonJ n (p + q + source.val) := by
     unfold MetricCodes.johnsonJ
@@ -6553,7 +6545,7 @@ theorem clebschCenteredExpectation_eq_johnsonDiagonal
   have hden'' :
       (n : ℝ) - 2 * (p : ℝ) - 2 * (q : ℝ) -
         2 * (source.val : ℝ) + 2 ≠ 0 := by
-    nlinarith [hden']
+    linarith [hden']
   have hJ' :
       (n : ℝ) / 2 -
         ((p : ℝ) + (q : ℝ) + (source.val : ℝ)) ≠ 0 := by
@@ -6772,7 +6764,6 @@ theorem johnsonMiddleSignedScalar_eq_sqrt_hattedDiagonal
   have hjw : j ≤ w := h.window_degree_le_weight source
   have hjhalf : 2 * j < n := by omega
   have hn : 0 < n := by omega
-  have hnreal : 0 < (n : ℝ) := by exact_mod_cast hn
   have hwreal : 0 < (w : ℝ) := by exact_mod_cast h.weight_pos
   have hcomp : 0 < ((n - w : ℕ) : ℝ) := by
     exact_mod_cast (Nat.sub_pos_of_lt h.weight_lt)
@@ -6786,13 +6777,7 @@ theorem johnsonMiddleSignedScalar_eq_sqrt_hattedDiagonal
       exact_mod_cast hstrict
     push_cast at hs
     linarith
-  have hF : 0 < F := by
-    dsimp [F]
-    positivity
   have hmiddle := johnsonMiddleScale_pos hsource hjhalf
-  have haxis : 0 < (n : ℝ) /
-      ((w : ℝ) * ((n - w : ℕ) : ℝ)) := by
-    positivity
   have hS : 0 < S := by
     dsimp [S]
     positivity
@@ -6887,7 +6872,7 @@ theorem booleanHarmonicDimension_mul_degreeComplement
       rw [Nat.cast_sub hk] at hchoose
       rw [MetricCodes.booleanHarmonicDimension_succ, Nat.cast_sub hmono]
       push_cast at hchoose ⊢
-      nlinarith [hchoose]
+      linarith [hchoose]
 
 theorem booleanHarmonicDimension_succ_div
     {n j : ℕ} (hhalf : 2 * (j + 1) ≤ n) :
@@ -7139,7 +7124,7 @@ theorem clebschSignedAdjacentCross_sum_comm
   rw [← clebschSignedAdjacentCross_sum hsupport hcomplement]
   apply Finset.sum_congr rfl
   intro r _
-  ring
+  ac_rfl
 
 theorem johnsonAxisLower_coupledHarmonic_dot_closed
     {n w p q t : ℕ}
@@ -7317,7 +7302,7 @@ theorem johnsonMiddleChannel_coupled_axis_inner_of_split_proportional
       htsupport htcomplement a)]
   rw [johnsonAxisMembership_coupledHarmonic_dot_of_split_proportional
     x hp hq htsupport htcomplement a f hpair]
-  ring
+  ac_rfl
 
 theorem johnsonMiddleChannel_coupled_axis_inner_closed
     {n w p q t : ℕ}
@@ -7937,7 +7922,6 @@ theorem johnsonAdjacentRawScalar_sq
   have hn : 0 < n := by omega
   have haxis :
       0 ≤ (n : ℝ) / ((w : ℝ) * ((n - w : ℕ) : ℝ)) := by
-    have hw' : 0 < (w : ℝ) := by exact_mod_cast hw
     have hN : 0 < ((n - w : ℕ) : ℝ) := by
       exact_mod_cast Nat.sub_pos_of_lt hwn
     positivity
@@ -7984,8 +7968,6 @@ theorem johnsonAdjacentRawScalar_pos
     (hcomplement : 2 * q + (t + 1) ≤ n - w) :
     0 < johnsonAdjacentRawScalar n w p q t := by
   have hn : 0 < n := by omega
-  have hnreal : 0 < (n : ℝ) := by exact_mod_cast hn
-  have hwreal : 0 < (w : ℝ) := by exact_mod_cast hw
   have hcomp : 0 < ((n - w : ℕ) : ℝ) := by
     exact_mod_cast Nat.sub_pos_of_lt hwn
   have hcoefficient :
@@ -8204,8 +8186,6 @@ theorem johnsonLowerOffDiagonalScalar_eq_sqrt_source
       2 * q + (target.val + 1) ≤ n - w := by omega
   have hraw := johnsonAdjacentRawScalar_pos
     h.weight_pos h.weight_lt hsupport' hcomplement'
-  have hdegree : 0 < ((p + q + (target.val + 1) : ℕ) : ℝ) := by
-    positivity
   have hleft :
       0 < johnsonLowerOffDiagonalScalar n w p q target.val := by
     unfold johnsonLowerOffDiagonalScalar
diff --git a/LeanPool/MetricCodes/Branching.lean b/LeanPool/MetricCodes/Branching.lean
index 91e0d4e9c..496f177e7 100644
--- a/LeanPool/MetricCodes/Branching.lean
+++ b/LeanPool/MetricCodes/Branching.lean
@@ -296,7 +296,7 @@ theorem polarization_chain_commutator
     intro a _
     rw [SpherePacking.mvPolynomial_pderiv_commute
       (variableIndex j a) (variableIndex k b) p]
-    ring
+    ac_rfl
   have hleft :
       polarization r n i j (polarization r n j k p) =
         polarization r n i k p +
@@ -1005,7 +1005,7 @@ theorem actual_weyl_detailed_balance {r n : ℕ}
         minusProbability n (raiseWeight lam ℓ) mu ℓ := by
   rw [plusProbability_eq_weylEdgeRatio_mul_minus ℓ h hraise,
     weylDimension_raiseWeight ℓ h]
-  ring
+  ac_rfl
 
 end
 
@@ -1832,7 +1832,7 @@ theorem tendsto_plusProbability_of_ratios {r : ℕ}
     rw [mul_div_mul_comm, ← Finset.prod_div_distrib]
     congr 2
     funext j
-    ring
+    ac_rfl
   · unfold lagrangeWeight lagrangeNumerator lagrangeDenominator
     rw [Finset.prod_div_distrib]
 
@@ -2114,8 +2114,8 @@ theorem reachable_zero {r m : ℕ} (v : Vertex r m) :
         exact hreach.trans hedge
 
 theorem grid_connected (r m : ℕ) : (grid r m).Connected := by
-  apply (grid r m).connected_iff_exists_forall_reachable.mpr
-  exact ⟨(fun _ => 0), reachable_zero⟩
+  exact (SimpleGraph.connected_iff_exists_forall_reachable (grid r m)).mpr
+      (Exists.intro (fun _ ↦ 0) reachable_zero)
 
 private theorem forwardMatrix_nextVertex_pos_metriccodes2_dbb1ceac {r m : ℕ}
     (edge : Vertex r m → Fin (r + 1) → ℝ)
@@ -2901,7 +2901,7 @@ theorem eventually_const_mul_lt_rpow_of_logRate
       (Real.exp_log hpositive).symm
     _ < Real.exp (Real.log 2 * (R * (n : ℝ))) := by
       apply Real.exp_lt_exp.mpr
-      nlinarith
+      linarith
     _ = (2 : ℝ) ^ (R * (n : ℝ)) :=
       (Real.rpow_def_of_pos (by norm_num) _).symm
 
@@ -2982,7 +2982,7 @@ theorem fixedLevelHierarchyCodeBound_levelZero
   refine (hn C).trans ?_
   apply Real.strictMono_rpow_of_base_gt_one (by norm_num : (1 : ℝ) < 2)
   have hnreal : 0 < (n : ℝ) := by exact_mod_cast hnpos
-  nlinarith [mul_pos (sub_pos.mpr hR) hnreal]
+  linarith [mul_pos (sub_pos.mpr hR) hnreal]
 
 theorem fixedLevelHierarchyCodeBound_of_positiveTerminal
     (hpositive : ∀ {r : ℕ} {s R : ℝ}, 0 < s → s < 1 →
@@ -3008,7 +3008,7 @@ theorem fixedLevelHierarchyCodeBound_of_positiveTerminal
           exists_sameLevel_opening_strict_refinement
             (Nat.succ_pos r) hinterlacing hzero
         exact hpositive hs hs' A b hA hApos
-          (by nlinarith) (by linarith)
+          (by linarith) (by linarith)
 
 theorem fixedLevelHierarchyCodeBound_of_actualRectangularGraphs
     (hboxes : ∀ {r : ℕ} {s R : ℝ}, 0 < s → s < 1 →
@@ -3189,6 +3189,22 @@ def liftChannel {I : Type*} [Fintype I] [DecidableEq I]
   ((coordinateInclusion (n := n) lam target).toLinearMap.comp T.toLinearMap).comp
     (vertexInclusion (YoungVertex (n := n) lam) source).adjoint
 
+private theorem adjoint_comp_composed_maps
+    {E F F' G G' H : Type*}
+    [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
+    [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F]
+    [AddCommMonoid F'] [Module ℝ F']
+    [NormedAddCommGroup G] [InnerProductSpace ℝ G] [FiniteDimensional ℝ G]
+    [NormedAddCommGroup G'] [InnerProductSpace ℝ G'] [FiniteDimensional ℝ G']
+    [NormedAddCommGroup H] [InnerProductSpace ℝ H] [FiniteDimensional ℝ H]
+    (C : F →ₗ[ℝ] E) (T : G →ₗ[ℝ] F) (V : G →ₗ[ℝ] H)
+    (C' : F' →ₗ[ℝ] E) (T' : G' →ₗ[ℝ] F') (V' : G' →ₗ[ℝ] H) :
+    ((C.comp T).comp V.adjoint).adjoint.comp
+      ((C'.comp T').comp V'.adjoint) =
+        V.comp ((T.adjoint.comp ((C.adjoint.comp C').comp T')).comp V'.adjoint) := by
+  rw [LinearMap.adjoint_comp, LinearMap.adjoint_comp, LinearMap.adjoint_adjoint]
+  simp only [LinearMap.comp_assoc]
+
 theorem liftChannel_adjoint_comp_self
     {I : Type*} [Fintype I] [DecidableEq I]
     {r n : ℕ} (lam : I → Fin (r + 1) → ℕ) (target source : I)
@@ -3197,36 +3213,11 @@ theorem liftChannel_adjoint_comp_self
     (liftChannel lam target source T).adjoint.comp
       (liftChannel lam target source T) =
         vertexProjection (YoungVertex (n := n) lam) source := by
-  apply LinearMap.ext
-  intro v
-  apply ext_inner_right ℝ
-  intro w
-  change
-    ⟪(liftChannel lam target source T).adjoint
-        (liftChannel lam target source T v), w⟫_ℝ =
-      ⟪vertexInclusion (YoungVertex (n := n) lam) source
-        ((vertexInclusion (YoungVertex (n := n) lam) source).adjoint v),
-          w⟫_ℝ
-  rw [LinearMap.adjoint_inner_left]
-  calc
-    ⟪liftChannel lam target source T v,
-      liftChannel lam target source T w⟫_ℝ =
-        ⟪(vertexInclusion (YoungVertex (n := n) lam) source).adjoint v,
-          (vertexInclusion (YoungVertex (n := n) lam) source).adjoint w⟫_ℝ := by
-            change
-              ⟪coordinateInclusion (n := n) lam target
-                  (T ((vertexInclusion (YoungVertex (n := n) lam) source).adjoint v)),
-                coordinateInclusion (n := n) lam target
-                  (T ((vertexInclusion (YoungVertex (n := n) lam) source).adjoint w))⟫_ℝ = _
-            rw [(coordinateInclusion (n := n) lam target).inner_map_map,
-              T.inner_map_map]
-            rfl
-    _ = ⟪vertexInclusion (YoungVertex (n := n) lam) source
-          ((vertexInclusion (YoungVertex (n := n) lam) source).adjoint v),
-            w⟫_ℝ :=
-            LinearMap.adjoint_inner_right
-              (vertexInclusion (YoungVertex (n := n) lam) source).toLinearMap
-                ((vertexInclusion (YoungVertex (n := n) lam) source).adjoint v) w
+  unfold liftChannel
+  rw [adjoint_comp_composed_maps,
+    (coordinateInclusion (n := n) lam target).adjoint_comp_self',
+    LinearMap.id_comp, T.adjoint_comp_self', LinearMap.id_comp]
+  rfl
 
 theorem liftChannel_orthogonal_of_target_ne
     {I : Type*} [Fintype I] [DecidableEq I]
@@ -3239,40 +3230,10 @@ theorem liftChannel_orthogonal_of_target_ne
     (hne : target ≠ target') :
     (liftChannel lam target source T).adjoint.comp
       (liftChannel lam target' source' T') = 0 := by
-  apply LinearMap.ext
-  intro v
-  apply ext_inner_right ℝ
-  intro w
-  rw [LinearMap.zero_apply, inner_zero_left]
-  change
-    ⟪(liftChannel lam target source T).adjoint
-      (liftChannel lam target' source' T' v), w⟫_ℝ = 0
-  rw [LinearMap.adjoint_inner_left]
-  change
-    ⟪coordinateInclusion (n := n) lam target'
-        (T' ((vertexInclusion (YoungVertex (n := n) lam) source').adjoint v)),
-      coordinateInclusion (n := n) lam target
-        (T ((vertexInclusion (YoungVertex (n := n) lam) source).adjoint w))⟫_ℝ = 0
-  let z := T' ((vertexInclusion (YoungVertex (n := n) lam) source').adjoint v)
-  let q := T ((vertexInclusion (YoungVertex (n := n) lam) source).adjoint w)
-  have hzero := congrArg
-    (fun A : (SpherePacking.Euclidean n ⊗[ℝ]
-      YoungVertex (n := n) lam target') →ₗ[ℝ]
-        (SpherePacking.Euclidean n ⊗[ℝ]
-          YoungVertex (n := n) lam target) => A z)
-    (coordinateInclusion_orthogonal lam hne)
-  change
-    (coordinateInclusion (n := n) lam target).adjoint
-      (coordinateInclusion (n := n) lam target' z) = 0 at hzero
-  calc
-    ⟪coordinateInclusion (n := n) lam target' z,
-      coordinateInclusion (n := n) lam target q⟫_ℝ =
-        ⟪(coordinateInclusion (n := n) lam target).adjoint
-          (coordinateInclusion (n := n) lam target' z), q⟫_ℝ :=
-            (LinearMap.adjoint_inner_left
-              (coordinateInclusion (n := n) lam target).toLinearMap q
-                (coordinateInclusion (n := n) lam target' z)).symm
-    _ = 0 := by simp only [hzero, inner_zero_left]
+  unfold liftChannel
+  rw [adjoint_comp_composed_maps, coordinateInclusion_orthogonal lam hne,
+    LinearMap.zero_comp, LinearMap.comp_zero, LinearMap.zero_comp,
+    LinearMap.comp_zero]
 
 theorem liftChannel_orthogonal_of_clebsch
     {I : Type*} [Fintype I] [DecidableEq I]
@@ -3285,34 +3246,10 @@ theorem liftChannel_orthogonal_of_clebsch
     (horthogonal : T.adjoint.comp T'.toLinearMap = 0) :
     (liftChannel lam target source T).adjoint.comp
       (liftChannel lam target source' T') = 0 := by
-  apply LinearMap.ext
-  intro v
-  apply ext_inner_right ℝ
-  intro w
-  rw [LinearMap.zero_apply, inner_zero_left]
-  change
-    ⟪(liftChannel lam target source T).adjoint
-      (liftChannel lam target source' T' v), w⟫_ℝ = 0
-  rw [LinearMap.adjoint_inner_left]
-  change
-    ⟪coordinateInclusion (n := n) lam target
-        (T' ((vertexInclusion (YoungVertex (n := n) lam) source').adjoint v)),
-      coordinateInclusion (n := n) lam target
-        (T ((vertexInclusion (YoungVertex (n := n) lam) source).adjoint w))⟫_ℝ = 0
-  rw [(coordinateInclusion (n := n) lam target).inner_map_map]
-  let z := (vertexInclusion (YoungVertex (n := n) lam) source').adjoint v
-  let q := (vertexInclusion (YoungVertex (n := n) lam) source).adjoint w
-  have hzero := congrArg
-    (fun A : YoungVertex (n := n) lam source' →ₗ[ℝ]
-      YoungVertex (n := n) lam source => A z)
-    horthogonal
-  change T.adjoint (T' z) = 0 at hzero
-  calc
-    ⟪T' z, T q⟫_ℝ = ⟪T.adjoint (T' z), q⟫_ℝ :=
-      (LinearMap.adjoint_inner_left T.toLinearMap q (T' z)).symm
-    _ = 0 := by
-      rw [hzero]
-      exact inner_zero_left q
+  unfold liftChannel
+  rw [adjoint_comp_composed_maps,
+    (coordinateInclusion (n := n) lam target).adjoint_comp_self',
+    LinearMap.id_comp, horthogonal, LinearMap.zero_comp, LinearMap.comp_zero]
 
 theorem liftChannel_adjoint_axis_self
     {I : Type*} [Fintype I] [DecidableEq I]
@@ -4372,8 +4309,6 @@ theorem gramPivot_pair_eq_of_variable_eq {r n : ℕ}
     a b (gramPivot hn z) (gramPivot hn w)).mp h
   have hrow := congrArg Fin.val hindex.1
   have hcolumn := congrArg Fin.val hindex.2
-  have hz := z.property
-  have hw := w.property
   apply Subtype.ext
   apply Prod.ext
   · apply Fin.ext
@@ -4671,7 +4606,7 @@ theorem gramVariableWeight_nonneg {r n : ℕ}
     push_cast
     omega
   unfold gramVariableWeight
-  nlinarith [mul_nonneg (sub_nonneg.mpr hupp)
+  linarith [mul_nonneg (sub_nonneg.mpr hupp)
     (show 0 ≤ ((n + 2 * r : ℕ) : ℤ) +
       ((k.val : ℤ) - 2 * (i.val : ℤ)) by linarith)]
 
@@ -5114,10 +5049,8 @@ theorem representationDegree_add_sum_sum_lt
       representationDegree m b (g a a') < d) :
     representationDegree m b
       (r + ∑ a ∈ s, ∑ a' ∈ t, g a a') < d := by
-  apply lt_of_le_of_lt
-    (representationDegree_add_le m b r (∑ a ∈ s, ∑ a' ∈ t, g a a'))
-  exact (max_lt_iff).mpr
-    ⟨hr, representationDegree_sum_sum_lt m b s t g d hd hterm⟩
+  exact lt_of_le_of_lt (representationDegree_add_le m b r (∑ a ∈ s, ∑ a' ∈ t, g a a'))
+      (max_lt_iff.mpr ⟨hr, representationDegree_sum_sum_lt m b s t g d hd hterm⟩)
 
 theorem representationDegree_single_lt
     {σ K ι : Type*} [Field K]
@@ -5526,7 +5459,7 @@ private theorem buchberger_correction_pair_lt
               MvPolynomial.C cij *
                 (Pij * ((m.leadingTerm (b j) - b j) * b i)) := by
           simp only [Algebra.smul_def, algebraMap_eq]
-          ring
+          ac_rfl
         rw [hrewrite, m.degree_mul
           (MvPolynomial.C_ne_zero.mpr hc0) hterm,
           m.degree_C, zero_add]
@@ -5603,7 +5536,7 @@ private theorem buchberger_correction_pair_lt
               MvPolynomial.C cij *
                 (Pij * ((b i - m.leadingTerm (b i)) * b j)) := by
           simp only [Algebra.smul_def, algebraMap_eq]
-          ring
+          ac_rfl
         rw [hrewrite, m.degree_mul
           (MvPolynomial.C_ne_zero.mpr hc0) hterm,
           m.degree_C, zero_add]
@@ -6330,7 +6263,7 @@ theorem rowEuler_axialCoordinate_lowerPolarizationPath
   · subst a
     by_cases hstart : row = polarizationPathStart row S
     · rw [← hstart]
-      simp only [ite_true, raiseWeight, Function.update_self,
+      simp only [ite_true, raiseWeight.eq_1, Function.update_self,
         Nat.cast_add, Nat.cast_one, MvPolynomial.smul_eq_C_mul,
         map_add, map_one]
       ring
@@ -6346,7 +6279,7 @@ theorem rowEuler_axialCoordinate_lowerPolarizationPath
       ring
     · simp only [hastart, ↓reduceIte, zero_mul, MvPolynomial.smul_eq_C_mul, map_natCast, harow,
         add_zero, sub_zero, zero_add, raiseWeight, ne_eq, not_false_eq_true, Function.update_of_ne]
-      ring
+      ac_rfl
 
 theorem arbitraryRowAxialRaise_rowEuler
     {r n : ℕ} (lam : Fin (r + 1) → ℕ)
@@ -6902,7 +6835,7 @@ theorem simpleRoot_coordinate_both
     a b j hab hbj q A B hq hqa hqb]
   simp only [polarization_apply, MvPolynomial.smul_eq_C_mul, MvPolynomial.C_add, MvPolynomial.C_sub,
     MvPolynomial.C_1]
-  ring
+  ac_rfl
 
 end ArbitraryRowMickelssonSimpleRoot
 
diff --git a/LeanPool/MetricCodes/Conclusion.lean b/LeanPool/MetricCodes/Conclusion.lean
index 709b90209..7645fe129 100644
--- a/LeanPool/MetricCodes/Conclusion.lean
+++ b/LeanPool/MetricCodes/Conclusion.lean
@@ -26,6 +26,23 @@ namespace Spherical
 
 section
 
+open scoped BigOperators
+
+private theorem linearMap_comp_sum_comp
+    {ι E F G : Type*} [Fintype ι]
+    [AddCommMonoid E] [Module ℝ E]
+    [AddCommMonoid F] [Module ℝ F]
+    [AddCommMonoid G] [Module ℝ G]
+    (C : F →ₗ[ℝ] E) (T : ι → F →ₗ[ℝ] F) (J : G →ₗ[ℝ] F) :
+    C.comp ((∑ i, T i).comp J) = ∑ i, C.comp ((T i).comp J) := by
+  apply LinearMap.ext
+  intro x
+  simp only [LinearMap.comp_apply, LinearMap.sum_apply, map_sum]
+
+end
+
+section
+
 
 open scoped BigOperators InnerProductSpace
 
@@ -425,10 +442,10 @@ theorem gtAxisCompressedCharacteristicMinor_eval_neg_wallShift_eq_zero_of_last_e
     (p q : HarmonicYoungSpace (n := n) mu) :
     (gtAxisCompressedCharacteristicMinor lam mu h hgram p q).eval
       (-(wallShift (n + 1) (r + 1))) = 0 := by
-  apply gtAxisCompressedCharacteristicMinor_eval_neg_wallShift_eq_zero
-    lam mu h hgram hfinite hlast p q
-  exact gtAxisCompressedSignedProjectorCoefficient_last_false_eq_zero
-    hfinite.1 lam mu h hgram h.antitone_ambient hlast hfinite p q
+  exact gtAxisCompressedCharacteristicMinor_eval_neg_wallShift_eq_zero lam mu h hgram hfinite hlast
+    p q
+      (gtAxisCompressedSignedProjectorCoefficient_last_false_eq_zero hfinite.left lam mu h hgram
+        (Interlaces.antitone_ambient h) hlast hfinite p q)
 
 end AllRankGTAbsentWallCharacteristicFactor
 
@@ -978,7 +995,6 @@ theorem gtStabilizerArrowheadNode_injective_of_interlacing
     unfold stabilizerShift wallShift
     have hi : i.val ≤ r := by have := i.isLt; omega
     have hireal : (i.val : ℝ) ≤ r := by exact_mod_cast hi
-    have hmu : (0 : ℝ) ≤ mu i := Nat.cast_nonneg _
     push_cast
     linarith
   · have hanti : Antitone mu := by
@@ -2091,7 +2107,7 @@ theorem isometric_scaled_sector_adjoint_compression
       (T (I (C p))), hnode p, real_inner_smul_right]
   rw [map_smul, real_inner_smul_left, real_inner_smul_right,
     hpair, ← hinner]
-  ring
+  ac_rfl
 
 theorem normalizedGTTransverseNegativeSector_relativeCasimir_adjoint_compression
     {r n : ℕ} (lam : Fin (r + 2) → ℕ)
@@ -2508,28 +2524,15 @@ theorem appendedPaddedPieriLower_adjoint_paddedPhysicalStabilizerTensor_eq_zero
   let C := (normalizedPaddedPieriLower (n := n + 1)
     (appendZeroWeight lam) hdominant (Fin.last (r + 2)) hsource).toLinearMap
   let A := paddedPhysicalStabilizerTensor lam mu B
-  apply LinearMap.ext
-  intro p
-  change C.adjoint (A p) = 0
-  apply ext_inner_left ℝ
-  intro q
-  rw [inner_zero_right]
-  have hsum := canonicalFullBranch_sum_projection source hn hsource q
-  rw [← hsum, sum_inner]
-  apply Finset.sum_eq_zero
-  intro nu _
-  let F := (canonicalFullBranchFibre source hn nu).toLinearMap
-  obtain ⟨z, hz⟩ := Submodule.starProjection_apply_mem
-    (LinearMap.range F) q
-  change F z = (LinearMap.range F).starProjection q at hz
-  rw [← hz]
-  calc
-    ⟪F z, C.adjoint (A p)⟫_ℝ = ⟪C (F z), A p⟫_ℝ :=
-      LinearMap.adjoint_inner_right C (F z) (A p)
-    _ = ⟪A p, C (F z)⟫_ℝ := real_inner_comm (A p) (C (F z))
-    _ = 0 :=
-      appendedFullBranchPieriLower_paddedPhysicalStabilizerTensor_inner_eq_zero
-        lam mu hmu B hB hdominant hsource hn nu p z
+  apply adjoint_composition_eq_zero_of_spanning_ranges
+    (fun nu : FullBranchWeight source =>
+      HarmonicYoungSpace (n := n) (fullBranchSignature nu))
+    (fun nu => (canonicalFullBranchFibre source hn nu).toLinearMap)
+    (canonicalFullBranch_iSup_range_eq_top_unconditional source hn hsource) C A
+  intro nu x p
+  exact (real_inner_comm _ _).trans
+    (appendedFullBranchPieriLower_paddedPhysicalStabilizerTensor_inner_eq_zero
+      lam mu hmu B hB hdominant hsource hn nu p x)
 
 theorem paddedOrthogonalTensorPieriChannel_appended_stabilizerIntertwiner_orthogonal
     {r n : ℕ} (lam : Fin (r + 2) → ℕ)
@@ -3234,10 +3237,7 @@ theorem transverseTensorEmbeddingOfStabilizerIsometry_mixedRotation
             (transverseTensorEmbeddingOfStabilizerIsometry lam nu F).toLinearMap) = 0
     rw [transverseTensorEmbeddingOfStabilizerIsometry_rotationTerm,
       gtTransverseEuclideanIsometry_cross_rotation_compression]
-    apply TensorProduct.ext
-    ext v p
-    simp only [TensorProduct.map_zero_left, LinearMap.compr₂ₛₗ_apply, TensorProduct.mk_apply,
-      LinearMap.zero_apply]
+    exact TensorProduct.map_zero_left _
   have hcrossSwap (a : Fin n) :
       I.adjoint.comp
           ((TensorProduct.map
@@ -3251,10 +3251,7 @@ theorem transverseTensorEmbeddingOfStabilizerIsometry_mixedRotation
             (transverseTensorEmbeddingOfStabilizerIsometry lam nu F).toLinearMap) = 0
     rw [transverseTensorEmbeddingOfStabilizerIsometry_rotationTerm,
       gtTransverseEuclideanIsometry_cross_rotation_compression_swap]
-    apply TensorProduct.ext
-    ext v p
-    simp only [TensorProduct.map_zero_left, LinearMap.compr₂ₛₗ_apply, TensorProduct.mk_apply,
-      LinearMap.zero_apply]
+    exact TensorProduct.map_zero_left _
   have hlast :
       I.adjoint.comp
           ((TensorProduct.map
@@ -3278,20 +3275,11 @@ theorem transverseTensorEmbeddingOfStabilizerIsometry_mixedRotation
   change I.adjoint.comp ((∑ a : Fin (n + 1), ∑ b : Fin (n + 1),
     TensorProduct.map (euclideanAmbientRotation a b)
       (youngAmbientRotation lam a b)).comp I) = _
-  apply TensorProduct.ext'
-  intro v p
-  simp only [LinearMap.comp_apply, LinearMap.sum_apply, map_sum]
+  rw [linearMap_comp_sum_comp]
+  simp_rw [linearMap_comp_sum_comp]
   rw [Fin.sum_univ_castSucc]
   simp_rw [Fin.sum_univ_castSucc]
-  have htangential' (a b : Fin n) :=
-    LinearMap.congr_fun (htangential a b) (v ⊗ₜ[ℝ] p)
-  have hcross' (a : Fin n) :=
-    LinearMap.congr_fun (hcross a) (v ⊗ₜ[ℝ] p)
-  have hcrossSwap' (a : Fin n) :=
-    LinearMap.congr_fun (hcrossSwap a) (v ⊗ₜ[ℝ] p)
-  have hlast' := LinearMap.congr_fun hlast (v ⊗ₜ[ℝ] p)
-  simp only [LinearMap.comp_apply, LinearMap.zero_apply] at htangential' hcross' hcrossSwap' hlast'
-  simp only [htangential', hcross', hcrossSwap', hlast',
+  simp only [htangential, hcross, hcrossSwap, hlast,
     Finset.sum_const_zero, add_zero]
 
 theorem transverseTensorEmbeddingOfStabilizerIsometry_relativeCasimir
@@ -4529,17 +4517,8 @@ theorem physicalPaddedPieriChannel_relativeCasimir
         (HigherChannel.ambientShift n (appendZeroWeight lam))
         (paddedPieriSignedChannel (appendZeroWeight lam) i) •
         physicalPaddedPieriChannel hn lam hdom i p := by
-  exact zeroRowTransportPaddedPieriChannel_eigen_of_intertwine
-    lam
-    (paddedOrthogonalTensorPieriChannel hn
-      (appendZeroWeight lam) hdom i)
-    (zeroRowTensorIsometryEquiv_symm_relativeCasimir_intertwine lam).symm
-    (HigherChannel.signedNode
-      (HigherChannel.ambientShift n (appendZeroWeight lam))
-      (paddedPieriSignedChannel (appendZeroWeight lam) i))
-    p
-    (paddedOrthogonalTensorPieriChannel_relativeCasimir hn
-      (appendZeroWeight lam) hdom i p)
+  exact @AllRankGTSelectedPhysicalChannelRange.physicalPaddedPieriChannel_relativeCasimir r n hn lam
+    hdom i p
 
 end AllRankGTPhysicalPaddedPieriFullEigen
 
@@ -4950,7 +4929,7 @@ theorem negativeSector_shortTensor_adjoint_eigen
     I.toLinearMap.adjoint
       (gtRelativeCasimir (n := n + 1) lam (B p)) =
         d • I.toLinearMap.adjoint (B p) := by
-  dsimp
+  dsimp only
   let c : ℝ := (Real.sqrt
     (internalRowLowerGramScalar (raiseWeight mu row) row *
       weylEdgeRatio n mu row))⁻¹
@@ -4997,7 +4976,7 @@ theorem negativeSector_selectedFullTensor_adjoint_eigen
     J.toLinearMap.adjoint
       (gtRelativeCasimir (n := n + 1) lam (B p)) =
         d • J.toLinearMap.adjoint (B p) := by
-  dsimp
+  dsimp only
   apply adjoint_eigen_of_range_le
     (gtTransverseTensorEmbedding lam (raiseWeight mu row) hnu hnuGram).toLinearMap
     (gtFullTransverseEmbedding lam hn
@@ -5032,7 +5011,7 @@ theorem positiveSector_shortTensor_adjoint_eigen
     I.toLinearMap.adjoint
       (gtRelativeCasimir (n := n + 1) lam (B p)) =
         d • I.toLinearMap.adjoint (B p) := by
-  dsimp
+  dsimp only
   let c : ℝ := (Real.sqrt (internalRowLowerGramScalar mu row))⁻¹
   let C := youngClebschLower (n := n) nu mu
     (by rw [hmunu]; exact sum_raiseWeight nu row) row
@@ -5075,7 +5054,7 @@ theorem positiveSector_selectedFullTensor_adjoint_eigen
     J.toLinearMap.adjoint
       (gtRelativeCasimir (n := n + 1) lam (B p)) =
         d • J.toLinearMap.adjoint (B p) := by
-  dsimp
+  dsimp only
   apply adjoint_eigen_of_range_le
     (gtTransverseTensorEmbedding lam nu hnu hnuGram).toLinearMap
     (gtFullTransverseEmbedding lam hn
@@ -5269,7 +5248,7 @@ theorem normalizedWallSector_selectedFullTensor_adjoint_eigen
     J.toLinearMap.adjoint
       (gtRelativeCasimir (n := n + 1) lam (B p)) =
         d • J.toLinearMap.adjoint (B p) := by
-  dsimp
+  dsimp only
   apply adjoint_eigen_of_range_le
     (gtTransverseWallTensorEmbedding lam mu h hn hlast).toLinearMap
     (gtFullTransverseEmbedding lam hn
@@ -5333,16 +5312,8 @@ theorem transverseStabilizerIsometry_adjoint_comp_of_orthogonal
       F.toLinearMap).adjoint.comp
       (TensorProduct.map (gtTransverseEuclideanIsometry n).toLinearMap
         G.toLinearMap) = 0
-  rw [TensorProduct.adjoint_map]
-  apply TensorProduct.ext
-  ext v p
-  have hz := LinearMap.congr_fun horth p
-  change F.toLinearMap.adjoint (G p) = 0 at hz
-  change
-    (gtTransverseEuclideanIsometry n).toLinearMap.adjoint
-        (gtTransverseEuclideanIsometry n v) ⊗ₜ[ℝ]
-      F.toLinearMap.adjoint (G p) = 0
-  rw [hz, TensorProduct.tmul_zero]
+  rw [TensorProduct.adjoint_map, ← TensorProduct.map_comp, horth,
+    TensorProduct.map_zero_right]
 
 theorem transverseStabilizerIsometry_rotationTerm_cross
     {s t r n : ℕ} (lam : Fin (r + 1) → ℕ)
@@ -5414,10 +5385,7 @@ theorem transverseStabilizerIsometry_mixedRotation_cross_eq_zero
         ((youngAmbientRotation lam a.castSucc b.castSucc).comp G.toLinearMap) = 0 := by
       rw [← hG a b, ← LinearMap.comp_assoc, horth, LinearMap.zero_comp]
     rw [hz]
-    apply TensorProduct.ext
-    ext v p
-    simp only [TensorProduct.map_zero_right, LinearMap.compr₂ₛₗ_apply, TensorProduct.mk_apply,
-      LinearMap.zero_apply]
+    exact TensorProduct.map_zero_right _
   have hcross (a : Fin n) :
       I.adjoint.comp
         ((TensorProduct.map
@@ -5433,10 +5401,7 @@ theorem transverseStabilizerIsometry_mixedRotation_cross_eq_zero
             lam target G).toLinearMap) = 0
     rw [transverseStabilizerIsometry_rotationTerm_cross,
       gtTransverseEuclideanIsometry_cross_rotation_compression]
-    apply TensorProduct.ext
-    ext v p
-    simp only [TensorProduct.map_zero_left, LinearMap.compr₂ₛₗ_apply, TensorProduct.mk_apply,
-      LinearMap.zero_apply]
+    exact TensorProduct.map_zero_left _
   have hcrossSwap (a : Fin n) :
       I.adjoint.comp
         ((TensorProduct.map
@@ -5452,10 +5417,7 @@ theorem transverseStabilizerIsometry_mixedRotation_cross_eq_zero
             lam target G).toLinearMap) = 0
     rw [transverseStabilizerIsometry_rotationTerm_cross,
       gtTransverseEuclideanIsometry_cross_rotation_compression_swap]
-    apply TensorProduct.ext
-    ext v p
-    simp only [TensorProduct.map_zero_left, LinearMap.compr₂ₛₗ_apply, TensorProduct.mk_apply,
-      LinearMap.zero_apply]
+    exact TensorProduct.map_zero_left _
   have hlast :
       I.adjoint.comp
         ((TensorProduct.map
@@ -5480,22 +5442,11 @@ theorem transverseStabilizerIsometry_mixedRotation_cross_eq_zero
   change I.adjoint.comp ((∑ a : Fin (n + 1), ∑ b : Fin (n + 1),
     TensorProduct.map (euclideanAmbientRotation a b)
       (youngAmbientRotation lam a b)).comp J) = 0
-  apply TensorProduct.ext'
-  intro v p
-  simp only [LinearMap.comp_apply, LinearMap.sum_apply, map_sum,
-    LinearMap.zero_apply]
+  rw [linearMap_comp_sum_comp]
+  simp_rw [linearMap_comp_sum_comp]
   rw [Fin.sum_univ_castSucc]
   simp_rw [Fin.sum_univ_castSucc]
-  have htangential' (a b : Fin n) :=
-    LinearMap.congr_fun (htangential a b) (v ⊗ₜ[ℝ] p)
-  have hcross' (a : Fin n) :=
-    LinearMap.congr_fun (hcross a) (v ⊗ₜ[ℝ] p)
-  have hcrossSwap' (a : Fin n) :=
-    LinearMap.congr_fun (hcrossSwap a) (v ⊗ₜ[ℝ] p)
-  have hlast' := LinearMap.congr_fun hlast (v ⊗ₜ[ℝ] p)
-  simp only [LinearMap.comp_apply, LinearMap.zero_apply]
-    at htangential' hcross' hcrossSwap' hlast'
-  simp only [htangential', hcross', hcrossSwap', hlast',
+  simp only [htangential, hcross, hcrossSwap, hlast,
     Finset.sum_const_zero, add_zero]
 
 theorem transverseStabilizerIsometry_relativeCasimir_cross_eq_zero
@@ -5905,7 +5856,7 @@ theorem gtRelativeCasimir_normalizedWallSector_additiveColumn
       (-(wallShift (n + 1) (r + 1))) • B.toLinearMap +
         (canonicalGelfandTsetlinAxisTensor lam mu h hgram).comp
           (gtTransverseAxisCrossBlock lam mu h hgram B.toLinearMap) := by
-  dsimp
+  dsimp only
   let hw : 2 * (r + 1) + 5 ≤ n + 1 := by omega
   let B := normalizedGTTransverseWallSector lam mu h hw hlast
   let A := canonicalGelfandTsetlinAxisTensor lam mu h hgram
@@ -6070,9 +6021,6 @@ theorem negativeStabilizerNode_ne_signedAmbientNode_of_valid_raise
   · simp only [gtStabilizerArrowheadNode_neg, one_div, signedNode, Bool.false_eq_true, ↓reduceIte,
       ne_eq]
     intro heq
-    have hcoord :
-        stabilizerShift (n + 1) mu row + 1 / 2 =
-          ambientShift (n + 1) lam j := by linarith
     by_cases hji : j ≤ row.castSucc
     · have hmono := hfinite.ambientShift_strictAnti.antitone hji
       linarith
@@ -6747,7 +6695,7 @@ theorem gtAxisCompressedCharacteristicMinor_eval_eq_adjugate_inner
     gtAxisCompressedSignedProjectorCoefficient]
   apply Finset.sum_congr rfl
   intro channel _
-  ring
+  ac_rfl
 
 theorem gtAxisCompressedCharacteristicMinor_eval_eq_zero_of_arrowhead_operator_row
     {r n : ℕ} (lam : Fin (r + 2) → ℕ)
@@ -6997,11 +6945,9 @@ theorem gtSignedArrowheadCoupling_injective
     (hnode : ∀ i : Fin (r + 1) × Bool,
       d ≠ signedNode (HigherChannel.ambientShift n lam) i) :
     Function.Injective K := by
-  apply arrowhead_isometric_coupling_injective_of_characteristic
-    (gtRelativeCasimir (n := n) lam) B A K
-    (gtChannelCharacteristicPolynomial n lam) d hrow hchar
-  exact gtChannelCharacteristicPolynomial_eval_ne_zero_of_forall_ne
-    lam d hnode
+  exact arrowhead_isometric_coupling_injective_of_characteristic (gtRelativeCasimir lam) B A K
+      (gtChannelCharacteristicPolynomial n lam) d hrow hchar
+      (gtChannelCharacteristicPolynomial_eval_ne_zero_of_forall_ne lam d hnode)
 
 end AllRankGTCartanSpectralCouplingNonvanishing
 
@@ -7356,7 +7302,7 @@ theorem isometricSector_operator_column_of_mem_axis_sector
       rw [hx, hy]
     _ = d • B p + A (A.toLinearMap.adjoint (T (B p))) := by
       rw [map_smul]
-      module
+      ac_rfl
 
 theorem gtAxisCompressedCharacteristicMinor_eval_negativeValidNode_eq_zero
     {r n : ℕ} (lam : Fin (r + 2) → ℕ)
diff --git a/LeanPool/MetricCodes/Foundations.lean b/LeanPool/MetricCodes/Foundations.lean
index fedca76d3..f44b724a1 100644
--- a/LeanPool/MetricCodes/Foundations.lean
+++ b/LeanPool/MetricCodes/Foundations.lean
@@ -34,6 +34,15 @@ noncomputable section MetricCodesNoncomputable
 
 namespace MetricCodes
 
+private theorem one_sub_centered_square (x : ℝ) :
+    1 - (1 - 2 * x) ^ 2 = 4 * (x * (1 - x)) := by
+  ring
+
+private theorem centered_square_difference (x y : ℝ) :
+    (1 - 2 * x) ^ 2 - (1 - 2 * y) ^ 2 =
+      4 * (y * (1 - y) - x * (1 - x)) := by
+  ring
+
 section
 
 open scoped BigOperators InnerProductSpace
@@ -85,8 +94,8 @@ theorem hammingCorrelation_le_of_dist_le
   have hdiv :
       2 * (d : ℝ) / (n : ℝ) ≤
         2 * (hammingDist x y : ℝ) / (n : ℝ) := by
-    exact (div_le_div_iff_of_pos_right hn').2 (by linarith)
-  linarith
+    exact (div_le_div_iff_of_pos_right hn').2 (by linarith only [hd'])
+  linarith only [hdiv]
 
 /-- The spherical entropy used in the metric-code argument. -/
 def sphericalEntropy (u : ℝ) : ℝ :=
@@ -100,7 +109,7 @@ theorem sphericalEntropy_eq_log_add {u : ℝ} (hu : 0 < u) :
       Real.logb 2 (1 + u) +
         u * Real.logb 2 ((1 + u) / u) := by
   have hu' : u ≠ 0 := hu.ne'
-  have hone : 1 + u ≠ 0 := by linarith
+  have hone : 1 + u ≠ 0 := by linarith only [hu]
   rw [Real.logb_div hone hu']
   unfold sphericalEntropy
   ring
@@ -124,8 +133,8 @@ theorem binaryEntropy_nonneg {u : ℝ} (hu : 0 ≤ u) (hu' : u ≤ 1) :
   have hleft : u * Real.logb 2 u ≤ 0 :=
     mul_nonpos_of_nonneg_of_nonpos hu h₁
   have hright : (1 - u) * Real.logb 2 (1 - u) ≤ 0 :=
-    mul_nonpos_of_nonneg_of_nonpos (by linarith) h₂
-  linarith
+    mul_nonpos_of_nonneg_of_nonpos (by linarith only [hu']) h₂
+  linarith only [hleft, hright]
 
 /-- The gamma used in the metric-code argument. -/
 def Gamma (a b : ℝ) : ℝ :=
@@ -150,16 +159,16 @@ def classicalThreshold (s : ℝ) : ℝ :=
 
 theorem classicalThreshold_pos {s : ℝ} (hs : 0 < s) (hs' : s < 1) :
     0 < classicalThreshold s := by
-  have hrad : 0 < 1 - s ^ 2 := by nlinarith
+  have hrad : 0 < 1 - s ^ 2 := by nlinarith only [hs', hs]
   have hsqrt : 0 < Real.sqrt (1 - s ^ 2) := Real.sqrt_pos.2 hrad
   have hsqrt' : Real.sqrt (1 - s ^ 2) < 1 := by
     apply (Real.sqrt_lt' (by norm_num : (0 : ℝ) < 1)).2
-    nlinarith
+    nlinarith only [hs]
   have hrecip : 1 < 1 / Real.sqrt (1 - s ^ 2) := by
     apply (lt_div_iff₀ hsqrt).2
     simpa only [one_mul] using hsqrt'
   unfold classicalThreshold
-  linarith
+  linarith only [hrecip]
 
 /-- The boolean harmonic dimension used in the metric-code argument. -/
 def booleanHarmonicDimension (n : ℕ) : ℕ → ℕ
@@ -208,7 +217,6 @@ theorem hammingJacobiEntry_pos {n k i : ℕ}
   have hn' : (0 : ℝ) < n := by exact_mod_cast hn
   have hki' : (k : ℝ) ≤ i := by exact_mod_cast hki
   have hi' : (i : ℝ) + (k : ℝ) < n := by exact_mod_cast hi
-  have hzero : (0 : ℝ) ≤ k := Nat.cast_nonneg k
   unfold hammingJacobiEntry
   apply div_pos
   · apply mul_pos <;> linarith
@@ -737,7 +745,7 @@ theorem dot_smul_right (f g : Function n) (c : ℝ) :
   simp only [dot, Pi.smul_apply, smul_eq_mul, Finset.mul_sum]
   apply Finset.sum_congr rfl
   intro S _
-  ring
+  ac_rfl
 
 theorem normSq_nonneg (f : Function n) : 0 ≤ (dot f f) := by
   unfold dot
@@ -1228,7 +1236,7 @@ theorem sum_dot_raiseAt (f g : Function n) :
       apply Finset.sum_congr rfl
       intro S _
       rw [← Finset.mul_sum, sum_not_mem_indicator]
-      ring
+      ac_rfl
 
 theorem sum_dot_lowerAt_of_level {i : ℕ} (f g : Function n)
     (hf : IsLevel i f) :
@@ -1248,7 +1256,7 @@ theorem sum_dot_lowerAt_of_level {i : ℕ} (f g : Function n)
       simp only [dot, Finset.mul_sum]
       apply Finset.sum_congr rfl
       intro S _
-      ring
+      ac_rfl
 
 theorem sum_dot_raiseAt_of_level {i : ℕ} (f g : Function n)
     (hf : IsLevel i f) :
@@ -1270,7 +1278,7 @@ theorem sum_dot_raiseAt_of_level {i : ℕ} (f g : Function n)
       simp only [dot, Finset.mul_sum]
       apply Finset.sum_congr rfl
       intro S _
-      ring
+      ac_rfl
 
 theorem dot_smul_left (f g : Function n) (c : ℝ) :
     dot (c • f) g = c * dot f g := by
@@ -1320,10 +1328,9 @@ theorem raise_eq_zero_iff_of_level {i : ℕ}
     rw [hzero] at hidentity
     have hi' : (2 : ℝ) * (i : ℝ) < (n : ℝ) := by
       exact_mod_cast hi
-    have hpos : 0 < (n : ℝ) - 2 * (i : ℝ) := by linarith
     have hnonneg := normSq_nonneg (lower f)
     have hfn := normSq_nonneg f
-    have hnorm : (dot f f) = 0 := by nlinarith
+    have hnorm : (dot f f) = 0 := by nlinarith only [hfn, hidentity, hnonneg, hi']
     exact (normSq_eq_zero_iff f).mp hnorm
   · intro hfzero
     subst f
@@ -1609,7 +1616,7 @@ theorem deleteChannel_isometry {i : ℕ} (hi : 0 < i)
         dot_smul_right, Finset.mul_sum]
       apply Finset.sum_congr rfl
       intro a _
-      ring
+      ac_rfl
     _ = ((Real.sqrt (i : ℝ))⁻¹ * (Real.sqrt (i : ℝ))⁻¹) *
           ((i : ℝ) * dot f g) := by rw [sum_dot_lowerAt_of_level f g hf]
     _ = ((Real.sqrt (i : ℝ))⁻¹ * (Real.sqrt (i : ℝ))⁻¹ *
@@ -1650,7 +1657,7 @@ theorem addChannel_isometry {i : ℕ} (hi : i < n)
         dot_smul_right, Finset.mul_sum]
       apply Finset.sum_congr rfl
       intro a _
-      ring
+      ac_rfl
     _ = ((Real.sqrt ((n : ℝ) - (i : ℝ)))⁻¹ *
           (Real.sqrt ((n : ℝ) - (i : ℝ)))⁻¹) *
             (((n : ℝ) - (i : ℝ)) * dot f g) := by
@@ -1695,7 +1702,7 @@ theorem harmonicCoefficient_pos {n k r : ℕ}
     exact_mod_cast hbound
   unfold harmonicCoefficient
   apply mul_pos hr'
-  linarith
+  linarith only [hbound']
 
 theorem harmonicNormFactor_pos {n k r : ℕ}
     (hbound : 2 * k + r ≤ n) :
@@ -1734,7 +1741,7 @@ theorem dot_raised_of_harmonic {k : ℕ}
   | succ r ih =>
       rw [dot_raised_succ_of_harmonic f g hg r, ih]
       simp only [harmonicNormFactor, Finset.prod_range_succ]
-      ring
+      ac_rfl
 
 /-- The harmonic embedding used in the binary-code argument. -/
 def harmonicEmbedding (k r : ℕ) (f : Function n) : Function n :=
@@ -1778,7 +1785,7 @@ theorem harmonicEmbedding_isometry {k : ℕ}
           (Real.sqrt (harmonicNormFactor n k r))⁻¹) *
             dot (raised f r) (raised g r) := by
       simp only [harmonicEmbedding, dot_smul_left, dot_smul_right]
-      ring
+      ac_rfl
     _ = ((Real.sqrt (harmonicNormFactor n k r))⁻¹ *
           (Real.sqrt (harmonicNormFactor n k r))⁻¹) *
             (harmonicNormFactor n k r * dot f g) := by
@@ -2326,7 +2333,7 @@ theorem hammingAxis_inner {n : ℕ} (hn : 0 < n)
         rw [sum_sign_mul_eq_hammingDist]
     _ = MetricCodes.hammingCorrelation x y := by
         unfold MetricCodes.hammingCorrelation
-        field_simp [hsne, hn'.ne']; nlinarith [hs]
+        field_simp [hsne, hn'.ne']; nlinarith only [hs]
 
 theorem character_insert
     (x : BinaryWord n) (a : Fin n) (S : Finset (Fin n))
@@ -2372,7 +2379,7 @@ theorem raiseAt_twist
       _ = sign (x a) *
           ((sign (x a) * character x (S.erase a)) *
             f (S.erase a)) := by
-        ring
+        ac_rfl
   · simp only [raiseAt, ha, ↓reduceIte, Pi.smul_apply, twist, mul_zero, smul_eq_mul]
 
 /-- The Euclidean vector of matrix entries used to express the Hilbert–Schmidt inner product. -/
@@ -2438,7 +2445,7 @@ theorem matrixAxisLift_transpose_mul
   rw [Finset.mul_sum]
   apply Finset.sum_congr rfl
   intro r _
-  ring
+  ac_rfl
 
 /-- The axis-lifted projection minus the scaled channel image of that projection. -/
 def matrixAxisResidual
@@ -2610,7 +2617,7 @@ theorem lower_harmonicEmbedding {k : ℕ}
     Real.sqrt_mul hfactor.le]
   congr 1
   field_simp [hfactorne, hcoefficientne];
-    nlinarith [Real.sq_sqrt hcoefficient.le]
+    linarith only [Real.sq_sqrt hcoefficient.le]
 
 /-- The Hamming Jacobi entry rescaled by the square roots of the source and target layer
 dimensions. -/
@@ -2693,13 +2700,13 @@ theorem hammingRecurrenceWeight_eigenrecurrence
                   (Real.sqrt (n.choose (k + i.val) : ℝ) * v i) =
                 (hammingSourceChannelCoefficient n k L m i *
                   Real.sqrt (n.choose (k + i.val) : ℝ)) * v i := by
-                    ring
+                    ac_rfl
               _ = (MetricCodes.hammingJacobiMatrix n k L m i *
                     Real.sqrt (n.choose (k + m.val) : ℝ)) * v i := by
                       rw [h]
               _ = (MetricCodes.hammingJacobiMatrix n k L m i * v i) *
                     Real.sqrt (n.choose (k + m.val) : ℝ) := by
-                      ring
+                      ac_rfl
     _ = (∑ i : Fin (L - k + 1),
           MetricCodes.hammingJacobiMatrix n k L m i * v i) *
         Real.sqrt (n.choose (k + m.val) : ℝ) := by
@@ -2707,7 +2714,7 @@ theorem hammingRecurrenceWeight_eigenrecurrence
     _ = lam * hammingRecurrenceWeight n k L v m := by
           rw [hcoordinate]
           unfold hammingRecurrenceWeight
-          ring
+          ac_rfl
 
 theorem hammingRecurrenceWeight_pos_of_pos
     {n k L : ℕ} (hkL : k ≤ L) (hLn : L + k ≤ n)
@@ -2846,7 +2853,7 @@ theorem hammingAdjacentBlockCoefficient_amplitude_identity
       hammingFibreAmplitude_sq
         hkL hLn v hunit (fun i => (hv i).le) source]
     field_simp [hlam.ne', hsource.ne', hnormal.ne']
-  nlinarith
+  exact (sq_eq_sq₀ hleft hright).mp hsquare
 
 theorem hammingAdjacentBlockCoefficient_amplitude_sum
     {n k L : ℕ} (hn : 0 < n)
@@ -2974,7 +2981,6 @@ theorem hammingAdjacentChannel_isometry
       (hammingAdjacentChannel n k L target source f)
       (hammingAdjacentChannel n k L target source g) =
       dot f g := by
-  have hs := source.isLt
   have ht := target.isLt
   rcases hadjacent with hdelete | hadd
   · simp only [hammingAdjacentChannel, hdelete, ↓reduceIte]
@@ -3150,7 +3156,7 @@ theorem hammingWindowChannelMatrix_pairing
           rw [Finset.mul_sum]
           apply Finset.sum_congr rfl
           intro S _
-          ring
+          ac_rfl
     _ = (hammingAdjacentBlockCoefficient n k L v lam target Q.1 *
         hammingAdjacentBlockCoefficient n k L v lam target R.1) *
         coordinateDot
@@ -3446,8 +3452,6 @@ theorem hammingSourceChannelCoefficient_eq_deletion
     (hadjacent : target.val + 1 = source.val) :
     hammingSourceChannelCoefficient n k L source target =
       hammingDeletionChannelSquare n k (k + target.val) := by
-  have hs := source.isLt
-  have ht := target.isLt
   have hboundary : k + target.val + k < n := by omega
   have hdimension : 0 < (n.choose (k + target.val) : ℝ) := by
     exact_mod_cast Nat.choose_pos (show k + target.val ≤ n by omega)
@@ -3469,8 +3473,6 @@ theorem hammingSourceChannelCoefficient_eq_insertion
     (hadjacent : source.val + 1 = target.val) :
     hammingSourceChannelCoefficient n k L source target =
       hammingInsertionChannelSquare n k (k + source.val) := by
-  have hs := source.isLt
-  have ht := target.isLt
   have hboundary : k + source.val + k < n := by omega
   have hdimension : 0 < (n.choose (k + target.val) : ℝ) := by
     exact_mod_cast Nat.choose_pos (show k + target.val ≤ n by omega)
@@ -3548,7 +3550,7 @@ theorem coordinateDot_deleteChannel_hammingAxisTensor
                 (Real.sqrt (n : ℝ))⁻¹) *
                   ((sign (x a) * sign (x a)) *
                     dot f (twist x (raiseAt a g))) := by
-                      ring
+                      ac_rfl
             _ = ((Real.sqrt (i : ℝ))⁻¹ *
                 (Real.sqrt (n : ℝ))⁻¹) *
                   dot f (twist x (raiseAt a g)) := by
@@ -3590,7 +3592,7 @@ theorem coordinateDot_addChannel_hammingAxisTensor
                 (Real.sqrt (n : ℝ))⁻¹) *
                   ((sign (x a) * sign (x a)) *
                     dot f (twist x (lowerAt a g))) := by
-                      ring
+                      ac_rfl
             _ = ((Real.sqrt ((n : ℝ) - (i : ℝ)))⁻¹ *
                 (Real.sqrt (n : ℝ))⁻¹) *
                   dot f (twist x (lowerAt a g)) := by
@@ -3617,7 +3619,6 @@ theorem hammingSourceChannelCoefficient_sqrt_deletion
       (Real.sqrt (n : ℝ))⁻¹ *
         (Real.sqrt (k + source.val : ℝ))⁻¹ *
           Real.sqrt (harmonicCoefficient n k source.val) := by
-  have hs := source.isLt
   have hcoefficient : 0 < harmonicCoefficient n k source.val :=
     harmonicCoefficient_pos (by omega) (by omega)
   have hnreal : 0 < (n : ℝ) := by exact_mod_cast hn
@@ -3651,8 +3652,6 @@ theorem hammingSourceChannelCoefficient_sqrt_insertion
       (Real.sqrt (n : ℝ))⁻¹ *
         (Real.sqrt ((n : ℝ) - (k + source.val : ℝ)))⁻¹ *
           Real.sqrt (harmonicCoefficient n k target.val) := by
-  have hs := source.isLt
-  have ht := target.isLt
   have hcoefficient : 0 < harmonicCoefficient n k target.val :=
     harmonicCoefficient_pos (by omega) (by omega)
   have hnreal : 0 < (n : ℝ) := by exact_mod_cast hn
@@ -3825,7 +3824,7 @@ theorem hammingWindowChannelMatrix_transpose_axis_fibre
           rw [Finset.mul_sum]
           apply Finset.sum_congr rfl
           intro S _
-          ring
+          ac_rfl
     _ = (∑ target : Fin (L - k + 1),
           hammingAdjacentBlockCoefficient n k L v lam target Q.1 *
             hammingFibreAmplitude n k L v target *
@@ -3839,14 +3838,14 @@ theorem hammingWindowChannelMatrix_transpose_axis_fibre
           rw [Finset.sum_mul]
           apply Finset.sum_congr rfl
           intro target _
-          ring
+          ac_rfl
     _ = Real.sqrt lam *
           (hammingFibreAmplitude n k L v Q.1 *
             wordHarmonicEmbedding x k Q.1.val h Q.2.val) := by
           rw [hammingAdjacentBlockCoefficient_amplitude_sum
             hn hkL hLn v hunit hv lam hlam heigen Q.1,
             dot_hammingWindowBasis_apply]
-          ring
+          ac_rfl
 
 theorem hammingChannelMatrix_transpose_axis_fibre
     {n k L : ℕ} (hn : 0 < n) (hk : 2 * k ≤ n)
@@ -3922,7 +3921,7 @@ theorem matrixAxisLift_mul
   rw [Finset.mul_sum]
   apply Finset.sum_congr rfl
   intro r _
-  ring
+  ac_rfl
 
 theorem hammingChannelMatrix_transpose_axis_projection
     {n k L : ℕ} (hn : 0 < n) (hk : 2 * k ≤ n)
@@ -4473,13 +4472,10 @@ theorem factorialLogError_eq (n : ℕ) (hn : n ≠ 0) :
         (1 / 2 : ℝ) * Real.log (2 * (n : ℝ)) / (n : ℝ) := by
   have hn' : (n : ℝ) ≠ 0 := by exact_mod_cast hn
   have he : Real.exp (1 : ℝ) ≠ 0 := (Real.exp_pos 1).ne'
-  have hst := Stirling.log_stirlingSeq_formula n
-  rw [Real.log_div hn' he, Real.log_exp] at hst
-  have hlog : Real.log ((n : ℝ) * 2) = Real.log (2 * (n : ℝ)) := by
-    rw [mul_comm]
   unfold factorialLogError
+  rw [Stirling.log_stirlingSeq_formula, Real.log_div hn' he, Real.log_exp]
   field_simp [hn']
-  nlinarith [hst, hlog]
+  ring
 
 theorem tendsto_factorialLogError :
     Tendsto factorialLogError atTop (nhds 0) := by
@@ -4704,11 +4700,9 @@ def threshold (n d : ℕ) : ℝ :=
 
 theorem threshold_lt_one {n d : ℕ} (hn : 0 < n) (hd : 0 < d) :
     threshold n d < 1 := by
-  have hn' : (0 : ℝ) < (n : ℝ) := by exact_mod_cast hn
-  have hd' : (0 : ℝ) < (d : ℝ) := by exact_mod_cast hd
   unfold threshold
   have hpos : 0 < 2 * (d : ℝ) / (n : ℝ) := by positivity
-  linarith
+  linarith only [hpos]
 
 /-- The index used in the binary-code argument. -/
 abbrev Index (k L : ℕ) := Fin (L - k + 1)
@@ -5132,10 +5126,10 @@ theorem rateSet_bddBelow (δ : ℝ) : BddBelow (rateSet δ) := by
   rintro r ⟨a, b, hfeasible, rfl⟩
   obtain ⟨hb, hba, ha, hgamma⟩ := hfeasible
   have ha0 : 0 ≤ a := (lt_of_le_of_lt hb hba).le
-  have ha1 : a ≤ 1 := by linarith
+  have ha1 : a ≤ 1 := by linarith only [ha]
   have hapos := MetricCodes.binaryEntropy_nonneg ha0 ha1
   have hble := binaryEntropy_le_one b
-  linarith
+  linarith only [hapos, hble]
 
 theorem variationalRate_le_of_feasible {δ a b : ℝ}
     (h : Feasible δ a b) :
@@ -5153,33 +5147,33 @@ def classicalRate (δ : ℝ) : ℝ :=
 theorem classicalParameter_lt_half {δ : ℝ}
     (hδ : 0 < δ) (hδ' : δ < (1 : ℝ) / 2) :
     classicalParameter δ < (1 : ℝ) / 2 := by
-  have hrad : 0 < δ * (1 - δ) := mul_pos hδ (by linarith)
+  have hrad : 0 < δ * (1 - δ) := mul_pos hδ (by linarith only [hδ'])
   unfold classicalParameter
   have hsqrt := Real.sqrt_pos.2 hrad
-  linarith
+  linarith only [hsqrt]
 
 theorem classicalParameter_pos {δ : ℝ}
     (hδ : 0 < δ) (hδ' : δ < (1 : ℝ) / 2) :
     0 < classicalParameter δ := by
   have hsquare : 0 < (δ - (1 : ℝ) / 2) ^ 2 :=
-    sq_pos_of_ne_zero (by linarith)
-  have hrad : 0 < δ * (1 - δ) := mul_pos hδ (by linarith)
+    sq_pos_of_ne_zero (by linarith only [hδ'])
+  have hrad : 0 < δ * (1 - δ) := mul_pos hδ (by linarith only [hδ'])
   have hrad_lt : δ * (1 - δ) < ((1 : ℝ) / 2) ^ 2 := by
-    nlinarith
+    linarith only [hsquare]
   have hroot : Real.sqrt (δ * (1 - δ)) < (1 : ℝ) / 2 := by
     exact (Real.sqrt_lt' (by norm_num)).2 hrad_lt
   unfold classicalParameter
-  linarith
+  linarith only [hroot]
 
 theorem classicalParameter_mul_one_sub {δ : ℝ}
     (hδ : 0 < δ) (hδ' : δ < (1 : ℝ) / 2) :
     classicalParameter δ * (1 - classicalParameter δ) =
       ((1 : ℝ) / 2 - δ) ^ 2 := by
   have hrad : 0 ≤ δ * (1 - δ) :=
-    (mul_pos hδ (by linarith)).le
+    (mul_pos hδ (by linarith only [hδ'])).le
   have hsqrt := Real.sq_sqrt hrad
   unfold classicalParameter
-  nlinarith
+  linarith only [hsqrt]
 
 theorem hammingGamma_zero {a : ℝ} (ha : 0 < a) (ha' : a < 1) :
     MetricCodes.hammingGamma a 0 = 2 * Real.sqrt (a * (1 - a)) := by
@@ -5190,7 +5184,7 @@ theorem hammingGamma_zero {a : ℝ} (ha : 0 < a) (ha' : a < 1) :
   unfold MetricCodes.hammingGamma
   simp only [sub_zero]
   field_simp [hsqrt]
-  nlinarith
+  linarith only [hsquare]
 
 theorem hammingGamma_classicalParameter {δ : ℝ}
     (hδ : 0 < δ) (hδ' : δ < (1 : ℝ) / 2) :
@@ -5200,7 +5194,7 @@ theorem hammingGamma_classicalParameter {δ : ℝ}
     (classicalParameter_lt_half hδ hδ').trans (by norm_num)
   rw [hammingGamma_zero ha ha',
     classicalParameter_mul_one_sub hδ hδ',
-    Real.sqrt_sq (by linarith : 0 ≤ (1 : ℝ) / 2 - δ)]
+    Real.sqrt_sq (by linarith only [hδ'] : 0 ≤ (1 : ℝ) / 2 - δ)]
   ring
 
 /-- The slope `2 / (1 - 2 * a) + 1` used to perturb the Hamming bound parameters. -/
@@ -5213,10 +5207,10 @@ def improvementPath (a b : ℝ) : ℝ :=
 
 theorem improvementSlope_gt_one {a : ℝ} (ha : a < (1 : ℝ) / 2) :
     1 < improvementSlope a := by
-  have hden : 0 < 1 - 2 * a := by linarith
+  have hden : 0 < 1 - 2 * a := by linarith only [ha]
   have hfrac : 0 < 2 / (1 - 2 * a) := by positivity
   unfold improvementSlope
-  linarith
+  linarith only [hfrac]
 
 theorem tendsto_improvementPath_zero (a : ℝ) :
     Tendsto (improvementPath a) (𝓝[>] (0 : ℝ)) (nhds a) := by
@@ -5252,7 +5246,7 @@ theorem spectralMarginPolynomial_improvement_zero {a : ℝ}
     (ha : a < (1 : ℝ) / 2) :
     spectralMarginPolynomial a (improvementSlope a) 0 =
       a * (1 - a) * (1 - 2 * a) := by
-  have hden : 1 - 2 * a ≠ 0 := by linarith
+  have hden : 1 - 2 * a ≠ 0 := by linarith only [ha]
   have hden' : 1 - a * 2 ≠ 0 := by
     simpa only [ne_eq, mul_comm] using hden
   unfold spectralMarginPolynomial improvementSlope
@@ -5265,8 +5259,8 @@ theorem eventually_hammingGamma_improvement {a : ℝ}
         MetricCodes.hammingGamma (improvementPath a b) b := by
   have hzero : 0 < spectralMarginPolynomial a (improvementSlope a) 0 := by
     rw [spectralMarginPolynomial_improvement_zero ha']
-    have ha1 : 0 < 1 - a := by linarith
-    have hhalf : 0 < 1 - 2 * a := by linarith
+    have ha1 : 0 < 1 - a := by linarith only [ha']
+    have hhalf : 0 < 1 - 2 * a := by linarith only [ha']
     exact mul_pos (mul_pos ha ha1) hhalf
   have hpoly :
       ∀ᶠ b : ℝ in 𝓝[>] 0,
@@ -5286,13 +5280,13 @@ theorem eventually_hammingGamma_improvement {a : ℝ}
     with b hbpoly hbhalf (hb : 0 < b)
   have hbpath : b < improvementPath a b := by
     unfold improvementPath
-    nlinarith [mul_pos (sub_pos.mpr hc) hb]
+    linarith only [ha, mul_pos (sub_pos.mpr hc) hb]
   have hpath : 0 < improvementPath a b := lt_trans hb hbpath
   have hpath1 : improvementPath a b < 1 :=
     hbhalf.trans (by norm_num)
   have htail : 0 < 1 - improvementPath a b - b := by
-    linarith
-  have hbase : 0 < a * (1 - a) := mul_pos ha (by linarith)
+    linarith only [hbhalf, hbpath]
+  have hbase : 0 < a * (1 - a) := mul_pos ha (by linarith only [ha'])
   have hrad : 0 < improvementPath a b * (1 - improvementPath a b) :=
     mul_pos hpath (sub_pos.mpr hpath1)
   have hfactor :
@@ -5331,7 +5325,9 @@ theorem eventually_hammingGamma_improvement {a : ℝ}
           Real.sqrt (improvementPath a b *
             (1 - improvementPath a b)) :=
       mul_nonneg (Real.sqrt_nonneg _) (Real.sqrt_nonneg _)
-    nlinarith
+    apply (sq_lt_sq₀ hnonneg htarget.le).mp
+    rw [hsquare]
+    exact sub_pos.mp hmargin
   have hquot :
       Real.sqrt (a * (1 - a)) <
         ((improvementPath a b - b) *
@@ -5339,7 +5335,7 @@ theorem eventually_hammingGamma_improvement {a : ℝ}
             Real.sqrt
               (improvementPath a b * (1 - improvementPath a b)) :=
     (lt_div_iff₀ (Real.sqrt_pos.2 hrad)).2 hroot
-  rw [hammingGamma_zero ha (by linarith)]
+  rw [hammingGamma_zero ha (by linarith only [ha'])]
   unfold MetricCodes.hammingGamma
   calc
     2 * Real.sqrt (a * (1 - a)) <
@@ -5370,9 +5366,9 @@ theorem neg_mul_logb_le_binaryEntropy {b : ℝ}
   have hlog : Real.logb 2 (1 - b) ≤ 0 :=
     Real.logb_nonpos (by norm_num) (by linarith) (by linarith)
   have hterm : (1 - b) * Real.logb 2 (1 - b) ≤ 0 :=
-    mul_nonpos_of_nonneg_of_nonpos (by linarith) hlog
+    mul_nonpos_of_nonneg_of_nonpos (by linarith only [hb']) hlog
   unfold MetricCodes.binaryEntropy
-  nlinarith
+  linarith only [hterm]
 
 theorem eventually_binaryEntropy_improvement {a : ℝ}
     (ha : 0 < a) (ha' : a < (1 : ℝ) / 2) :
@@ -5393,7 +5389,7 @@ theorem eventually_binaryEntropy_improvement {a : ℝ}
   let M : ℝ := deriv f 0 + 1
   have hM : deriv f 0 < M := by
     dsimp [M]
-    linarith
+    linarith only []
   have hslope := hf.hasDerivAt.tendsto_slope_zero_right
   have hupper :
       ∀ᶠ b : ℝ in 𝓝[>] 0,
@@ -5427,7 +5423,7 @@ theorem eventually_binaryEntropy_improvement {a : ℝ}
       b * M < b * (-Real.logb 2 b) :=
     mul_lt_mul_of_pos_left hlogb hb
   have hentropy := neg_mul_logb_le_binaryEntropy hb.le hb1.le
-  linarith
+  linarith only [houter_bound, hsingular, hentropy]
 
 theorem exists_strict_improving_feasible {δ : ℝ}
     (hδ : 0 < δ) (hδ' : δ < (1 : ℝ) / 2) :
@@ -5455,7 +5451,7 @@ theorem exists_strict_improving_feasible {δ : ℝ}
       with b hgamma' hentropy' hhalf' (hb : 0 < b)
     have hbpath : b < improvementPath a₀ b := by
       unfold improvementPath
-      nlinarith [mul_pos (sub_pos.mpr hc) hb]
+      linarith only [ha₀, mul_pos (sub_pos.mpr hc) hb]
     constructor
     · refine ⟨hb.le, hbpath, hhalf'.le, ?_⟩
       have hboundary : MetricCodes.hammingGamma a₀ 0 = 1 - 2 * δ := by
@@ -5831,10 +5827,10 @@ theorem eventually_transverse_add_le_longitudinal
   let ε : ℝ := (a - b) / 2
   have hε : 0 < ε := by
     dsimp [ε]
-    linarith
+    linarith only [hba]
   have hεlt : ε < a - b := by
     dsimp [ε]
-    linarith
+    linarith only [hba]
   have ha : 0 ≤ a := (lt_of_le_of_lt hb hba).le
   have hratio :=
     ((tendsto_longitudinal_ratio ha).sub
@@ -5859,7 +5855,7 @@ theorem eventually_transverse_add_le_longitudinal
   have hreal :
       (transverseDegree b n : ℝ) + (m : ℝ) ≤
         (longitudinalDegree a n : ℝ) := by
-    linarith
+    linarith only [hnlarge, hmul]
   exact_mod_cast hreal
 
 /-- The terminal edge rayleigh used in the binary-code argument. -/
@@ -5927,7 +5923,7 @@ theorem eventually_topEigenvalue_gt {a b s : ℝ}
         (transverseDegree b n) (longitudinalDegree a n) := by
   have ha0 : 0 < a := lt_of_le_of_lt hb hba
   have ha1 : a < 1 := lt_of_le_of_lt ha (by norm_num)
-  have hsecond : 0 < 1 - a - b := by linarith
+  have hsecond : 0 < 1 - a - b := by linarith only [hba, ha]
   have hgamma : 0 < MetricCodes.hammingGamma a b := by
     unfold MetricCodes.hammingGamma
     apply div_pos
@@ -5936,14 +5932,14 @@ theorem eventually_topEigenvalue_gt {a b s : ℝ}
   let ε : ℝ := (MetricCodes.hammingGamma a b - s) / 2
   have hε : 0 < ε := by
     dsimp [ε]
-    linarith
+    linarith only [hs]
   obtain ⟨m, hm⟩ := exists_nat_gt (MetricCodes.hammingGamma a b / ε)
   have hprod : MetricCodes.hammingGamma a b < (m : ℝ) * ε :=
     (div_lt_iff₀ hε).mp hm
   have hden : 0 < (m : ℝ) + 1 := by positivity
   have hrem : MetricCodes.hammingGamma a b / ((m : ℝ) + 1) < ε := by
     apply (div_lt_iff₀ hden).2
-    nlinarith
+    linarith only [hε, hprod]
   have hidentity :
       ((m : ℝ) / ((m : ℝ) + 1)) * MetricCodes.hammingGamma a b =
         MetricCodes.hammingGamma a b -
@@ -5953,7 +5949,7 @@ theorem eventually_topEigenvalue_gt {a b s : ℝ}
       s < ((m : ℝ) / ((m : ℝ) + 1)) * MetricCodes.hammingGamma a b := by
     rw [hidentity]
     dsimp [ε] at hrem
-    linarith
+    linarith only [hs, hrem]
   have hquot := (tendsto_terminalEdgeRayleigh ha0 ha1 hb m).eventually
     (lt_mem_nhds hbelow)
   filter_upwards [hquot,
@@ -5980,7 +5976,7 @@ theorem complement_longitudinalDegree_floor_le {a : ℝ}
   change (1 - a) * (n : ℝ) ≤
     ((n - longitudinalDegree a n : ℕ) : ℝ)
   rw [Nat.cast_sub hk]
-  nlinarith
+  linarith only [hfloor]
 
 theorem tendsto_complement_longitudinalDegree {a : ℝ}
     (ha : 0 ≤ a) (ha' : a < 1) :
@@ -6068,7 +6064,7 @@ theorem longitudinalDegree_le_half {a : ℝ}
     Nat.floor (a * (n : ℝ)) ≤ Nat.floor ((n : ℝ) / 2) := by
       apply Nat.floor_mono
       have h := mul_le_mul_of_nonneg_right ha (Nat.cast_nonneg n)
-      linarith
+      linarith only [h]
     _ = n / 2 := by
       simpa only [Nat.cast_ofNat] using (Nat.floor_div_eq_div (K := ℝ) n 2)
 
@@ -6144,13 +6140,13 @@ theorem choose_le_mul_hammingFibreDimension
           norm_num
         rw [hcastj]
         push_cast at hcast
-        linarith
+        linarith only [hcast]
       have hlower :
           (n.choose (j + 1) : ℝ) ≤
             ((n.choose (j + 1) - n.choose j : ℕ) : ℝ) *
               ((n - j : ℕ) : ℝ) := by
         rw [hidentity]
-        nlinarith [Nat.cast_nonneg (α := ℝ) (n.choose (j + 1))]
+        exact le_mul_of_one_le_right (Nat.cast_nonneg _) hfactorReal
       have hrange :
           ((n - j : ℕ) : ℝ) ≤ ((n + 1 : ℕ) : ℝ) := by
         exact_mod_cast (show n - j ≤ n + 1 by omega)
@@ -6363,7 +6359,7 @@ theorem tendsto_logb_hammingFibreDimension {b : ℝ}
           (MetricCodes.hammingFibreDimension n
             (transverseDegree b n) : ℝ) / (n : ℝ) := by
         apply (div_le_div_iff_of_pos_right hnreal).2
-        linarith
+        linarith only [hlog]
   have hupper :
       ∀ᶠ n : ℕ in atTop,
         Real.logb 2
@@ -6438,7 +6434,7 @@ theorem windowFibreQuotient_pos_of_fit {a b : ℝ} {n : ℕ}
   have hfibre :
       0 < (MetricCodes.hammingFibreDimension n (transverseDegree b n) : ℝ) := by
     exact_mod_cast hammingFibreDimension_pos hkhalf
-  have haone : a ≤ 1 := by linarith
+  have haone : a ≤ 1 := by linarith only [ha]
   have hchoose : 0 < (n.choose (longitudinalDegree a n) : ℝ) := by
     exact_mod_cast
       Nat.choose_pos (longitudinalDegree_le_dimension haone n)
@@ -6476,7 +6472,7 @@ theorem tendsto_logb_windowFibreQuotient {a b : ℝ}
   have hfibre :
       0 < (MetricCodes.hammingFibreDimension n (transverseDegree b n) : ℝ) := by
     exact_mod_cast hammingFibreDimension_pos hkhalf
-  have haone : a ≤ 1 := by linarith
+  have haone : a ≤ 1 := by linarith only [ha]
   have hchoose : 0 < (n.choose (longitudinalDegree a n) : ℝ) := by
     exact_mod_cast
       Nat.choose_pos (longitudinalDegree_le_dimension haone n)
@@ -6559,7 +6555,7 @@ theorem threshold_ceil_numerator_le_two {δ : ℝ}
     apply (div_le_iff₀ hnreal).2
     exact mul_le_mul_of_nonneg_left hdreal (by norm_num)
   unfold threshold
-  linarith
+  linarith only [hquot]
 
 /-- The spectral gap used in the binary-code argument. -/
 def spectralGap (δ a b : ℝ) : ℝ :=
@@ -6568,7 +6564,7 @@ def spectralGap (δ a b : ℝ) : ℝ :=
 theorem spectralGap_pos {δ a b : ℝ} (h : Feasible δ a b) :
     0 < spectralGap δ a b := by
   unfold spectralGap
-  linarith [h.2.2.2]
+  linarith only [h.2.2.2]
 
 theorem eventually_topEigenvalue_uniform_gap_ceil {δ a b : ℝ}
     (hδ : 0 ≤ δ) (h : Feasible δ a b) :
@@ -6584,7 +6580,7 @@ theorem eventually_topEigenvalue_uniform_gap_ceil {δ a b : ℝ}
       1 - 2 * δ + 2 * spectralGap δ a b <
         MetricCodes.hammingGamma a b := by
     unfold spectralGap
-    linarith
+    linarith only [hgamma]
   have heigen :=
     eventually_topEigenvalue_gt hb hba ha hstrict
   have hthreshold :
@@ -6592,9 +6588,9 @@ theorem eventually_topEigenvalue_uniform_gap_ceil {δ a b : ℝ}
         threshold n (Nat.ceil (δ * (n : ℝ))) <
           1 - 2 * δ + spectralGap δ a b :=
     (tendsto_threshold_ceil hδ).eventually
-      (gt_mem_nhds (by linarith))
+      (gt_mem_nhds (by linarith only [hgap]))
   filter_upwards [heigen, hthreshold] with n he hn
-  linarith
+  linarith only [he, hn]
 
 /-- The spectral prefactor used in the binary-code argument. -/
 def spectralPrefactor (δ a b : ℝ) : ℝ :=
@@ -6706,7 +6702,7 @@ theorem binaryRate_le_variationalRate_of_positive_feasible_bounds
       have hcunit : c ≤ 1 := by
         have hca := hfeasible'.2.1
         have hahalf := hfeasible'.2.2.1
-        linarith
+        linarith only [hca, hahalf]
       have hentropy : 0 ≤ MetricCodes.binaryEntropy c :=
         MetricCodes.binaryEntropy_nonneg hc.le hcunit
       calc
@@ -6761,7 +6757,7 @@ theorem eventually_codeNumber_le_spectralPrefactor_mul_windowFibreQuotient
     ∀ᶠ n : ℕ in atTop,
       (codeNumber n (Nat.ceil (δ * (n : ℝ))) : ℝ) ≤
         spectralPrefactor δ a b * windowFibreQuotient a b n := by
-  have hδone : δ ≤ 1 := by linarith
+  have hδone : δ ≤ 1 := by linarith only [hδ']
   filter_upwards [
     eventually_gt_atTop (0 : ℕ),
     eventually_transverse_add_le_longitudinal h.1 h.2.1 1,
@@ -6781,7 +6777,7 @@ theorem eventually_codeNumber_le_spectralPrefactor_mul_windowFibreQuotient
         topEigenvalue n (transverseDegree b n)
           (longitudinalDegree a n) := by
     have hpositive := spectralGap_pos h
-    linarith
+    linarith only [hmargin, hpositive]
   have hfinite := finite_bound hn hd hkL hLn hgap
   have hquotient : 0 ≤ windowFibreQuotient a b n :=
     (windowFibreQuotient_pos_of_fit h.2.2.1 hkL.le).le
@@ -7031,7 +7027,7 @@ theorem threshold_lt_one {n w d : ℕ}
     positivity
   unfold threshold
   have hfrac := div_pos hnum hden
-  linarith
+  linarith only [hfrac]
 
 theorem correlation_le_threshold {n w d : ℕ}
     (hw : 0 < w) (hwn : w < n)
@@ -7053,7 +7049,7 @@ theorem correlation_le_threshold {n w d : ℕ}
   have hfrac :=
     (div_le_div_iff_of_pos_right hden).mpr hnum
   unfold threshold
-  linarith
+  linarith only [hfrac]
 
 theorem correlation_le_threshold_of_code {n w d : ℕ}
     (hw : 0 < w) (hwn : w < n)
@@ -7462,29 +7458,15 @@ theorem zonalDiagonal_eq {n w j : ℕ}
     exact_mod_cast hhalf
   have hj' : (j : ℝ) ≤ (w : ℝ) := by
     exact_mod_cast hj
-  have hsmall : (2 : ℝ) * (j : ℝ) < (n : ℝ) := by
-    nlinarith
   have hw' : (w : ℝ) ≠ 0 := by
     exact_mod_cast Nat.ne_of_gt hw
   have hN' : (n : ℝ) - (w : ℝ) ≠ 0 := by
-    have hwn' : (w : ℝ) < (n : ℝ) := by exact_mod_cast hwn
-    linarith
+    linarith only [hhalf']
   have hJ : 0 < MetricCodes.johnsonJ n j := by
     unfold MetricCodes.johnsonJ
-    nlinarith
+    linarith only [hhalf', hj']
   have hJone : MetricCodes.johnsonJ n j + 1 ≠ 0 := by
-    linarith
-  have hD :
-      (n : ℝ) * 2 - (n : ℝ) * (j : ℝ) * 4 +
-          (n : ℝ) ^ 2 - (j : ℝ) * 4 + (j : ℝ) ^ 2 * 4 ≠ 0 := by
-    have hfactor :
-        (n : ℝ) * 2 - (n : ℝ) * (j : ℝ) * 4 +
-            (n : ℝ) ^ 2 - (j : ℝ) * 4 + (j : ℝ) ^ 2 * 4 =
-          ((n : ℝ) - 2 * (j : ℝ)) *
-            ((n : ℝ) - 2 * (j : ℝ) + 2) := by
-      ring
-    rw [hfactor]
-    exact (mul_pos (by linarith) (by linarith)).ne'
+    linarith only [hJ]
   unfold MetricCodes.johnsonZonalDiagonal MetricCodes.johnsonDiagonal
     MetricCodes.johnsonMu MetricCodes.johnsonM
     MetricCodes.johnsonJ1 MetricCodes.johnsonJ2
@@ -7505,13 +7487,13 @@ theorem zonalDiagonal_pos {n w j : ℕ}
     exact_mod_cast hjw
   have hM : 0 < MetricCodes.johnsonM n w := by
     unfold MetricCodes.johnsonM
-    nlinarith
+    linarith only [hhalf']
   have hJ : 0 < MetricCodes.johnsonJ n j := by
     unfold MetricCodes.johnsonJ
-    nlinarith
+    linarith only [hhalf', hjw']
   have hN : 0 < n - w := Nat.sub_pos_of_lt hwn
   have hlast : 0 < (n : ℝ) - (j : ℝ) + 1 := by
-    nlinarith
+    linarith only [hhalf', hjw']
   positivity
 
 theorem associatedEdge_pos {n w p q j : ℕ}
@@ -8052,7 +8034,7 @@ theorem johnsonRecurrenceWeight_eigenrecurrence
                   Real.sqrt
                     (MetricCodes.booleanHarmonicDimension
                       n (p + q + i.val) : ℝ)) * v i := by
-                    ring
+                    ac_rfl
               _ = (matrix n w p q L m i *
                     Real.sqrt
                       (MetricCodes.booleanHarmonicDimension
@@ -8062,7 +8044,7 @@ theorem johnsonRecurrenceWeight_eigenrecurrence
                     Real.sqrt
                       (MetricCodes.booleanHarmonicDimension
                         n (p + q + m.val) : ℝ) := by
-                      ring
+                      ac_rfl
     _ = (∑ i : Index p q L,
           matrix n w p q L m i * v i) *
         Real.sqrt
@@ -8071,7 +8053,7 @@ theorem johnsonRecurrenceWeight_eigenrecurrence
     _ = lam * johnsonRecurrenceWeight n w p q L v m := by
           rw [hcoordinate]
           unfold johnsonRecurrenceWeight
-          ring
+          ac_rfl
 
 /-- The johnson adjacent block coefficient used in the Johnson-code argument. -/
 def johnsonAdjacentBlockCoefficient
@@ -8233,7 +8215,7 @@ theorem johnsonAdjacentBlockCoefficient_amplitude_identity
       Real.sq_sqrt hlam.le,
       johnsonFibreAmplitude_sq h v hv source]
     field_simp [hlam.ne', hsource.ne', hnormal.ne']
-  nlinarith
+  exact (sq_eq_sq₀ hleft hright).mp hsquare
 
 theorem johnsonAdjacentBlockCoefficient_amplitude_sum
     {n w p q L : ℕ} (h : AdmissibleDegrees n w p q L)
@@ -8362,12 +8344,10 @@ theorem sqrt_one_sub_centeredDegree_sq {u : ℝ}
     Real.sqrt (1 - (1 - 2 * u) ^ 2) =
       2 * Real.sqrt (u * (1 - u)) := by
   have hvariance : 0 < u * (1 - u) :=
-    mul_pos hu (by linarith)
-  have hrad : 0 ≤ 1 - (1 - 2 * u) ^ 2 := by
-    nlinarith
-  have hsquare := Real.sq_sqrt hvariance.le
-  apply (Real.sqrt_eq_iff_eq_sq hrad (by positivity)).mpr
-  nlinarith
+    mul_pos hu (by linarith only [hhalf])
+  rw [one_sub_centered_square, Real.sqrt_mul' _ hvariance.le,
+    show (4 : ℝ) = (2 : ℝ) ^ 2 by norm_num,
+    Real.sqrt_sq (by norm_num : (0 : ℝ) ≤ 2)]
 
 theorem zero_fibre_spectral_algebra
     (z m s : ℝ)
@@ -8382,7 +8362,7 @@ theorem zero_fibre_spectral_algebra
       (z ^ 2 - m ^ 2) /
         ((1 - m ^ 2) * (1 + 2 * s)) := by
   have hzsq : z ^ 2 = 1 - 4 * s ^ 2 := by
-    nlinarith [hsquare]
+    linarith only [hsquare]
   have hzsqne : 1 - 4 * s ^ 2 ≠ 0 := by
     rw [← hzsq]
     exact pow_ne_zero 2 hz
@@ -8406,16 +8386,16 @@ theorem spectralLimit_zero_fibre_boundary {α u : ℝ}
   let m : ℝ := (1 - 2 * α)
   let s : ℝ := Real.sqrt (u * (1 - u))
   have ha0 : 0 < α := lt_trans hu hua
-  have haone : 0 < 1 - α := by linarith
-  have huone : 0 < 1 - u := by linarith
+  have haone : 0 < 1 - α := by linarith only [ha]
+  have huone : 0 < 1 - u := by linarith only [hua, ha]
   have hvariance : 0 < u * (1 - u) := mul_pos hu huone
   have hz : z ≠ 0 := by
     dsimp [z]
-    nlinarith
+    linarith only [hua, ha]
   have hm : 1 - m ^ 2 ≠ 0 := by
-    have hprod : 0 < α * (1 - α) := mul_pos ha0 haone
     dsimp [m]
-    nlinarith
+    rw [one_sub_centered_square]
+    exact mul_ne_zero (by norm_num) (mul_pos ha0 haone).ne'
   have hs : s ≠ 0 := by
     dsimp [s]
     exact (Real.sqrt_pos.mpr hvariance).ne'
@@ -8423,11 +8403,10 @@ theorem spectralLimit_zero_fibre_boundary {α u : ℝ}
     have hsnonneg : 0 ≤ s := by
       dsimp [s]
       exact Real.sqrt_nonneg _
-    nlinarith
+    linarith only [hsnonneg]
   have hsquare : 4 * s ^ 2 = 1 - z ^ 2 := by
-    have hsqrt := Real.sq_sqrt hvariance.le
     dsimp [s, z]
-    nlinarith
+    rw [one_sub_centered_square, Real.sq_sqrt hvariance.le]
   have hradical : Real.sqrt (1 - z ^ 2) = 2 * s := by
     simpa only using sqrt_one_sub_centeredDegree_sq hu (lt_trans hua ha)
   have hsig : (1 - 2 * 0 - 2 * 0 : ℝ) = 1 := by
@@ -8459,16 +8438,6 @@ theorem spectralLimit_zero_fibre_boundary {α u : ℝ}
         ((z ^ 2 - m ^ 2) * (1 - z ^ 2)) /
           (z ^ 2 * (1 - m ^ 2) * Real.sqrt (1 - z ^ 2)) = _
     rw [hradical]
-  have hA : α * (1 - α) ≠ 0 :=
-    (mul_pos ha0 haone).ne'
-  have hfourA : α * 4 - α ^ 2 * 4 ≠ 0 := by
-    have hp : 0 < α * (1 - α) := mul_pos ha0 haone
-    nlinarith
-  have hmraw : 1 - (1 - 2 * α) ^ 2 ≠ 0 := by
-    simpa [m] using hm
-  have hplusraw :
-      1 + 2 * Real.sqrt (u * (1 - u)) ≠ 0 := by
-    simpa only [ne_eq] using hplus
   calc
     1 - spectralLimit α 0 0 u =
         (z ^ 2 - m ^ 2) /
@@ -8479,7 +8448,8 @@ theorem spectralLimit_zero_fibre_boundary {α u : ℝ}
         (α * (1 - α) *
           (1 + 2 * Real.sqrt (u * (1 - u)))) := by
       dsimp [z, m, s]
-      field_simp [hmraw, hA, hfourA, hplusraw]; ring
+      rw [centered_square_difference, one_sub_centered_square,
+        mul_assoc, mul_div_mul_left _ _ (by norm_num : (4 : ℝ) ≠ 0)]
 
 /-- The asymptotic threshold used in the Johnson-code argument. -/
 def asymptoticThreshold (δ α : ℝ) : ℝ :=
@@ -8527,32 +8497,31 @@ variable {δ α β γ u : ℝ}
 
 theorem weight_pos (h : AsymptoticParameters δ α β γ u) :
     0 < α := by
-  nlinarith [h.distance_pos, h.weight_gt_distance]
+  linarith only [h.distance_pos, h.weight_gt_distance]
 
 theorem weight_complement_pos
     (h : AsymptoticParameters δ α β γ u) :
     0 < 1 - α := by
-  nlinarith [h.weight_lt_half]
+  linarith only [h.weight_lt_half]
 
 theorem degree_pos (h : AsymptoticParameters δ α β γ u) :
     0 < u := by
-  nlinarith [h.support_nonneg, h.complement_nonneg,
-    h.first_lt_degree]
+  linarith only [h.support_nonneg, h.complement_nonneg, h.first_lt_degree]
 
 theorem centeredDegree_pos
     (h : AsymptoticParameters δ α β γ u) :
     0 < (1 - 2 * u) := by
-  nlinarith [h.degree_lt_weight, h.weight_lt_half]
+  linarith only [h.degree_lt_weight, h.weight_lt_half]
 
 theorem centeredWeight_pos
     (h : AsymptoticParameters δ α β γ u) :
     0 < (1 - 2 * α) := by
-  nlinarith [h.weight_lt_half]
+  linarith only [h.weight_lt_half]
 
 theorem centeredSigma_gt_degree
     (h : AsymptoticParameters δ α β γ u) :
     (1 - 2 * u) < (1 - 2 * β - 2 * γ) := by
-  nlinarith [h.first_lt_degree]
+  linarith only [h.first_lt_degree]
 
 theorem centeredSigma_pos
     (h : AsymptoticParameters δ α β γ u) :
@@ -8563,28 +8532,28 @@ theorem centeredEta_lt_degree
     (h : AsymptoticParameters δ α β γ u) :
     centeredEta α β γ < (1 - 2 * u) := by
   unfold centeredEta
-  nlinarith [h.degree_lt_left]
+  linarith only [h.degree_lt_left]
 
 theorem neg_degree_lt_centeredEta
     (h : AsymptoticParameters δ α β γ u) :
     -(1 - 2 * u) < centeredEta α β γ := by
   unfold centeredEta
-  nlinarith [h.degree_lt_right]
+  linarith only [h.degree_lt_right]
 
 theorem one_sub_centeredWeight_sq_pos
     (h : AsymptoticParameters δ α β γ u) :
     0 < 1 - (1 - 2 * α) ^ 2 := by
   have hprod : 0 < α * (1 - α) :=
     mul_pos h.weight_pos h.weight_complement_pos
-  nlinarith
+  linarith only [hprod]
 
 theorem one_sub_centeredDegree_sq_pos
     (h : AsymptoticParameters δ α β γ u) :
     0 < 1 - (1 - 2 * u) ^ 2 := by
   have huone : 0 < 1 - u := by
-    nlinarith [h.degree_lt_weight, h.weight_lt_half]
+    linarith only [h.degree_lt_weight, h.weight_lt_half]
   have hprod : 0 < u * (1 - u) := mul_pos h.degree_pos huone
-  nlinarith
+  linarith only [hprod]
 
 theorem degree_sq_sub_eta_sq_pos
     (h : AsymptoticParameters δ α β γ u) :
@@ -8592,8 +8561,9 @@ theorem degree_sq_sub_eta_sq_pos
   have hleft : 0 < (1 - 2 * u) - centeredEta α β γ :=
     sub_pos.mpr h.centeredEta_lt_degree
   have hright : 0 < (1 - 2 * u) + centeredEta α β γ := by
-    linarith [h.neg_degree_lt_centeredEta]
-  nlinarith [mul_pos hleft hright]
+    linarith only [h.neg_degree_lt_centeredEta]
+  rw [sq_sub_sq]
+  exact mul_pos hright hleft
 
 theorem sigma_sq_sub_degree_sq_pos
     (h : AsymptoticParameters δ α β γ u) :
@@ -8602,7 +8572,8 @@ theorem sigma_sq_sub_degree_sq_pos
     sub_pos.mpr h.centeredSigma_gt_degree
   have hright : 0 < (1 - 2 * β - 2 * γ) + (1 - 2 * u) :=
     add_pos h.centeredSigma_pos h.centeredDegree_pos
-  nlinarith [mul_pos hleft hright]
+  rw [sq_sub_sq]
+  exact mul_pos hright hleft
 
 theorem rankPenalty_le_one
     (h : AsymptoticParameters δ α β γ u) :
@@ -8613,17 +8584,17 @@ theorem rankPenalty_le_one
     (binaryEntropy_le_one (γ / (1 - α)))
     h.weight_complement_pos.le
   unfold rankPenalty
-  nlinarith
+  linarith only [hsupport, hcomplement]
 
 theorem shellRate_lower
     (h : AsymptoticParameters δ α β γ u) :
     -1 ≤ shellRate α β γ u := by
   have hαentropy := binaryEntropy_le_one α
   have huone : u ≤ 1 := by
-    nlinarith [h.degree_lt_weight, h.weight_lt_half]
+    linarith only [h.degree_lt_weight, h.weight_lt_half]
   have huentropy := MetricCodes.binaryEntropy_nonneg h.degree_pos.le huone
   unfold shellRate
-  nlinarith [h.rankPenalty_le_one]
+  linarith only [hαentropy, huentropy, h.rankPenalty_le_one]
 
 end AsymptoticParameters
 
@@ -8681,13 +8652,13 @@ theorem mrrwG_continuous : Continuous mrrwG := by
 theorem mrrwG_nonneg {v : ℝ} (hv : 0 ≤ v) :
     0 ≤ mrrwG v := by
   have hroot : Real.sqrt (1 - v) ≤ 1 :=
-    Real.sqrt_le_one.mpr (by linarith)
+    Real.sqrt_le_one.mpr (by linarith only [hv])
   have hroot' : 0 ≤ Real.sqrt (1 - v) :=
     Real.sqrt_nonneg _
   unfold mrrwG
   apply MetricCodes.binaryEntropy_nonneg
-  · linarith
-  · linarith
+  · linarith only [hroot]
+  · linarith only [hroot']
 
 /-- The mrrw objective used in the Johnson-code argument. -/
 def mrrwObjective (δ r : ℝ) : ℝ :=
@@ -8738,7 +8709,7 @@ theorem mrrwG_variance {u : ℝ}
     ring
   have hroot :
       Real.sqrt (1 - 4 * u * (1 - u)) = 1 - 2 * u := by
-    rw [hrad, Real.sqrt_sq (by linarith)]
+    rw [hrad, Real.sqrt_sq (by linarith only [hu])]
   unfold mrrwG
   rw [hroot]
   congr 1
@@ -8753,14 +8724,14 @@ theorem zero_fibre_boundary_variance {δ α u : ℝ}
       (2 * Real.sqrt (u * (1 - u))) ^ 2 +
         2 * δ * (2 * Real.sqrt (u * (1 - u))) + 2 * δ := by
   have hα : 0 < α := lt_trans hu hua
-  have hαone : 0 < 1 - α := by linarith
-  have huone : 0 < 1 - u := by linarith
+  have hαone : 0 < 1 - α := by linarith only [ha]
+  have huone : 0 < 1 - u := by linarith only [hua, ha]
   have hA : α * (1 - α) ≠ 0 :=
     (mul_pos hα hαone).ne'
   have hs : 0 ≤ Real.sqrt (u * (1 - u)) :=
     Real.sqrt_nonneg _
   have hplus : 1 + 2 * Real.sqrt (u * (1 - u)) ≠ 0 := by
-    nlinarith
+    linarith only [hs]
   have hrel :
       δ / (2 * α * (1 - α)) =
         (α * (1 - α) - u * (1 - u)) /
@@ -8775,7 +8746,7 @@ theorem zero_fibre_boundary_variance {δ α u : ℝ}
       _ = _ := spectralLimit_zero_fibre_boundary hu hua ha
   have hsquare := Real.sq_sqrt (mul_pos hu huone).le
   field_simp [hA, hα.ne', hαone.ne', hplus] at hrel
-  nlinarith [hsquare]
+  linarith only [hrel, hsquare]
 
 theorem mrrwObjective_zero_fibre_boundary {δ α u : ℝ}
     (hu : 0 < u) (hua : u < α)
@@ -8784,22 +8755,22 @@ theorem mrrwObjective_zero_fibre_boundary {δ α u : ℝ}
       spectralLimit α 0 0 u = asymptoticThreshold δ α) :
     mrrwObjective δ (2 * Real.sqrt (u * (1 - u))) =
       shellRate α 0 0 u := by
-  have huone : 0 < 1 - u := by linarith
+  have huone : 0 < 1 - u := by linarith only [hua, ha]
   have hsquare := Real.sq_sqrt (mul_pos hu huone).le
   have hvariance :
       (2 * Real.sqrt (u * (1 - u))) ^ 2 =
         4 * u * (1 - u) := by
-    nlinarith [hsquare]
+    linarith only [hsquare]
   have hboundary' :=
     zero_fibre_boundary_variance hu hua ha hboundary
   have hargument :
       4 * u * (1 - u) +
         2 * δ * (2 * Real.sqrt (u * (1 - u))) + 2 * δ =
         4 * α * (1 - α) := by
-    nlinarith [hboundary', hvariance]
+    linarith only [hboundary', hvariance]
   unfold mrrwObjective
   rw [hvariance, hargument,
-    mrrwG_variance (by linarith : u ≤ (1 : ℝ) / 2),
+    mrrwG_variance (by linarith only [hua, ha] : u ≤ (1 : ℝ) / 2),
     mrrwG_variance ha.le, shellRate_zero_fibre]
   ring
 
@@ -8818,7 +8789,7 @@ theorem mrrwRateSet_bddBelow (δ : ℝ) :
     unfold mrrwG
     exact binaryEntropy_le_one _
   unfold mrrwObjective
-  linarith
+  linarith only [hfirst, hsecond]
 
 /-- The mrrw rate used in the Johnson-code argument. -/
 def mrrwRate (δ : ℝ) : ℝ :=
@@ -8839,7 +8810,7 @@ theorem exists_mrrw_minimizer {δ : ℝ}
     refine ⟨0, ?_⟩
     constructor
     · exact le_rfl
-    · linarith
+    · linarith only [hhalf]
   obtain ⟨r, hr, hmin⟩ :=
     isCompact_Icc.exists_isMinOn hnonempty
       (mrrwObjective_continuous δ).continuousOn
@@ -8865,7 +8836,7 @@ theorem mrrwRate_le_classicalRate {δ : ℝ}
     mrrwRate δ ≤ MetricCodes.Hamming.classicalRate δ := by
   calc
     mrrwRate δ ≤ mrrwObjective δ (1 - 2 * δ) :=
-      mrrwRate_le_objective (by linarith) (le_refl _)
+      mrrwRate_le_objective (by linarith only [hhalf]) (le_refl _)
     _ = MetricCodes.Hamming.classicalRate δ :=
       mrrwObjective_endpoint hδ hhalf
 
diff --git a/LeanPool/MetricCodes/HarmonicAnalysis.lean b/LeanPool/MetricCodes/HarmonicAnalysis.lean
index 56ac018d8..04dab16d8 100644
--- a/LeanPool/MetricCodes/HarmonicAnalysis.lean
+++ b/LeanPool/MetricCodes/HarmonicAnalysis.lean
@@ -103,16 +103,20 @@ theorem sphericalCodeNumber_lt_top {n : ℕ} {s : ℝ}
       have hinner : sphericalInner u v ≤ s := C.inner_le hu hv huv
       have hsq : 2 * (1 - s) ≤ dist (u : Ambient n) (v : Ambient n) ^ 2 := by
         rw [dist_eq_norm, spherical_dist_sq]
-        nlinarith
+        linarith only [hinner]
       have hsqrt : 0 < Real.sqrt (2 * (1 - s)) := by
         positivity
       have hsqrt_sq : Real.sqrt (2 * (1 - s)) ^ 2 = 2 * (1 - s) :=
-        Real.sq_sqrt (by nlinarith)
+        Real.sq_sqrt (by linarith only [hs])
       have hreal : ((2 * ε : NNReal) : ℝ) <
           dist (u : Ambient n) (v : Ambient n) := by
         change 2 * (Real.sqrt (2 * (1 - s)) / 4) <
           dist (u : Ambient n) (v : Ambient n)
-        nlinarith [show 0 ≤ dist (u : Ambient n) (v : Ambient n) from dist_nonneg]
+        nlinarith only [
+          hsq,
+          hsqrt_sq,
+          hsqrt,
+          show 0 ≤ dist (u : Ambient n) (v : Ambient n) from dist_nonneg]
       calc
         (↑(2 * ε) : ENNReal) =
             ENNReal.ofReal (((2 * ε : NNReal) : ℝ)) :=
@@ -144,22 +148,22 @@ theorem sphericalEntropy_pos {u : ℝ} (hu : 0 < u) :
     0 < sphericalEntropy u := by
   rw [sphericalEntropy_eq_log_add hu]
   have hfirst : 0 < Real.logb 2 (1 + u) :=
-    Real.logb_pos (by norm_num : (1 : ℝ) < 2) (by linarith)
+    Real.logb_pos (by norm_num : (1 : ℝ) < 2) (by linarith only [hu])
   have hratio : 1 < (1 + u) / u := by
     apply (lt_div_iff₀ hu).2
-    linarith
+    linarith only []
   have hsecond : 0 < Real.logb 2 ((1 + u) / u) :=
     Real.logb_pos (by norm_num : (1 : ℝ) < 2) hratio
-  nlinarith
+  nlinarith only [hfirst, hu, hsecond]
 
 theorem Gamma_pos {a b : ℝ} (hb : 0 ≤ b) (hab : b < a) :
     0 < Gamma a b := by
   have ha : 0 < a := lt_of_le_of_lt hb hab
   unfold Gamma
   apply div_pos
-  · exact mul_pos (sub_pos.mpr hab) (by linarith)
-  · exact mul_pos (by linarith)
-      (Real.sqrt_pos.2 (mul_pos ha (by linarith)))
+  · exact mul_pos (sub_pos.mpr hab) (by linarith only [hb, hab])
+  · exact mul_pos (by linarith only [hb, hab])
+      (Real.sqrt_pos.2 (mul_pos ha (by linarith only [hb, hab])))
 
 /-- The harmonic dimension used in the metric-code argument. -/
 def harmonicDimension (n : ℕ) : ℕ → ℕ
@@ -277,7 +281,7 @@ theorem finite_linear_programming_bound {α : Type*}
   · have hcard : 0 < (C.card : ℝ) := by
       exact_mod_cast Nat.pos_of_ne_zero hC
     apply (le_div_iff₀ hc).2
-    nlinarith [hpositive, hsum]
+    nlinarith only [hcard, hpositive, hsum]
 
 /-- The polynomial laplacian used in the spherical-code argument. -/
 def polynomialLaplacian (n : ℕ) :
@@ -481,7 +485,7 @@ theorem recurrenceDenominator_pos {n i : ℕ}
   have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn
   have hi' : (0 : ℝ) < i := by exact_mod_cast hi
   unfold recurrenceDenominator
-  linarith
+  linarith only [hn', hi']
 
 theorem forwardCoefficient_pos {n i : ℕ}
     (hn : 2 ≤ n) (hi : 0 < i) : 0 < forwardCoefficient n i := by
@@ -489,7 +493,7 @@ theorem forwardCoefficient_pos {n i : ℕ}
   have hi' : (0 : ℝ) < i := by exact_mod_cast hi
   unfold forwardCoefficient
   apply div_pos
-  · linarith
+  · linarith only [hn', hi']
   · exact recurrenceDenominator_pos hn hi
 
 theorem backwardCoefficient_pos {n i : ℕ}
@@ -638,8 +642,6 @@ theorem alphaSq_pos {n k i : ℕ}
     (hn : 3 ≤ n) (hki : k ≤ i) : 0 < alphaSq n k i := by
   have hn' : (3 : ℝ) ≤ n := by exact_mod_cast hn
   have hki' : (k : ℝ) ≤ i := by exact_mod_cast hki
-  have hi' : (0 : ℝ) ≤ i := Nat.cast_nonneg i
-  have hk' : (0 : ℝ) ≤ k := Nat.cast_nonneg k
   unfold alphaSq
   apply div_pos
   · apply mul_pos <;> linarith
@@ -649,16 +651,14 @@ theorem jacobiCoefficient_pos {n k i : ℕ}
     (hn : 3 ≤ n) (hki : k ≤ i) : 0 < jacobiCoefficient n k i := by
   have hn' : (3 : ℝ) ≤ n := by exact_mod_cast hn
   have hki' : (k : ℝ) ≤ i := by exact_mod_cast hki
-  have hi' : (0 : ℝ) ≤ i := Nat.cast_nonneg i
-  have hk' : (0 : ℝ) ≤ k := Nat.cast_nonneg k
   unfold jacobiCoefficient
   apply div_pos
   · apply mul_pos <;> linarith
   · apply Real.sqrt_pos.2
-    have h₁ : 0 < (i : ℝ) + 1 := by linarith
-    have h₂ : 0 < (i : ℝ) + (n : ℝ) - 2 := by linarith
-    have h₃ : 0 < 2 * (i : ℝ) + (n : ℝ) - 2 := by linarith
-    have h₄ : 0 < 2 * (i : ℝ) + (n : ℝ) := by linarith
+    have h₁ : 0 < (i : ℝ) + 1 := by linarith only []
+    have h₂ : 0 < (i : ℝ) + (n : ℝ) - 2 := by linarith only [hn']
+    have h₃ : 0 < 2 * (i : ℝ) + (n : ℝ) - 2 := by linarith only [hn']
+    have h₄ : 0 < 2 * (i : ℝ) + (n : ℝ) := by linarith only [hn']
     exact mul_pos (mul_pos (mul_pos h₁ h₂) h₃) h₄
 
 /-- The symmetric tridiagonal matrix with Gegenbauer Jacobi coefficients between adjacent
@@ -761,7 +761,7 @@ theorem directionalDerivative_mul
               smul_add] using h
     _ = directionalDerivative n x p * q +
           p * directionalDerivative n x q := by
-            ring
+            ac_rfl
 
 /-- The axis polynomial used in the spherical-code argument. -/
 def axisPolynomial (n : ℕ) (x : Euclidean n) :
@@ -1073,7 +1073,7 @@ theorem polynomialInner_comm (n : ℕ)
           rw [Finset.union_comm]
           apply Finset.sum_congr rfl
           intro a ha
-          ring
+          ac_rfl
     _ = ∑ a ∈ q.support,
         multiFactorial a *
           q.coeff a * p.coeff a := by
@@ -1141,7 +1141,7 @@ theorem polynomialInner_smul_left (n : ℕ) (c : ℝ)
       change
         multiFactorial a * (c * d) * q.coeff a =
           c * (multiFactorial a * d * q.coeff a)
-      ring
+      ac_rfl
   | add p q hp hq =>
       rw [smul_add, polynomialInner_add_left,
         polynomialInner_add_left, hp, hq]
@@ -1156,7 +1156,7 @@ theorem polynomialInner_smul_right (n : ℕ) (c : ℝ)
   rw [Finset.mul_sum]
   apply Finset.sum_congr rfl
   intro a ha
-  ring
+  ac_rfl
 
 theorem polynomialInner_sum_left {ι : Type*} (n : ℕ)
     (s : Finset ι) (f : ι → MvPolynomial (Fin n) ℝ)
@@ -1203,7 +1203,7 @@ theorem polynomialInner_X_mul (n : ℕ) (i : Fin n)
       rw [hmono, polynomialInner_monomial,
         polynomialInner_monomial, coeff_pderiv, hshift,
         multiFactorial_add_single]
-      ring
+      ac_rfl
   | add p q hp hq =>
       rw [mul_add, polynomialInner_add_left,
         polynomialInner_add_left, hp, hq]
@@ -1261,7 +1261,7 @@ theorem polynomialInner_axis_directional (n : ℕ)
           congr 1
           apply Finset.sum_congr rfl
           intro i hi
-          ring
+          ac_rfl
     _ = ∑ i : Fin n,
         polynomialInner n
           (MvPolynomial.C (x i) *
@@ -2100,7 +2100,7 @@ theorem polynomialLaplacian_radialPolynomial (n : ℕ) :
     MvPolynomial.pderiv_X, Pi.single_eq_same, smul_eq_mul, mul_one, MvPolynomial.derivation_C,
     mul_zero, add_zero, Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul,
     MvPolynomial.C_mul, map_natCast]
-  ring
+  ac_rfl
 
 theorem polynomialLaplacian_mul {n : ℕ}
     (p q : MvPolynomial (Fin n) ℝ) :
@@ -2152,7 +2152,7 @@ theorem polynomialLaplacian_radial_mul {n m : ℕ}
               rw [Finset.mul_sum]
               apply Finset.sum_congr rfl
               intro i hi
-              ring
+              ac_rfl
       _ = _ := by rw [heuler]
   rw [polynomialLaplacian_mul,
     polynomialLaplacian_radialPolynomial]
@@ -2202,7 +2202,7 @@ theorem directionalDerivative_radialPolynomial
           apply Finset.sum_congr rfl
           intro i hi
           rw [← MvPolynomial.C_mul']
-          ring
+          ac_rfl
     _ = 2 * axisPolynomial n x := rfl
 
 theorem directionalDerivative_radial_mul
@@ -2325,17 +2325,7 @@ theorem fischer_harmonicInner_eq_polynomialInner
       Fischer.polynomialInner n
         (p : MvPolynomial (Fin n) ℝ)
         (q : MvPolynomial (Fin n) ℝ) := by
-  change
-    Fischer.homogeneousInner n k
-      (Submodule.inclusion
-        (show harmonicHomogeneousSubmodule n k ≤
-          MvPolynomial.homogeneousSubmodule (Fin n) ℝ k from
-          inf_le_left) p)
-      (Submodule.inclusion
-        (show harmonicHomogeneousSubmodule n k ≤
-          MvPolynomial.homogeneousSubmodule (Fin n) ℝ k from
-          inf_le_left) q) = _
-  exact Fischer.homogeneousInner_eq_polynomialInner n k _ _
+  exact @Fischer.harmonicInner_eq_polynomialInner n k p q
 
 theorem fischer_harmonicInner_self_nonneg
     (n k : ℕ) (p : harmonicHomogeneousSubmodule n k) :
@@ -2416,14 +2406,7 @@ theorem polynomialLaplacian_axis_mul_for_tangent
     polynomialLaplacian n (axisPolynomial n x * p) =
       2 * directionalDerivative n x p +
         axisPolynomial n x * polynomialLaplacian n p := by
-  rw [polynomialLaplacian_mul,
-    polynomialLaplacian_axisPolynomial]
-  simp only [zero_mul, zero_add, pderiv_axisPolynomial]
-  rw [directionalDerivative_apply]
-  congr 2
-  apply Finset.sum_congr rfl
-  intro i hi
-  exact MvPolynomial.C_mul'
+  exact @solidHarmonicAxis_polynomialLaplacian_axis_mul n x p
 
 /-- The denominator `2 * k + n` used to remove the radial part of an axis projection. -/
 def harmonicAxisProjectionDenominator (n k : ℕ) : ℝ :=
@@ -2658,17 +2641,16 @@ theorem harmonicAxisProjection_correction_nonneg
     0 ≤ 1 - 2 * (harmonicAxisProjectionDenominator n k)⁻¹ := by
   have hnreal : (2 : ℝ) ≤ (n : ℝ) := by
     exact_mod_cast hn
-  have hkreal : 0 ≤ (k : ℝ) := by positivity
   have hden :
       (2 : ℝ) ≤ harmonicAxisProjectionDenominator n k := by
     unfold harmonicAxisProjectionDenominator
-    nlinarith
+    linarith only [hnreal]
   have hpos := harmonicAxisProjectionDenominator_pos
     (show 0 < n by omega) k
   apply sub_nonneg.mpr
   rw [← div_eq_mul_inv]
   apply (div_le_iff₀ hpos).mpr
-  nlinarith
+  linarith only [hden]
 
 theorem harmonicAxisLift_injective
     {n : ℕ} (hn : 2 ≤ n) (k : ℕ)
@@ -3009,30 +2991,30 @@ def boundaryDegree (s a : ℝ) : ℝ :=
 theorem spectral_iff_quadratic {s a b : ℝ} (ha : 0 < a) :
     s < 2 * MetricCodes.Gamma a b ↔
       b * (1 + b) < boundaryQuadratic s a := by
-  have hrad : 0 < a * (1 + a) := mul_pos ha (by linarith)
+  have hrad : 0 < a * (1 + a) := mul_pos ha (by linarith only [ha])
   have hden : 0 < (1 + 2 * a) * Real.sqrt (a * (1 + a)) :=
-    mul_pos (by linarith) (Real.sqrt_pos.2 hrad)
+    mul_pos (by linarith only [ha]) (Real.sqrt_pos.2 hrad)
   rw [MetricCodes.Gamma_eq_sub, ← mul_div_assoc, lt_div_iff₀ hden]
   unfold boundaryQuadratic
-  constructor <;> intro h <;> nlinarith
+  constructor <;> intro h <;> linarith only [h]
 
 theorem boundaryDegree_pos {s a : ℝ}
     (hQ : 0 < boundaryQuadratic s a) :
     0 < boundaryDegree s a := by
-  have hrad : 0 ≤ 1 + 4 * boundaryQuadratic s a := by linarith
+  have hrad : 0 ≤ 1 + 4 * boundaryQuadratic s a := by linarith only [hQ]
   have hsquare := Real.sq_sqrt hrad
   have hsqrt := Real.sqrt_nonneg (1 + 4 * boundaryQuadratic s a)
   unfold boundaryDegree
-  nlinarith
+  nlinarith only [hQ, hsquare, hsqrt]
 
 theorem boundaryDegree_mul_one_add {s a : ℝ}
     (hQ : 0 ≤ boundaryQuadratic s a) :
     boundaryDegree s a * (1 + boundaryDegree s a) =
       boundaryQuadratic s a := by
-  have hrad : 0 ≤ 1 + 4 * boundaryQuadratic s a := by linarith
+  have hrad : 0 ≤ 1 + 4 * boundaryQuadratic s a := by linarith only [hQ]
   have hsquare := Real.sq_sqrt hrad
   unfold boundaryDegree
-  nlinarith
+  linarith only [hsquare]
 
 theorem quadratic_iff_boundaryDegree {s a b : ℝ}
     (hb : 0 ≤ b) (hQ : 0 < boundaryQuadratic s a) :
@@ -3042,10 +3024,9 @@ theorem quadratic_iff_boundaryDegree {s a b : ℝ}
   constructor
   · intro h
     by_contra hnot
-    have hge : boundaryDegree s a ≤ b := le_of_not_gt hnot
-    nlinarith [sq_nonneg (b - boundaryDegree s a)]
+    nlinarith only [hroot, h, hnot, hpositive]
   · intro h
-    nlinarith [sq_nonneg (b - boundaryDegree s a)]
+    nlinarith only [hroot, h, hb]
 
 end
 
@@ -3108,15 +3089,11 @@ theorem Phi_oneRow (a b : ℝ) :
 theorem spectralAtom_lt_half {u : ℝ} (hu : 0 ≤ u) :
     spectralAtom u < (1 / 2 : ℝ) := by
   have hrad : 0 ≤ u * (1 + u) := by positivity
-  have hroot : 0 ≤ Real.sqrt (u * (1 + u)) := Real.sqrt_nonneg _
   have hlinear : 0 < 1 + 2 * u := by positivity
   have hsquare := Real.sq_sqrt hrad
-  have hgap :
-      (2 * Real.sqrt (u * (1 + u))) ^ 2 < (1 + 2 * u) ^ 2 := by
-    nlinarith
   unfold spectralAtom
   apply (div_lt_iff₀ hlinear).mpr
-  nlinarith
+  nlinarith only [hsquare, hu]
 
 theorem Interlacing.Gamma_nonneg {r : ℕ}
     {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
@@ -3334,13 +3311,7 @@ open scoped BigOperators Nat Topology
 theorem tendsto_natCast_div_self :
     Tendsto (fun n : ℕ => (n : ℝ) / (n : ℝ))
       atTop (nhds 1) := by
-  have heq :
-      (fun _ : ℕ => (1 : ℝ)) =ᶠ[atTop]
-        (fun n : ℕ => (n : ℝ) / (n : ℝ)) := by
-    filter_upwards [eventually_ne_atTop (0 : ℕ)] with n hn
-    have hn' : (n : ℝ) ≠ 0 := by exact_mod_cast hn
-    exact (div_self hn').symm
-  exact tendsto_const_nhds.congr' heq
+  exact MetricCodes.Johnson.Asymptotics.tendsto_dimension_ratio
 
 theorem tendsto_nat_succ_cast_div :
     Tendsto (fun n : ℕ => ((n + 1 : ℕ) : ℝ) / (n : ℝ))
@@ -4187,7 +4158,7 @@ theorem finiteGramFibreAmplitudeVector_unit
     simpa only [finiteGramFibreAmplitudeVector] using finiteGramFibreAmplitude_sq_sum hn k L v
       hunit hv
   rw [← EuclideanSpace.real_norm_sq_eq] at hsum
-  nlinarith [norm_nonneg (finiteGramFibreAmplitudeVector n k L v)]
+  nlinarith only [hsum, norm_nonneg (finiteGramFibreAmplitudeVector n k L v)]
 
 /-- Data encoding the finite gram certificate construction. -/
 structure FiniteGramCertificate (n k L : ℕ) where
@@ -4314,7 +4285,7 @@ theorem finiteGramCertificate_shift_identity
   rw [certificate.lift_kernel x hx y hy,
     certificate.bulk_kernel x hx y hy,
     certificate.spectralCoefficient_sq] at hdecomp
-  linarith
+  linarith only [hdecomp]
 
 theorem isometricPackingKernel_nonneg
     {n k L : ℕ}
@@ -4450,7 +4421,7 @@ theorem finiteGramCertificate_sphericalCode_bound
         ∑ x ∈ C.points, ∑ y ∈ C.points,
           isometricPackingKernel certificate.fibre x y := by
     apply (div_le_iff₀ hD).2
-    nlinarith [htrace]
+    linarith only [htrace]
   have hpositive :
       (C.points.card : ℝ) ^ 2 *
         ((Jacobi.topEigenvalue n k L - s) *
@@ -5128,7 +5099,7 @@ theorem solidHarmonicAxisFischerScale_succ
         ((r : ℝ) + 1) *
         ((r : ℝ) + harmonicAxisParameter n k - 2) := by
   simp only [solidHarmonicAxisFischerScale, Finset.prod_range_succ]
-  ring
+  ac_rfl
 
 theorem solidHarmonicAxisFischerScale_pos
     {n : ℕ} (hn : 3 ≤ n) (k r : ℕ) :
@@ -5137,17 +5108,15 @@ theorem solidHarmonicAxisFischerScale_pos
   apply Finset.prod_pos
   intro j hj
   have hnreal : (3 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn
-  have hjreal : (0 : ℝ) ≤ (j : ℝ) := by positivity
-  have hkreal : (0 : ℝ) ≤ (k : ℝ) := by positivity
   have hfirst :
       0 < 2 * (j : ℝ) + harmonicAxisParameter n k - 2 := by
     unfold harmonicAxisParameter
-    linarith
+    linarith only [hnreal]
   have hmiddle : 0 < (j : ℝ) + 1 := by positivity
   have hlast :
       0 < (j : ℝ) + harmonicAxisParameter n k - 2 := by
     unfold harmonicAxisParameter
-    linarith
+    linarith only [hnreal]
   exact mul_pos (mul_pos hfirst hmiddle) hlast
 
 theorem fischer_polynomialInner_constant_mul_left
@@ -5254,7 +5223,7 @@ theorem solidHarmonicAxisLift_fischer_inner
   | succ r ihr =>
       rw [solidHarmonicAxisLift_fischer_inner_succ x hx p q r,
         ihr, solidHarmonicAxisFischerScale_succ]
-      ring
+      ac_rfl
 
 /-- The solid harmonic axis lift restricted from tangent harmonics to the target harmonic
 degree. -/
@@ -5472,7 +5441,7 @@ theorem positiveRadicalSymmetrization
   have hright : 0 ≤
       (r / Real.sqrt (a * b * p * q)) * Real.sqrt e := by
     positivity
-  nlinarith
+  nlinarith only [hsquare, hleft, hright]
 
 theorem alphaSq_sqrt_harmonicDimension
     {n k i : ℕ} (hn : 3 ≤ n) (hki : k ≤ i) :
@@ -5481,7 +5450,6 @@ theorem alphaSq_sqrt_harmonicDimension
       Gegenbauer.jacobiCoefficient n k i *
         Real.sqrt (Gegenbauer.harmonicDimension n (i + 1) : ℝ) := by
   have hnreal : (3 : ℝ) ≤ n := by exact_mod_cast hn
-  have hireal : (0 : ℝ) ≤ i := Nat.cast_nonneg _
   have hkreal : (k : ℝ) ≤ i := by exact_mod_cast hki
   have hd : (0 : ℝ) < Gegenbauer.harmonicDimension n i := by
     exact_mod_cast
@@ -5490,12 +5458,11 @@ theorem alphaSq_sqrt_harmonicDimension
     exact_mod_cast
       Gegenbauer.harmonicDimension_pos (by omega : 2 ≤ n) (i + 1)
   have ha : 0 < (i : ℝ) + 1 := by positivity
-  have hb : 0 < (i : ℝ) + n - 2 := by linarith
-  have hp : 0 < 2 * (i : ℝ) + n - 2 := by linarith
-  have hq : 0 < 2 * (i : ℝ) + n := by linarith
+  have hb : 0 < (i : ℝ) + n - 2 := by linarith only [hnreal]
+  have hp : 0 < 2 * (i : ℝ) + n - 2 := by linarith only [hnreal]
+  have hq : 0 < 2 * (i : ℝ) + n := by linarith only [hnreal]
   have hr : 0 ≤
       ((i : ℝ) - k + 1) * ((i : ℝ) + k + n - 2) := by
-    have hkzero : (0 : ℝ) ≤ k := Nat.cast_nonneg _
     exact mul_nonneg (by linarith) (by linarith)
   have hcross0 :=
     JacobiSymmetrization.harmonicDimension_cross_mul hn i
@@ -5519,7 +5486,6 @@ theorem betaSq_sqrt_harmonicDimension
       Gegenbauer.jacobiCoefficient n k i *
         Real.sqrt (Gegenbauer.harmonicDimension n i : ℝ) := by
   have hnreal : (3 : ℝ) ≤ n := by exact_mod_cast hn
-  have hireal : (0 : ℝ) ≤ i := Nat.cast_nonneg _
   have hkreal : (k : ℝ) ≤ i := by exact_mod_cast hki
   have hd : (0 : ℝ) < Gegenbauer.harmonicDimension n i := by
     exact_mod_cast
@@ -5528,12 +5494,11 @@ theorem betaSq_sqrt_harmonicDimension
     exact_mod_cast
       Gegenbauer.harmonicDimension_pos (by omega : 2 ≤ n) (i + 1)
   have ha : 0 < (i : ℝ) + 1 := by positivity
-  have hb : 0 < (i : ℝ) + n - 2 := by linarith
-  have hp : 0 < 2 * (i : ℝ) + n - 2 := by linarith
-  have hq : 0 < 2 * (i : ℝ) + n := by linarith
+  have hb : 0 < (i : ℝ) + n - 2 := by linarith only [hnreal]
+  have hp : 0 < 2 * (i : ℝ) + n - 2 := by linarith only [hnreal]
+  have hq : 0 < 2 * (i : ℝ) + n := by linarith only [hnreal]
   have hr : 0 ≤
       ((i : ℝ) - k + 1) * ((i : ℝ) + k + n - 2) := by
-    have hkzero : (0 : ℝ) ≤ k := Nat.cast_nonneg _
     exact mul_nonneg (by linarith) (by linarith)
   have hcross0 :=
     JacobiSymmetrization.harmonicDimension_cross_mul hn i
@@ -5620,7 +5585,7 @@ theorem finiteGramRecurrenceWeight_eigenrecurrence
             (sourceChannelCoefficient n k L m i *
               Real.sqrt
                 (Gegenbauer.harmonicDimension n (k + i.val) : ℝ)) * v i := by
-                ring
+                ac_rfl
           _ = (Gegenbauer.jacobiMatrix n k L m i *
                 Real.sqrt
                   (Gegenbauer.harmonicDimension n (k + m.val) : ℝ)) *
@@ -5629,7 +5594,7 @@ theorem finiteGramRecurrenceWeight_eigenrecurrence
           _ = (Gegenbauer.jacobiMatrix n k L m i * v i) *
                 Real.sqrt
                   (Gegenbauer.harmonicDimension n (k + m.val) : ℝ) := by
-                  ring
+                  ac_rfl
     _ = (∑ i : Jacobi.Index k L,
           Gegenbauer.jacobiMatrix n k L m i * v i) *
         Real.sqrt
@@ -5638,7 +5603,7 @@ theorem finiteGramRecurrenceWeight_eigenrecurrence
     _ = eigenvalue * finiteGramRecurrenceWeight n k L v m := by
         rw [hcoordinate]
         unfold finiteGramRecurrenceWeight
-        ring
+        ac_rfl
 
 theorem jacobiMatrix_adjacent_pos
     {n k L : ℕ} (hn : 3 ≤ n)
@@ -5937,7 +5902,7 @@ theorem sourceChannelCoefficient_nonneg
           (Gegenbauer.harmonicDimension n (k + i.val) : ℝ) := by
     rw [← hsource]
     exact mul_nonneg hmatrix (Real.sqrt_nonneg _)
-  nlinarith
+  nlinarith only [hproduct, hdimension]
 
 /-- The adjacent-block amplitude normalized by Perron recurrence weights and the top Jacobi
 eigenvalue. -/
@@ -6070,7 +6035,7 @@ theorem sourceAdjacentBlockCoefficient_amplitude_identity
       HarmonicCertificateAssembly.harmonicFibreAmplitudes_sq
         n k L hn source]
     field_simp [heigen.ne', hsource.ne', hnormal.ne']
-  nlinarith
+  nlinarith only [hsquare, hleft, hright]
 
 theorem sourceAdjacentBlockCoefficient_amplitude_sum
     {n k L : ℕ} (hn : 3 ≤ n) (hkl : k < L)
@@ -6529,7 +6494,7 @@ theorem sourceAdjacentChannel_inner_degreeAxisTensor
         (degreeFibre target x u)),
     adjacent.axis_projection x hx target source u,
     real_inner_smul_right]
-  ring
+  ac_rfl
 
 theorem harmonicWeightedFibre_ambient_inner
     (n k L : ℕ) (hn : 3 ≤ n) (hkl : k ≤ L)
@@ -6615,7 +6580,7 @@ theorem sourceAdjacentHarmonicRow_inner_axis_fibre
           (block source) u
           (HarmonicCertificateAssembly.harmonicFibreAmplitudes
             n k L hn target)]
-      ring
+      ac_rfl
     _ = ∑ source : Jacobi.Index k L,
           (∑ target : Jacobi.Index k L,
             sourceAdjacentBlockCoefficient
@@ -6639,7 +6604,7 @@ theorem sourceAdjacentHarmonicRow_inner_axis_fibre
       rw [Finset.mul_sum]
       apply Finset.sum_congr rfl
       intro source hsource
-      ring
+      ac_rfl
     _ = _ := by
       rw [harmonicWeightedFibre_ambient_inner]
 
@@ -6821,7 +6786,7 @@ theorem rawLiftChannel_inner {n k L : ℕ}
         rw [Finset.sum_mul]
         apply Finset.sum_congr rfl
         intro a ha
-        ring
+        ac_rfl
     _ = ⟪x, y⟫_ℝ * isometricPackingKernel f x y := by
       rw [projectionMatrixFeature_inner]
       congr 1
@@ -7197,7 +7162,7 @@ theorem sourceSpectralBulk_lift_basis_inner
     euclideanChannelFeatureMap_inner,
     sourceSpectralBulk_lift_inner fibre source x y hy,
     embeddedBulkChannel_inner]
-  ring
+  ac_rfl
 
 /-- Data encoding the spectral channel remainder construction. -/
 structure SpectralChannelRemainder
@@ -7676,7 +7641,6 @@ theorem upperChannelDenominator_pos
     {n : ℕ} (hn : 0 < n) (m : ℕ) :
     0 < upperChannelDenominator n m := by
   unfold upperChannelDenominator
-  have hnreal : 0 < (n : ℝ) := by exact_mod_cast hn
   positivity
 
 theorem upperChannelWeight_pos
@@ -7794,8 +7758,7 @@ theorem lowerChannelWeight_pos
     0 < lowerChannelWeight n m := by
   unfold lowerChannelWeight
   have hnreal : (3 : ℝ) ≤ n := by exact_mod_cast hn
-  have hmreal : 0 ≤ (m : ℝ) := by positivity
-  apply div_pos <;> linarith
+  apply div_pos <;> linarith only [hnreal]
 
 /-- The harmonic co-gradient scaled by the inverse square root of the channel denominator. -/
 def lowerChannelAdjoint (n m : ℕ) :
@@ -7863,7 +7826,7 @@ theorem radialGradient_derivative_sum
   apply congrArg₂ (· + ·)
   · apply Finset.sum_congr rfl
     intro j hj
-    ring
+    ac_rfl
   · rfl
 
 theorem harmonicCoordinateMultiplication_derivative_sum
@@ -7985,18 +7948,17 @@ theorem lowerFischerGradientWeight_eq
   cases m with
   | zero =>
       have hnne : (n : ℝ) ≠ 0 := by positivity
-      have hnsub : (n : ℝ) - 2 ≠ 0 := by linarith
+      have hnsub : (n : ℝ) - 2 ≠ 0 := by linarith only [hnreal]
       unfold lowerFischerGradientWeight upperChannelDenominator
         lowerChannelWeight
       norm_num
       field_simp
   | succ k =>
-      have hkreal : 0 ≤ (k : ℝ) := by positivity
       have hden : 2 * (k : ℝ) + n ≠ 0 := by
-        have : 0 < 2 * (k : ℝ) + n := by linarith
+        have : 0 < 2 * (k : ℝ) + n := by linarith only [hnreal]
         exact ne_of_gt this
       have hden' : (n : ℝ) + (k : ℝ) * 2 ≠ 0 := by
-        have : 0 < (n : ℝ) + (k : ℝ) * 2 := by nlinarith
+        have : 0 < (n : ℝ) + (k : ℝ) * 2 := by linarith only [hnreal]
         exact ne_of_gt this
       unfold lowerFischerGradientWeight upperChannelDenominator
         lowerChannelWeight
@@ -8074,7 +8036,7 @@ theorem harmonicCoordinateRaising_derivative_sum
             smul_smul, smul_smul, ← sub_smul]
           unfold lowerFischerGradientWeight
           congr 1
-          ring
+          ac_rfl
 
 theorem harmonicCoGradient_inner
     {n : ℕ} (hn : 3 ≤ n) (m : ℕ)
@@ -8271,7 +8233,7 @@ theorem upperChannel_harmonicAxisTensor_inner
       SpherePacking.Fischer.polynomialInner_smul_left,
       SpherePacking.Fischer.polynomialInner_smul_right,
       SpherePacking.Fischer.polynomialInner_smul_right]
-    ring
+    ac_rfl
   calc
     ⟪upperChannel n m (harmonicAxisTensor n m x p), q⟫_ℝ =
       ⟪harmonicAxisTensor n m x p,
@@ -8356,13 +8318,11 @@ theorem upperChannel_solidHarmonicSource
         SpherePacking.solidHarmonicAxisPolynomialIsometry
           hn k (r + 1) x hx p := by
   have hnreal : (3 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn
-  have hkreal : (0 : ℝ) ≤ (k : ℝ) := by positivity
-  have hrreal : (0 : ℝ) ≤ (r : ℝ) := by positivity
   have hraise :
       0 < 2 * (r : ℝ) +
         SpherePacking.harmonicAxisParameter n k - 2 := by
     unfold SpherePacking.harmonicAxisParameter
-    linarith
+    linarith only [hnreal]
   have hscale :=
     SpherePacking.solidHarmonicAxisFischerScale_pos hn k r
   have hscale_next :=
@@ -8423,13 +8383,11 @@ theorem solidHarmonicSourceUpperCoefficient_pos
     {n : ℕ} (hn : 3 ≤ n) (k r : ℕ) :
     0 < solidHarmonicSourceUpperCoefficient n k r := by
   have hnreal : (3 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn
-  have hkreal : (0 : ℝ) ≤ (k : ℝ) := by positivity
-  have hrreal : (0 : ℝ) ≤ (r : ℝ) := by positivity
   have hraise :
       0 < 2 * (r : ℝ) +
         SpherePacking.harmonicAxisParameter n k - 2 := by
     unfold SpherePacking.harmonicAxisParameter
-    linarith
+    linarith only [hnreal]
   unfold solidHarmonicSourceUpperCoefficient
   apply div_pos
   · exact Real.sqrt_pos.mpr
@@ -8456,13 +8414,11 @@ theorem solidHarmonicSourceUpperCoefficient_sq
           SpherePacking.harmonicAxisParameter n k - 2) *
           upperChannelDenominator n (k + r)) := by
   have hnreal : (3 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn
-  have hkreal : (0 : ℝ) ≤ (k : ℝ) := by positivity
-  have hrreal : (0 : ℝ) ≤ (r : ℝ) := by positivity
   have hraise :
       0 < 2 * (r : ℝ) +
         SpherePacking.harmonicAxisParameter n k - 2 := by
     unfold SpherePacking.harmonicAxisParameter
-    linarith
+    linarith only [hnreal]
   have hscale :=
     SpherePacking.solidHarmonicAxisFischerScale_pos hn k r
   have hscale_next :=
@@ -8483,15 +8439,6 @@ theorem solidHarmonicSourceUpperIsometryCoefficient_sq
     solidHarmonicSourceUpperIsometryCoefficient n k r ^ 2 =
       SpherePacking.Gegenbauer.alphaSq n k (k + r) := by
   have hnreal : (3 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn
-  have hkreal : (0 : ℝ) ≤ (k : ℝ) := by positivity
-  have hrreal : (0 : ℝ) ≤ (r : ℝ) := by positivity
-  have hraise :
-      0 < 2 * (r : ℝ) +
-        SpherePacking.harmonicAxisParameter n k - 2 := by
-    unfold SpherePacking.harmonicAxisParameter
-    linarith
-  have hden :=
-    upperChannelDenominator_pos (by omega : 0 < n) (k + r)
   have hweight :=
     upperChannelWeight_pos (by omega : 0 < n) (k + r)
   unfold solidHarmonicSourceUpperIsometryCoefficient
@@ -8525,7 +8472,7 @@ theorem solidHarmonicSourceUpperIsometryCoefficient_eq_sqrt
   have hroot := Real.sq_sqrt halpha.le
   have hroot_nonneg :=
     Real.sqrt_nonneg (SpherePacking.Gegenbauer.alphaSq n k (k + r))
-  nlinarith
+  nlinarith only [hsquare, hroot, hcoef, hroot_nonneg]
 
 theorem harmonic_inner_smul_right
     (n m : ℕ) (p q : HarmonicSpace n m) (c : ℝ) :
@@ -8668,7 +8615,7 @@ theorem lowerChannel_harmonicAxisTensor_inner
       SpherePacking.Fischer.harmonicInner_eq_polynomialInner,
       harmonicCoordinateDerivative_apply,
       SpherePacking.Fischer.polynomialInner_smul_left]
-    ring
+    ac_rfl
   calc
     ⟪lowerChannel n m (harmonicAxisTensor n (m + 1) x p), q⟫_ℝ =
       ⟪harmonicAxisTensor n (m + 1) x p,
@@ -8791,13 +8738,11 @@ theorem solidHarmonicSourceLowerCoefficient_pos
     {n : ℕ} (hn : 3 ≤ n) (k r : ℕ) :
     0 < solidHarmonicSourceLowerCoefficient n k r := by
   have hnreal : (3 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn
-  have hkreal : (0 : ℝ) ≤ (k : ℝ) := by positivity
-  have hrreal : (0 : ℝ) ≤ (r : ℝ) := by positivity
   have hlast :
       0 < (r : ℝ) +
         SpherePacking.harmonicAxisParameter n k - 2 := by
     unfold SpherePacking.harmonicAxisParameter
-    linarith
+    linarith only [hnreal]
   unfold solidHarmonicSourceLowerCoefficient
   apply div_pos
   · exact mul_pos
@@ -8820,13 +8765,11 @@ theorem solidHarmonicSourceLowerCoefficient_sq
           SpherePacking.harmonicAxisParameter n k - 2) *
           upperChannelDenominator n (k + r)) := by
   have hnreal : (3 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn
-  have hkreal : (0 : ℝ) ≤ (k : ℝ) := by positivity
-  have hrreal : (0 : ℝ) ≤ (r : ℝ) := by positivity
   have hraise :
       0 < 2 * (r : ℝ) +
         SpherePacking.harmonicAxisParameter n k - 2 := by
     unfold SpherePacking.harmonicAxisParameter
-    linarith
+    linarith only [hnreal]
   have hscale :=
     SpherePacking.solidHarmonicAxisFischerScale_pos hn k r
   have hscale_next :=
@@ -8847,39 +8790,30 @@ theorem solidHarmonicSourceLowerIsometryCoefficient_sq
     solidHarmonicSourceLowerIsometryCoefficient n k r ^ 2 =
       SpherePacking.Gegenbauer.betaSq n k (k + (r + 1)) := by
   have hnreal : (3 : ℝ) ≤ (n : ℝ) := by exact_mod_cast hn
-  have hkreal : (0 : ℝ) ≤ (k : ℝ) := by positivity
-  have hrreal : (0 : ℝ) ≤ (r : ℝ) := by positivity
-  have hraise :
-      0 < 2 * (r : ℝ) +
-        SpherePacking.harmonicAxisParameter n k - 2 := by
-    unfold SpherePacking.harmonicAxisParameter
-    linarith
-  have hden :=
-    upperChannelDenominator_pos (by omega : 0 < n) (k + r)
   have hweight := lowerChannelWeight_pos hn (k + r)
   have hmiddle :
       0 < (k : ℝ) + (r : ℝ) + (n : ℝ) - 2 := by
-    linarith
+    linarith only [hnreal]
   have hlowerden :
       0 < 2 * ((k : ℝ) + (r : ℝ)) + (n : ℝ) - 2 := by
-    linarith
+    linarith only [hnreal]
   have hupperden :
       0 < 2 * ((k : ℝ) + (r : ℝ)) + (n : ℝ) := by
-    linarith
+    linarith only [hnreal]
   have hfirstden :
       0 < 2 * (r : ℝ) +
         ((n : ℝ) + 2 * (k : ℝ)) - 2 := by
-    linarith
+    linarith only [hnreal]
   have hfirstden_rev :
       0 < (r : ℝ) * 2 +
         ((n : ℝ) + 2 * (k : ℝ)) - 2 := by
-    linarith
+    linarith only [hnreal]
   have hbetafirst :
       0 < (k : ℝ) + ((r : ℝ) + 1) + (n : ℝ) - 3 := by
-    linarith
+    linarith only [hnreal]
   have hbetasecond :
       0 < 2 * ((k : ℝ) + ((r : ℝ) + 1)) + (n : ℝ) - 2 := by
-    linarith
+    linarith only [hnreal]
   unfold solidHarmonicSourceLowerIsometryCoefficient
   rw [div_pow, Real.sq_sqrt hweight.le,
     solidHarmonicSourceLowerCoefficient_sq hn k r]
@@ -8916,7 +8850,7 @@ theorem solidHarmonicSourceLowerIsometryCoefficient_eq_sqrt
   have hroot := Real.sq_sqrt hbeta
   have hroot_nonneg := Real.sqrt_nonneg
     (SpherePacking.Gegenbauer.betaSq n k (k + (r + 1)))
-  nlinarith
+  nlinarith only [hsquare, hroot, hcoef, hroot_nonneg]
 
 theorem lowerChannelIsometry_adjoint_solidHarmonicSource
     {n : ℕ} (hn : 3 ≤ n) (k r : ℕ)
diff --git a/LeanPool/MetricCodes/Hierarchy.lean b/LeanPool/MetricCodes/Hierarchy.lean
index f66fce0cd..ff1c9749f 100644
--- a/LeanPool/MetricCodes/Hierarchy.lean
+++ b/LeanPool/MetricCodes/Hierarchy.lean
@@ -136,7 +136,7 @@ theorem normalizedCoefficient_eq_Gamma
           (a * (1 + a)) * (1 + 2 * a) ^ 2 := by
       ring
     rw [hpoly, Real.sqrt_mul hfactor, Real.sqrt_sq hlin]
-    ring
+    ac_rfl
   rw [hrad]
   congr 1
   ring
@@ -170,8 +170,8 @@ theorem tendsto_terminal_coefficient {a b : ℝ}
           (2 * a + 1 - 2 * 0) * (2 * a + 1)) ≠ 0 := by
     apply ne_of_gt
     apply Real.sqrt_pos.2
-    have hone : 0 < a + 1 := by linarith
-    have htwo : 0 < 2 * a + 1 := by linarith
+    have hone : 0 < a + 1 := by linarith only [ha]
+    have htwo : 0 < 2 * a + 1 := by linarith only [ha]
     simpa only [add_zero, mul_zero, sub_zero,
       gt_iff_lt] using mul_pos (mul_pos (mul_pos ha hone) htwo) htwo
   have hnorm := hnum.div hroot hrootne
@@ -206,10 +206,10 @@ theorem eventually_transverse_add_le_longitudinal
   let δ : ℝ := (a - b) / 2
   have hδ : 0 < δ := by
     dsimp [δ]
-    linarith
+    linarith only [hba]
   have hδlt : δ < a - b := by
     dsimp [δ]
-    linarith
+    linarith only [hba]
   have hratio :=
     ((tendsto_longitudinal_ratio ha.le).sub
       (tendsto_transverse_ratio hb)).eventually
@@ -234,7 +234,7 @@ theorem eventually_transverse_add_le_longitudinal
   have hreal :
       (transverseDegree b n : ℝ) + (m : ℝ) ≤
         (longitudinalDegree a n : ℝ) := by
-    linarith
+    linarith only [hnlarge, hmul]
   exact_mod_cast hreal
 
 /-- The terminal edge rayleigh used in the spherical-code argument. -/
@@ -307,20 +307,18 @@ theorem eventually_topEigenvalue_gt {a b s : ℝ}
       s < SpherePacking.Jacobi.topEigenvalue
         n (transverseDegree b n) (longitudinalDegree a n) := by
   have ha : 0 < a := lt_of_le_of_lt hb hba
-  have hgamma : 0 < 2 * MetricCodes.Gamma a b := by
-    exact mul_pos (by norm_num) (MetricCodes.Gamma_pos hb hba)
   let g : ℝ := 2 * MetricCodes.Gamma a b
   let ε : ℝ := (g - s) / 2
   have hε : 0 < ε := by
     dsimp [ε, g]
-    linarith
+    linarith only [hs]
   obtain ⟨m, hm⟩ := exists_nat_gt (g / ε)
   have hprod : g < (m : ℝ) * ε :=
     (div_lt_iff₀ hε).mp hm
   have hden : 0 < (m : ℝ) + 1 := by positivity
   have hrem : g / ((m : ℝ) + 1) < ε := by
     apply (div_lt_iff₀ hden).2
-    nlinarith [hgamma]
+    linarith only [hε, hprod]
   have hidentity :
       ((m : ℝ) / ((m : ℝ) + 1)) * g =
         g - g / ((m : ℝ) + 1) := by
@@ -329,7 +327,7 @@ theorem eventually_topEigenvalue_gt {a b s : ℝ}
       s < ((m : ℝ) / ((m : ℝ) + 1)) * g := by
     rw [hidentity]
     dsimp [ε] at hrem
-    linarith
+    linarith only [hs, hrem]
   have hquot :=
     (tendsto_terminalEdgeRayleigh ha hb m).eventually
       (lt_mem_nhds (show
@@ -349,7 +347,7 @@ theorem spectralGap_pos {s a b : ℝ}
     (h : s < 2 * MetricCodes.Gamma a b) :
     0 < spectralGap s a b := by
   unfold spectralGap
-  linarith
+  linarith only [h]
 
 theorem eventually_topEigenvalue_uniform_gap {a b s : ℝ}
     (hb : 0 ≤ b) (hba : b < a)
@@ -360,7 +358,7 @@ theorem eventually_topEigenvalue_uniform_gap {a b s : ℝ}
           n (transverseDegree b n) (longitudinalDegree a n) := by
   apply eventually_topEigenvalue_gt hb hba
   unfold spectralGap
-  linarith
+  linarith only [hs]
 
 /-- The spectral prefactor used in the spherical-code argument. -/
 def spectralPrefactor (s a b : ℝ) : ℝ :=
@@ -399,7 +397,7 @@ theorem eventually_sphericalCode_card_le_harmonicQuotient
   have heigengap :
       s < SpherePacking.Jacobi.topEigenvalue
         n (transverseDegree b n) (longitudinalDegree a n) := by
-    linarith
+    linarith only [hgap, heigen]
   have hactual := sphericalCode_bound_of_actual_harmonic_channels
     hn hkl hs.le heigengap C
   change
@@ -411,7 +409,7 @@ theorem eventually_sphericalCode_card_le_harmonicQuotient
   have hden :
       0 < SpherePacking.Jacobi.topEigenvalue
         n (transverseDegree b n) (longitudinalDegree a n) - s := by
-    linarith
+    linarith only [hgap, heigen]
   have hfrac :
       (1 - s) /
           (SpherePacking.Jacobi.topEigenvalue
@@ -420,8 +418,8 @@ theorem eventually_sphericalCode_card_le_harmonicQuotient
     unfold spectralPrefactor
     apply (div_le_div_iff₀ hden hgap).2
     apply mul_le_mul_of_nonneg_left
-    · linarith
-    · linarith
+    · linarith only [heigen]
+    · linarith only [hs]
   have htruncated :
       0 ≤ SpherePacking.truncatedDimensionQuotient a b n := by
     unfold SpherePacking.truncatedDimensionQuotient
@@ -755,9 +753,9 @@ theorem exists_oneRow_certificate_better_than_levelZero
       with b hγ hbentropy (hb : 0 < b)
     refine ⟨hb, ?_, ?_, hbentropy⟩
     · unfold MetricCodes.Spherical.sphericalImprovementPath
-      nlinarith [mul_pos (sub_pos.mpr hslope) hb]
+      linarith only [ha, mul_pos (sub_pos.mpr hslope) hb]
     · rw [hclassical] at hγ
-      nlinarith
+      linarith only [hspectral, hγ]
   obtain ⟨b, hb, hba, hgap, hcost⟩ := hgood.exists
   exact ⟨MetricCodes.Spherical.sphericalImprovementPath a b,
     b, hb, hba, hgap, hcost⟩
@@ -776,7 +774,7 @@ theorem eventually_sphericalCode_card_lt_levelZero
     have hazero : a 0 = 0 := le_antisymm (le_of_not_gt hnot) ha_nonneg
     rw [Gamma_zero, hazero] at hspectral
     norm_num [spectralAtom] at hspectral
-    linarith
+    linarith only [hs, hspectral]
   rw [Gamma_zero] at hspectral
   obtain ⟨A, B, hB, hBA, hgap, hcost⟩ :=
     exists_oneRow_certificate_better_than_levelZero ha hspectral
@@ -801,7 +799,7 @@ theorem zero_fibre_spectral_iff_classicalThreshold_lt
   have hroot : 0 < Real.sqrt (a * (1 + a)) :=
     Real.sqrt_pos.mpr hvariance
   have hroot_sq := Real.sq_sqrt hvariance.le
-  have hradical : 0 < 1 - s ^ 2 := by nlinarith
+  have hradical : 0 < 1 - s ^ 2 := by nlinarith only [hs', hs]
   have hclassical : 0 < Real.sqrt (1 - s ^ 2) :=
     Real.sqrt_pos.mpr hradical
   have hclassical_sq := Real.sq_sqrt hradical.le
@@ -828,27 +826,27 @@ theorem zero_fibre_spectral_iff_classicalThreshold_lt
         (mul_nonneg (by norm_num : (0 : ℝ) ≤ 2) hroot.le)).mpr h
       apply (sq_lt_sq₀ (by norm_num : (0 : ℝ) ≤ 1)
         (mul_nonneg hclassical.le hlinear.le)).mp
-      nlinarith [hsquares]
+      linarith only [hsq, hsquares]
     · intro h
       have hsq := (sq_lt_sq₀ (by norm_num : (0 : ℝ) ≤ 1)
         (mul_nonneg hclassical.le hlinear.le)).mpr h
       apply (sq_lt_sq₀
         (mul_nonneg hs.le hlinear.le)
         (mul_nonneg (by norm_num : (0 : ℝ) ≤ 2) hroot.le)).mp
-      nlinarith [hsquares]
+      linarith only [hsq, hsquares]
   rw [hcompare]
   unfold MetricCodes.classicalThreshold
   constructor
   · intro h
     have hdiv : 1 / Real.sqrt (1 - s ^ 2) < 1 + 2 * a := by
       apply (div_lt_iff₀ hclassical).mpr
-      nlinarith
-    linarith
+      linarith only [h]
+    linarith only [hdiv]
   · intro h
     have hdiv : 1 / Real.sqrt (1 - s ^ 2) < 1 + 2 * a := by
-      linarith
+      linarith only [h]
     have hmul := (div_lt_iff₀ hclassical).mp hdiv
-    nlinarith
+    linarith only [hmul]
 
 theorem boundaryQuadratic_pos_iff_classicalThreshold_lt
     {s a : ℝ} (hs : 0 < s) (hs' : s < 1) (ha : 0 < a) :
@@ -862,15 +860,13 @@ theorem boundaryDegree_lt_longitudinal
     {s a : ℝ} (hs : 0 < s) (ha : 0 < a)
     (hquadratic : 0 < boundaryQuadratic s a) :
     boundaryDegree s a < a := by
-  have hb := boundaryDegree_pos hquadratic
   have heq := boundaryDegree_mul_one_add hquadratic.le
-  have hroot : 0 < Real.sqrt (a * (1 + a)) := by positivity
   have hstrict : boundaryQuadratic s a < a * (1 + a) := by
     unfold boundaryQuadratic
     have hterm : 0 < (s / 2) * (1 + 2 * a) *
         Real.sqrt (a * (1 + a)) := by positivity
-    linarith
-  nlinarith [sq_nonneg (a - boundaryDegree s a)]
+    linarith only [hterm]
+  nlinarith only [heq, hstrict, ha]
 
 theorem feasible_iff_boundary
     {s a b : ℝ} (hs : 0 < s) (hs' : s < 1) :
@@ -882,7 +878,7 @@ theorem feasible_iff_boundary
     have ha : 0 < a := hb.trans hba
     have hquadratic : 0 < boundaryQuadratic s a := by
       have hstrict := (spectral_iff_quadratic ha).mp hspectral
-      nlinarith [mul_pos hb (by linarith : 0 < 1 + b)]
+      linarith only [hstrict, mul_pos hb (by linarith : 0 < 1 + b)]
     exact ⟨(boundaryQuadratic_pos_iff_classicalThreshold_lt
       hs hs' ha).mp hquadratic, hb,
       (quadratic_iff_boundaryDegree hb.le hquadratic).mp
@@ -1025,7 +1021,7 @@ theorem sphericalEntropy_upper_logb_add
         1 / Real.log 2 := by
     rw [← mul_div_assoc]
     exact (div_le_div_iff_of_pos_right hlogtwo).mpr hupper
-  linarith
+  linarith only [hfraction]
 
 end
 
@@ -1050,7 +1046,7 @@ theorem sliceCost_nonneg {s t : ℝ}
 @[simp] theorem sliceCost_self {s : ℝ} (hs : s < 1) :
     sliceCost s s = 0 := by
   unfold sliceCost
-  rw [div_self (by linarith : 1 - s ≠ 0)]
+  rw [div_self (by linarith only [hs] : 1 - s ≠ 0)]
   simp only [one_div, Real.logb_one, mul_zero]
 
 /-- The localized envelope used in the spherical-code argument. -/
@@ -1209,18 +1205,16 @@ theorem scaleCoordinate_nonneg {c u : ℝ}
   have hsquare := Real.sq_sqrt hrad
   have hsqrt := Real.sqrt_nonneg (1 + 4 * c * (u * (1 + u)))
   unfold scaleCoordinate
-  nlinarith [mul_nonneg hc hx]
+  nlinarith only [hsquare, hsqrt, mul_nonneg hc hx]
 
 theorem quadraticCoordinate_scaleCoordinate {c u : ℝ}
     (hc : 0 ≤ c) (hu : 0 ≤ u) :
     ((scaleCoordinate c u) * (1 + (scaleCoordinate c u))) =
       c * (u * (1 + u)) := by
-  have hx : 0 ≤ (u * (1 + u)) := by
-    positivity
   have hrad : 0 ≤ 1 + 4 * c * (u * (1 + u)) := by positivity
   have hsquare := Real.sq_sqrt hrad
   unfold scaleCoordinate at *
-  nlinarith
+  linarith only [hsquare]
 
 @[simp] theorem scaleCoordinate_one {u : ℝ} (hu : 0 ≤ u) :
     scaleCoordinate 1 u = u := by
@@ -1314,36 +1308,31 @@ theorem spectralAtom_strictMonoOn :
   change 0 ≤ u at hu
   change 0 ≤ v at hv
   have hquad := quadraticCoordinate_strictMonoOn hu hv huv
-  have hqu : 0 ≤ (u * (1 + u)) := by
-    positivity
-  have hqv : 0 ≤ (v * (1 + v)) := by
-    positivity
   have hdenu : 0 < 1 + 4 * (u * (1 + u)) := by positivity
   have hdenv : 0 < 1 + 4 * (v * (1 + v)) := by positivity
   have hsquare : spectralAtom u ^ 2 < spectralAtom v ^ 2 := by
     rw [spectralAtom_sq hu, spectralAtom_sq hv]
     apply (div_lt_div_iff₀ hdenu hdenv).mpr
-    nlinarith
-  have hnonneg := spectralAtom_nonneg hu
+    linarith only [hquad]
   have hnonneg' := spectralAtom_nonneg hv
-  nlinarith
+  nlinarith only [hsquare, hnonneg']
 
 theorem spectralAtom_scale_lt {c u : ℝ} (hc : 1 < c) (hu : 0 < u) :
     spectralAtom u < spectralAtom (scaleCoordinate c u) := by
   have hquad : (u * (1 + u)) <
       ((scaleCoordinate c u) * (1 + (scaleCoordinate c u))) := by
-    rw [quadraticCoordinate_scaleCoordinate (by linarith) hu.le]
+    rw [quadraticCoordinate_scaleCoordinate (by linarith only [hc]) hu.le]
     have hx : 0 < (u * (1 + u)) := by
       positivity
-    nlinarith
+    nlinarith only [hc, hx]
   have hcoord : u < scaleCoordinate c u := by
     by_contra hnot
     have hrev : scaleCoordinate c u ≤ u := le_of_not_gt hnot
     exact (not_le_of_gt hquad)
       (quadraticCoordinate_strictMonoOn.monotoneOn
-        (scaleCoordinate_nonneg (by linarith) hu.le) hu.le hrev)
+        (scaleCoordinate_nonneg (by linarith only [hc]) hu.le) hu.le hrev)
   exact spectralAtom_strictMonoOn hu.le
-    (scaleCoordinate_nonneg (by linarith) hu.le) hcoord
+    (scaleCoordinate_nonneg (by linarith only [hc]) hu.le) hcoord
 
 theorem Gamma_scale_gt {r : ℕ}
     {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
@@ -1354,7 +1343,7 @@ theorem Gamma_scale_gt {r : ℕ}
   apply Finset.sum_lt_sum_of_nonempty
     (Finset.univ_nonempty : (Finset.univ : Finset (Fin (r + 1))).Nonempty)
   intro i _
-  rw [lagrangeWeight_scale h (by linarith) i]
+  rw [lagrangeWeight_scale h (by linarith only [hc]) i]
   apply mul_lt_mul_of_pos_left _ (h.lagrangeWeight_pos i)
   exact spectralAtom_scale_lt hc
     (hlast.trans_le (h.strictAnti_ambient.antitone i.le_last))
@@ -1371,7 +1360,7 @@ theorem spectralAtom_scale_sub_lower_bound {c u : ℝ}
   have hx : 0 ≤ x := by
     dsimp [x]
     positivity
-  have hcpos : 0 < c := by linarith
+  have hcpos : 0 < c := by linarith only [hc]
   have hscale : 0 ≤ scaleCoordinate c u :=
     scaleCoordinate_nonneg hcpos.le hu
   have hsq : Q ^ 2 - q ^ 2 =
@@ -1384,8 +1373,6 @@ theorem spectralAtom_scale_sub_lower_bound {c u : ℝ}
     have hden₂ : 1 + 4 * x ≠ 0 := by positivity
     field_simp [hden₁, hden₂]
     ring
-  have hqu : 0 ≤ q := spectralAtom_nonneg hu
-  have hQ : 0 ≤ Q := spectralAtom_nonneg hscale
   have hqhalf : q < (1 / 2 : ℝ) := spectralAtom_lt_half hu
   have hQhalf : Q < (1 / 2 : ℝ) := spectralAtom_lt_half hscale
   have hqle : q ≤ Q := by
@@ -1393,7 +1380,7 @@ theorem spectralAtom_scale_sub_lower_bound {c u : ℝ}
         ((scaleCoordinate c u) * (1 + (scaleCoordinate c u))) := by
       rw [quadraticCoordinate_scaleCoordinate hcpos.le hu]
       dsimp [x] at hx ⊢
-      nlinarith
+      nlinarith only [hc, hx]
     by_contra hnot
     have hreverse : Q < q := lt_of_not_ge hnot
     have hreverse' : scaleCoordinate c u < u := by
@@ -1404,22 +1391,21 @@ theorem spectralAtom_scale_sub_lower_bound {c u : ℝ}
     exact (not_le_of_gt (quadraticCoordinate_strictMonoOn hscale hu hreverse'))
       hquad
   have hfactor : Q ^ 2 - q ^ 2 ≤ Q - q := by
-    nlinarith [mul_nonneg (sub_nonneg.mpr hqle)
-      (show 0 ≤ 1 - (Q + q) by linarith)]
+    nlinarith only [mul_nonneg (sub_nonneg.mpr hqle) (show 0 ≤ 1 - (Q + q) by linarith)]
   have hden₁ : 0 < (1 + 4 * c * x) * (1 + 4 * x) := by positivity
   have hden₂ : 0 < (1 + 8 * x) * (1 + 4 * x) := by positivity
   have hnum : 0 ≤ x * (c - 1) := mul_nonneg hx (sub_nonneg.mpr hc)
   have hdenle :
       (1 + 4 * c * x) * (1 + 4 * x) ≤
         (1 + 8 * x) * (1 + 4 * x) := by
-    nlinarith [mul_nonneg (sub_nonneg.mpr hc') hx]
+    nlinarith only [hx, mul_nonneg (sub_nonneg.mpr hc') hx]
   have hfractions :
       x * (c - 1) / ((1 + 8 * x) * (1 + 4 * x)) ≤
         x * (c - 1) / ((1 + 4 * c * x) * (1 + 4 * x)) := by
     exact (div_le_div_iff₀ hden₂ hden₁).mpr
       (mul_le_mul_of_nonneg_left hdenle hnum)
   dsimp [x, Q, q] at hfractions hfactor hsq ⊢
-  linarith
+  linarith only [hsq, hfactor, hfractions]
 
 /-- The positive weighted coefficient controlling the spectral gain under parameter scaling. -/
 def scalingGainCoefficient {r : ℕ}
@@ -1459,7 +1445,7 @@ theorem Gamma_scale_sub_lower_bound {r : ℕ}
   rw [Finset.mul_sum, ← Finset.sum_sub_distrib]
   apply Finset.sum_le_sum
   intro i _
-  rw [lagrangeWeight_scale h (by linarith) i]
+  rw [lagrangeWeight_scale h (by linarith only [hc]) i]
   have hbound := spectralAtom_scale_sub_lower_bound
     hc hc' (h.ambient_nonneg i)
   have hweighted := mul_le_mul_of_nonneg_left hbound
@@ -1516,15 +1502,11 @@ theorem append_scaled_spectralLoss_le {r : ℕ}
     hlast.trans_le (h.strictAnti_ambient.antitone i.le_last)
   have hx : 0 < ((a i) * (1 + (a i))) := by
     positivity
-  have he : 0 ≤ (ε * (1 + ε)) := by
-    positivity
   have hatom := spectralAtom_lt_half
     (scaleCoordinate_nonneg hc.le hai.le)
   have hweight := h.lagrangeWeight_pos i
   field_simp [hc.ne', hx.ne']
-  nlinarith [mul_nonneg he hweight.le,
-    mul_nonneg (mul_nonneg he hweight.le)
-      (sub_nonneg.mpr hatom.le)]
+  nlinarith only [hε, hatom]
 
 /-- The scaling slope chosen to exceed the upper bound on the appended spectral loss. -/
 def compensatedScalingSlope {r : ℕ}
@@ -1536,7 +1518,7 @@ theorem compensatedScalingSlope_pos {r : ℕ}
     (h : Interlacing a b) (hlast : 0 < a (Fin.last r)) :
     0 < compensatedScalingSlope a b := by
   unfold compensatedScalingSlope
-  exact div_pos (by linarith [appendSpectralLossUpper_pos h hlast])
+  exact div_pos (by linarith only [appendSpectralLossUpper_pos h hlast])
     (scalingGainCoefficient_pos h hlast)
 
 /-- The parameter scaling factor compensating for an appended stabilizer coordinate `ε`. -/
@@ -1562,27 +1544,25 @@ theorem Gamma_compensated_append_gt {r : ℕ}
   have he : 0 < e := by
     dsimp [e]
     positivity
-  have hD : 0 < D := scalingGainCoefficient_pos h hlast
-  have hM : 0 < M := appendSpectralLossUpper_pos h hlast
   have hslope : 0 < compensatedScalingSlope a b :=
     compensatedScalingSlope_pos h hlast
   have hc : 1 < c := by
     dsimp [c, compensatedScalingFactor]
-    nlinarith
+    nlinarith only [he, hslope]
   have hscale : Interlacing (scaleAmbient c a) (scaleStabilizer c b) :=
-    h.scale (by linarith)
+    h.scale (by linarith only [hc])
   have hlast' : 0 < scaleAmbient c a (Fin.last r) :=
-    scaleCoordinate_pos (by linarith) hlast
+    scaleCoordinate_pos (by linarith only [hc]) hlast
   have hgain := Gamma_scale_sub_lower_bound h hc.le hmoderate
   have hloss := append_scaled_spectralLoss_le h hlast
-    (show 0 < c by linarith) hε.le
+    (show 0 < c by linarith only [hc]) hε.le
   have hrelation : (c - 1) * D = e * (M + 1) := by
     dsimp [c, D, M, compensatedScalingFactor, compensatedScalingSlope]
     field_simp [(scalingGainCoefficient_pos h hlast).ne']
     ring
   rw [Gamma_append hscale hlast']
   dsimp [c] at hgain hloss hrelation ⊢
-  nlinarith
+  linarith only [he, hgain, hloss, hrelation]
 
 @[simp] theorem compensatedScalingFactor_zero {r : ℕ}
     (a : Fin (r + 1) → ℝ) (b : Fin r → ℝ) :
@@ -1600,8 +1580,6 @@ theorem scaleCoordinate_compensated_differentiableAt_zero {r : ℕ}
     {u : ℝ} (hu : 0 ≤ u) :
     DifferentiableAt ℝ
       (fun ε => scaleCoordinate (compensatedScalingFactor a b ε) u) 0 := by
-  have hquad : 0 ≤ (u * (1 + u)) := by
-    positivity
   have hinner : DifferentiableAt ℝ
       (fun ε => 1 + 4 * compensatedScalingFactor a b ε * (u * (1 + u)))
       0 := by
@@ -1674,7 +1652,7 @@ theorem eventually_sub_sphericalEntropy_lt_of_differentiable
   let M : ℝ := deriv f 0 + 1
   have hM : deriv f 0 < M := by
     dsimp [M]
-    linarith
+    linarith only []
   have hslope := hf.hasDerivAt.tendsto_slope_zero_right
   have hupper :
       ∀ᶠ ε : ℝ in 𝓝[>] 0, ε⁻¹ * (f ε - f 0) < M := by
@@ -1698,21 +1676,21 @@ theorem eventually_sub_sphericalEntropy_lt_of_differentiable
     mul_lt_mul_of_pos_left hlogε hε
   have hentropy :=
     MetricCodes.Spherical.neg_mul_logb_le_sphericalEntropy hε.le
-  linarith
+  linarith only [hold, hnew, hentropy]
 
 theorem le_scaleCoordinate_of_one_le {c u : ℝ}
     (hc : 1 ≤ c) (hu : 0 ≤ u) : u ≤ scaleCoordinate c u := by
   have hquad : (u * (1 + u)) ≤
       ((scaleCoordinate c u) * (1 + (scaleCoordinate c u))) := by
-    rw [quadraticCoordinate_scaleCoordinate (by linarith) hu]
+    rw [quadraticCoordinate_scaleCoordinate (by linarith only [hc]) hu]
     have hx : 0 ≤ (u * (1 + u)) := by
       positivity
-    nlinarith
+    nlinarith only [hc, hx]
   by_contra hnot
   have hreverse : scaleCoordinate c u < u := lt_of_not_ge hnot
   exact (not_le_of_gt
     (quadraticCoordinate_strictMonoOn
-      (scaleCoordinate_nonneg (by linarith) hu) hu hreverse)) hquad
+      (scaleCoordinate_nonneg (by linarith only [hc]) hu) hu hreverse)) hquad
 
 theorem eventually_Phi_compensated_append_lt {r : ℕ}
     {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
@@ -1733,19 +1711,20 @@ theorem eventually_Phi_compensated_append_lt {r : ℕ}
     positivity
   have hc : 1 ≤ c := by
     dsimp [c, compensatedScalingFactor]
-    nlinarith
+    nlinarith only [hslope, hquad]
   have hcoordinate : ε ≤ scaleCoordinate c ε :=
     le_scaleCoordinate_of_one_le hc hε.le
   have hentropy :
       MetricCodes.sphericalEntropy ε ≤
         MetricCodes.sphericalEntropy (scaleCoordinate c ε) :=
     MetricCodes.Spherical.sphericalEntropy_strictMono.monotoneOn
-      hε.le (scaleCoordinate_nonneg (by linarith) hε.le) hcoordinate
+      hε.le (scaleCoordinate_nonneg
+        (by linarith only [hc]) hε.le) hcoordinate
   rw [Phi_append]
   change compensatedPhiPath a b ε -
       MetricCodes.sphericalEntropy (scaleCoordinate c ε) < Phi a b
   rw [← compensatedPhiPath_zero h]
-  linarith
+  linarith only [hεbound, hentropy]
 
 theorem exists_nextLevel_strict_refinement {r : ℕ}
     {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
@@ -1778,7 +1757,7 @@ theorem exists_nextLevel_strict_refinement {r : ℕ}
       have hquad : 0 < (ε * (1 + ε)) := by
         positivity
       dsimp [c, compensatedScalingFactor]
-      nlinarith
+      nlinarith only [hslope, hquad]
     let A := appendAmbient (scaleAmbient c a)
     let B := appendStabilizer (scaleStabilizer c b) (scaleCoordinate c ε)
     refine ⟨A, B, ?_, ?_, ?_⟩
@@ -1851,20 +1830,20 @@ theorem classical_le_levelRate_zero {s : ℝ}
   have ha : Interlacing a b := by
     constructor
     · dsimp [a]
-      linarith
+      linarith only [ha₀]
     · intro i
       exact Fin.elim0 i
   have hboundary : spectralAtom (MetricCodes.classicalThreshold s) = s / 2 := by
     have h := MetricCodes.Spherical.classicalThreshold_spectral hs hs'
     rw [MetricCodes.Spherical.Gamma_zero ha₀] at h
     change 2 * spectralAtom (MetricCodes.classicalThreshold s) = s at h
-    linarith
+    linarith only [h]
   have hgap : s < 2 * Gamma a b := by
     rw [Gamma_zero]
     have hmono := spectralAtom_strictMonoOn ha₀.le
-      (show 0 ≤ a 0 by dsimp [a]; linarith)
-      (show MetricCodes.classicalThreshold s < a 0 by dsimp [a]; linarith)
-    linarith
+      (show 0 ≤ a 0 by dsimp [a]; linarith only [ha₀])
+      (show MetricCodes.classicalThreshold s < a 0 by dsimp [a]; linarith only [])
+    linarith only [hboundary, hmono]
   have hnonempty : (levelRateSet 0 s).Nonempty :=
     ⟨Phi a b, a, b, ha, hgap, rfl⟩
   unfold levelRate
@@ -1878,7 +1857,7 @@ theorem classical_le_levelRate_zero {s : ℝ}
     have hle : A 0 ≤ MetricCodes.classicalThreshold s := le_of_not_gt hnot
     have hmono := spectralAtom_strictMonoOn.monotoneOn
       hAnonneg ha₀.le hle
-    linarith
+    linarith only [hboundary, hgapA, hmono]
   exact (MetricCodes.Spherical.sphericalEntropy_strictMono
     ha₀.le hAnonneg hstrict).le
 
@@ -1896,7 +1875,7 @@ theorem localizedEnvelope_lt_of_minimizer
         κ' t + MetricCodes.Spherical.SidelnikovLocalization.sliceCost s t :=
       MetricCodes.Spherical.SidelnikovLocalization.localizedEnvelope_le hs ht hκ'
     _ < κ t + MetricCodes.Spherical.SidelnikovLocalization.sliceCost s t := by
-      linarith
+      linarith only [hstrict]
     _ = MetricCodes.Spherical.SidelnikovLocalization.localizedEnvelope κ s :=
       hmin.symm
 
@@ -2340,7 +2319,7 @@ theorem sum_sphericalEntropy_update {n : ℕ}
   rw [Finset.sdiff_singleton_eq_erase]
   have hsplit := Finset.add_sum_erase (Finset.univ : Finset (Fin n))
     (fun j => MetricCodes.sphericalEntropy (a j)) (Finset.mem_univ i)
-  linarith
+  linarith only [hsplit]
 
 theorem Phi_opening_eq {r : ℕ}
     (hr : 0 < r)
@@ -2374,7 +2353,7 @@ theorem eventually_Phi_opening_lt {r : ℕ}
     simpa [i] using hi
   have hlog : 0 < Real.logb 2 ((1 + a 0) / a 0) := by
     apply Real.logb_pos (by norm_num : (1 : ℝ) < 2)
-    exact (lt_div_iff₀ ha).mpr (by linarith)
+    exact (lt_div_iff₀ ha).mpr (by linarith only [])
   have hnegative : -(η * Real.logb 2 ((1 + a 0) / a 0)) < 0 := by
     exact neg_neg_of_pos (mul_pos hη hlog)
   have hlimit := tendsto_opening_entropy_slope (η := η) ha
@@ -2382,7 +2361,7 @@ theorem eventually_Phi_opening_lt {r : ℕ}
   filter_upwards [hevent, self_mem_nhdsWithin] with t ht (htpos : 0 < t)
   have hnegative' := (div_lt_iff₀ htpos).mp ht
   have hphi := Phi_opening_eq (b := b) hr hzero η t
-  linarith
+  linarith only [hnegative', hphi]
 
 /-- The positive opening speed bounded using the derivative of the regular spectral sum. -/
 def openingContractionSpeed {r : ℕ}
@@ -2413,7 +2392,7 @@ theorem opening_spectralSlope_pos {r : ℕ}
   have hbound : -|D| ≤ D := neg_abs_le D
   have hηbound := mul_le_mul_of_nonneg_left hbound hη.le
   dsimp [η, D, w] at hidentity hηbound ⊢
-  nlinarith [abs_nonneg (deriv (openingRegularGamma a b 1) 0)]
+  linarith only [hw, hη, hidentity, hηbound]
 
 theorem eventually_Gamma_opening_gt {r : ℕ}
     {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
@@ -2431,7 +2410,7 @@ theorem eventually_Gamma_opening_gt {r : ℕ}
     with t ht (htpos : 0 < t)
   have hdiff := (lt_div_iff₀ htpos).mp ht
   dsimp [η] at hdiff ⊢
-  linarith
+  linarith only [hdiff]
 
 theorem eventually_opening_interlacing {r : ℕ}
     {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
@@ -2501,7 +2480,7 @@ theorem exists_nextLevel_feasible_lt {r : ℕ} {s : ℝ}
   by_cases hlast : 0 < a (Fin.last r)
   · obtain ⟨A, B, hAB, hGamma, hPhi⟩ :=
       exists_nextLevel_strict_refinement h hlast
-    exact ⟨A, B, hAB, by linarith, hPhi⟩
+    exact ⟨A, B, hAB, by linarith only [hgap, hGamma], hPhi⟩
   · have hzero : a (Fin.last r) = 0 :=
       le_antisymm (le_of_not_gt hlast) (h.ambient_nonneg _)
     cases r with
@@ -2510,13 +2489,13 @@ theorem exists_nextLevel_feasible_lt {r : ℕ} {s : ℝ}
         have ha : a 0 = 0 := by simpa only [Fin.isValue, Fin.last_zero] using hzero
         simp only [spectralAtom, Fin.isValue, ha, add_zero, mul_one,
           Real.sqrt_zero, mul_zero, div_one] at hgap
-        linarith
+        linarith only [hs, hgap]
     | succ r =>
         obtain ⟨A, hlastA, hA, hGammaA, hPhiA⟩ :=
           exists_sameLevel_opening_strict_refinement (Nat.zero_lt_succ r) h hzero
         obtain ⟨C, D, hCD, hGammaC, hPhiC⟩ :=
           exists_nextLevel_strict_refinement hA hlastA
-        exact ⟨C, D, hCD, by linarith, lt_trans hPhiC hPhiA⟩
+        exact ⟨C, D, hCD, by linarith only [hgap, hGammaA, hGammaC], lt_trans hPhiC hPhiA⟩
 
 theorem levelRate_nonneg_of_nonempty {r : ℕ} {s : ℝ}
     (h : (levelRateSet r s).Nonempty) :
@@ -2546,20 +2525,20 @@ theorem levelRateSet_zero_nonempty_of_interior {s : ℝ}
   have ha : Interlacing a b := by
     constructor
     · dsimp [a]
-      linarith
+      linarith only [hthreshold]
     · intro i
       exact Fin.elim0 i
   have hboundary : spectralAtom (MetricCodes.classicalThreshold s) = s / 2 := by
     have h := MetricCodes.Spherical.classicalThreshold_spectral hs hs'
     rw [MetricCodes.Spherical.Gamma_zero hthreshold] at h
     change 2 * spectralAtom (MetricCodes.classicalThreshold s) = s at h
-    linarith
+    linarith only [h]
   refine ⟨Phi a b, a, b, ha, ?_, rfl⟩
   rw [Gamma_zero]
   have hmono := spectralAtom_strictMonoOn hthreshold.le
-    (show 0 ≤ a 0 by dsimp [a]; linarith)
-    (show MetricCodes.classicalThreshold s < a 0 by dsimp [a]; linarith)
-  linarith
+    (show 0 ≤ a 0 by dsimp [a]; linarith only [hthreshold])
+    (show MetricCodes.classicalThreshold s < a 0 by dsimp [a]; linarith only [])
+  linarith only [hboundary, hmono]
 
 theorem levelRateSet_nonempty_of_interior (r : ℕ) {s : ℝ}
     (hs : 0 < s) (hs' : s < 1) :
@@ -2662,7 +2641,7 @@ theorem exists_positive_localized_minimizer_of_lowerSemicontinuous
   have htpos : 0 < t := by
     rcases lt_or_eq_of_le ht.1 with htpos | rfl
     · exact htpos
-    · linarith
+    · linarith only [huzero, htu]
   exact ⟨t, ⟨htpos, ht.2⟩, heq⟩
 
 theorem sliceCost_sub_zero {s t : ℝ}
@@ -2670,8 +2649,8 @@ theorem sliceCost_sub_zero {s t : ℝ}
     MetricCodes.Spherical.SidelnikovLocalization.sliceCost s t -
         MetricCodes.Spherical.SidelnikovLocalization.sliceCost s 0 =
       (1 / 2 : ℝ) * Real.logb 2 (1 - t) := by
-  have hs' : 1 - s ≠ 0 := by linarith
-  have ht' : 1 - t ≠ 0 := by linarith
+  have hs' : 1 - s ≠ 0 := by linarith only [hs]
+  have ht' : 1 - t ≠ 0 := by linarith only [ht]
   unfold MetricCodes.Spherical.SidelnikovLocalization.sliceCost
   norm_num only [sub_zero]
   rw [Real.logb_div ht' hs',
@@ -2686,8 +2665,8 @@ theorem sliceCost_sub_zero_le_neg_linear {s t : ℝ}
       -t / (2 * Real.log 2) := by
   rw [sliceCost_sub_zero hs ht]
   have hlog : Real.log (1 - t) ≤ -t := by
-    simpa only [sub_sub_cancel_left] using Real.log_le_sub_one_of_pos (by linarith : 0 < 1 - t)
-  have hbase : 0 < Real.log 2 := Real.log_pos (by norm_num)
+    simpa only [sub_sub_cancel_left] using
+      Real.log_le_sub_one_of_pos (by linarith only [ht] : 0 < 1 - t)
   unfold Real.logb
   calc
     (1 / 2 : ℝ) * (Real.log (1 - t) / Real.log 2) =
@@ -2734,7 +2713,7 @@ theorem eventually_squared_spectralCertificate :
       (Ioi_mem_nhds (by norm_num : (1 / 2 : ℝ) < 1))
   filter_upwards [hratio, self_mem_nhdsWithin] with t hratio (ht : 0 < t)
   have hmul := (lt_div_iff₀ ht).mp hratio
-  linarith
+  linarith only [hmul]
 
 theorem eventually_levelRate_zero_le_squared_entropy :
     ∀ᶠ t : ℝ in 𝓝[>] 0,
@@ -2777,9 +2756,9 @@ theorem exists_sliceCost_endpoint_improvement_of_smallEntropy
     simpa only [neg_div] using hcost
   have hlinear : t / (4 * Real.log 2) < t / (2 * Real.log 2) := by
     exact (div_lt_div_iff_of_pos_left htpos (by positivity) (by positivity)).2
-      (by nlinarith)
+      (by linarith only [hbase])
   rw [hzero]
-  linarith
+  linarith only [hκ, hent', hcost', hlinear]
 
 theorem exists_positive_localized_minimizer_of_lsc_and_smallEntropy
     {κ : ℝ → ℝ} {s : ℝ} (hs : 0 < s) (hs' : s < 1)
@@ -2872,10 +2851,10 @@ theorem logb_one_add_le_sphericalEntropy {u : ℝ} (hu : 0 ≤ u) :
   · rw [MetricCodes.sphericalEntropy_eq_log_add hu]
     have hratio : 1 ≤ (1 + u) / u := by
       apply (le_div_iff₀ hu).2
-      linarith
+      linarith only []
     have hlog : 0 ≤ Real.logb 2 ((1 + u) / u) :=
       Real.logb_nonneg (by norm_num : (1 : ℝ) < 2) hratio
-    nlinarith [mul_nonneg hu.le hlog]
+    linarith only [mul_nonneg hu.le hlog]
 
 theorem entropy_gap_nonneg {r : ℕ}
     {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
@@ -2894,7 +2873,7 @@ theorem terminal_entropy_le_Phi {r : ℕ}
       (MetricCodes.sphericalEntropy (a i.castSucc) -
         MetricCodes.sphericalEntropy (b i)) :=
     Finset.sum_nonneg fun i _ => entropy_gap_nonneg h i
-  linarith
+  linarith only [hsum]
 
 theorem entropy_gap_le_Phi {r : ℕ}
     {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
@@ -2916,7 +2895,7 @@ theorem entropy_gap_le_Phi {r : ℕ}
             MetricCodes.sphericalEntropy (b j)) := by
     exact Finset.single_le_sum
       (fun j _ => entropy_gap_nonneg h j) (Finset.mem_univ i)
-  linarith
+  linarith only [hterminal, hsum]
 
 theorem degree_le_rpow_sub_one_of_sphericalEntropy_le
     {u C : ℝ} (hu : 0 ≤ u)
@@ -2926,8 +2905,8 @@ theorem degree_le_rpow_sub_one_of_sphericalEntropy_le
     (logb_one_add_le_sphericalEntropy hu).trans hC
   have hpow : 1 + u ≤ (2 : ℝ) ^ C :=
     (Real.logb_le_iff_le_rpow (by norm_num : (1 : ℝ) < 2)
-      (by linarith : 0 < 1 + u)).mp hlog
-  linarith
+      (by linarith only [hu] : 0 < 1 + u)).mp hlog
+  linarith only [hpow]
 
 theorem terminal_degree_le_rpow_sub_one_of_Phi_le {r : ℕ}
     {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
@@ -2949,7 +2928,7 @@ theorem shifted_degree_ratio_le_rpow_of_entropy_gap_le
   have hlog :
       Real.logb 2 (1 + a) - Real.logb 2 (1 + b) ≤
         C + 1 / Real.log 2 := by
-    linarith
+    linarith only [hC, hlower, hupper]
   have hratio :
       Real.logb 2 ((1 + a) / (1 + b)) ≤
         C + 1 / Real.log 2 := by
@@ -2971,10 +2950,10 @@ theorem degree_ratio_lower_bound_of_entropy_gap_le
     shifted_degree_ratio_le_rpow_of_entropy_gap_le hb₀ hba hC
   have hshift : 1 + a ≤ D * (1 + b) :=
     (div_le_iff₀ (by positivity : 0 < 1 + b)).mp hratio
-  have hdouble : 1 + b ≤ 2 * b := by linarith
+  have hdouble : 1 + b ≤ 2 * b := by linarith only [hb]
   have hbound : a ≤ (2 * D) * b := by
     have hmul := mul_le_mul_of_nonneg_left hdouble hD.le
-    nlinarith
+    linarith only [hshift, hmul]
   apply (le_div_iff₀ ha).2
   calc
     (2 * D)⁻¹ * a = a / (2 * D) := by ring
@@ -2992,7 +2971,7 @@ theorem tendsto_sphericalEntropy_sub_logb_one_add :
   rw [MetricCodes.sphericalEntropy_eq_log_add ha]
   have hratio : (1 + a) / a = 1 + 1 / a := by
     field_simp
-    ring
+    ac_rfl
   rw [hratio]
   unfold Real.logb
   ring
@@ -3050,7 +3029,7 @@ theorem sqrt_degree_mul_one_add_eq
     Real.sqrt (a * (1 + a)) = a * Real.sqrt (1 + a⁻¹) := by
   have hfactor : a * (1 + a) = a ^ 2 * (1 + a⁻¹) := by
     field_simp
-    ring
+    ac_rfl
   rw [hfactor, Real.sqrt_mul (sq_nonneg a), Real.sqrt_sq ha.le]
 
 theorem boundaryQuadratic_eq_sq_mul_normalized
@@ -3061,7 +3040,7 @@ theorem boundaryQuadratic_eq_sq_mul_normalized
     normalizedBoundaryQuadratic
   rw [sqrt_degree_mul_one_add_eq ha]
   field_simp
-  ring
+  ac_rfl
 
 theorem boundaryDegree_div_eq_normalized
     {s a : ℝ} (ha : 0 < a) :
@@ -3151,7 +3130,7 @@ def compactifiedHierarchyCoordinate (u : ℝ) : ℝ := (1 + u)⁻¹
 theorem compactifiedHierarchyCoordinate_mem_Ioc {u : ℝ} (hu : 0 ≤ u) :
     compactifiedHierarchyCoordinate u ∈ Set.Ioc (0 : ℝ) 1 := by
   constructor
-  · exact inv_pos.mpr (by linarith)
+  · exact inv_pos.mpr (by linarith only [hu])
   · exact (inv_le_one₀ (by linarith)).2 (by linarith)
 
 theorem compactifiedHierarchyCoordinate_mem_Icc {u : ℝ} (hu : 0 ≤ u) :
@@ -3189,18 +3168,18 @@ theorem tendsto_atTop_of_compactifiedHierarchyCoordinate_zero
   let T : ℝ := max M 0
   have hT : 0 < 1 + T := by
     dsimp [T]
-    linarith [le_max_right M 0]
+    linarith only [le_max_right M 0]
   have hevent :
       ∀ᶠ k in atTop,
         compactifiedHierarchyCoordinate (f k) < (1 + T)⁻¹ :=
     (tendsto_order.1 h).2 _ (inv_pos.mpr hT)
   filter_upwards [hevent] with k hk
-  have hfk : 0 < 1 + f k := by linarith [hf k]
+  have hfk : 0 < 1 + f k := by linarith only [hf k]
   have hlt : T < f k := by
     have hinv : (1 + f k)⁻¹ < (1 + T)⁻¹ := hk
     have hraw : 1 + T < 1 + f k :=
       (inv_lt_inv₀ hfk hT).mp hinv
-    linarith
+    linarith only [hraw]
   exact (le_max_left M 0).trans hlt.le
 
 theorem compactifiedHierarchyCoordinate_limit_pos_of_bddAbove
@@ -3211,7 +3190,7 @@ theorem compactifiedHierarchyCoordinate_limit_pos_of_bddAbove
       atTop (nhds x)) :
     0 < x := by
   have hC₀ : 0 ≤ C := (hf 0).trans (hC 0)
-  have hCpos : 0 < 1 + C := by linarith
+  have hCpos : 0 < 1 + C := by linarith only [hC₀]
   have hlower : (1 + C)⁻¹ ≤ x := by
     apply ge_of_tendsto h
     exact Filter.Eventually.of_forall fun k =>
@@ -3221,7 +3200,7 @@ theorem compactifiedHierarchyCoordinate_limit_pos_of_bddAbove
 theorem compactifiedHierarchyCoordinate_antitone
     {u v : ℝ} (hu : 0 ≤ u) (hv : 0 ≤ v) (h : u ≤ v) :
     compactifiedHierarchyCoordinate v ≤ compactifiedHierarchyCoordinate u := by
-  exact (inv_le_inv₀ (by linarith : 0 < 1 + v)
+  exact (inv_le_inv₀ (by linarith only [hv] : 0 < 1 + v)
     (by linarith : 0 < 1 + u)).mpr (by linarith)
 
 theorem compactifiedHierarchyCoordinate_recovered_nonneg
@@ -3229,13 +3208,13 @@ theorem compactifiedHierarchyCoordinate_recovered_nonneg
     0 ≤ x⁻¹ - 1 := by
   have hinv : (1 : ℝ) ≤ x⁻¹ := by
     simpa only [inv_one] using (inv_le_inv₀ (by norm_num : (0 : ℝ) < 1) hx).mpr hx'
-  linarith
+  linarith only [hinv]
 
 theorem compactifiedHierarchyCoordinate_recovered_antitone
     {x y : ℝ} (hx : 0 < x) (hy : 0 < y) (hxy : x ≤ y) :
     y⁻¹ - 1 ≤ x⁻¹ - 1 := by
   have hinv := (inv_le_inv₀ hy hx).mpr hxy
-  linarith
+  linarith only [hinv]
 
 theorem exists_levelRate_minimizing_sequence {r : ℕ} {s : ℝ}
     (hne : (levelRateSet r s).Nonempty) :
@@ -3337,7 +3316,7 @@ theorem compactifiedHierarchyTuple_limit_weak_interlacing {r : ℕ}
           ((h (φ k)).stabilizer_pos i).le
           ((h (φ k)).ambient_nonneg i.castSucc)
           ((h (φ k)).2 i).1.le)
-    linarith
+    linarith only [hle]
   · have hlim := (hB i).sub (hA i.succ)
     have hle : B i - A i.succ ≤ 0 := by
       apply le_of_tendsto hlim
@@ -3346,7 +3325,7 @@ theorem compactifiedHierarchyTuple_limit_weak_interlacing {r : ℕ}
           ((h (φ k)).ambient_nonneg i.succ)
           ((h (φ k)).stabilizer_pos i).le
           ((h (φ k)).2 i).2.le)
-    linarith
+    linarith only [hle]
 
 theorem compactifiedHierarchyTuple_recovered_weak_interlacing {r : ℕ}
     {A : Fin (r + 1) → ℝ} {B : Fin r → ℝ}
@@ -3607,8 +3586,6 @@ theorem tendsto_scaleCoordinate_sq_div_atTop {c : ℝ} (hc : 0 < c) :
       simp [hroot]
   apply hlimit.congr'
   filter_upwards [eventually_gt_atTop (0 : ℝ)] with u hu
-  have hquadratic : 0 ≤ (u * (1 + u)) := by
-    positivity
   have hrad : 0 ≤ 1 + 4 * c ^ 2 * (u * (1 + u)) := by positivity
   have hsmall :
       0 ≤ u⁻¹ ^ 2 + 4 * c ^ 2 * (u⁻¹ + 1) := by positivity
@@ -3628,7 +3605,7 @@ theorem tendsto_scaleCoordinate_sq_atTop {c : ℝ} (hc : 0 < c) :
   have hratio := tendsto_scaleCoordinate_sq_div_atTop hc
   have hevent :
       ∀ᶠ u : ℝ in atTop, c / 2 < scaleCoordinate (c ^ 2) u / u :=
-    hratio.eventually (Ioi_mem_nhds (by linarith))
+    hratio.eventually (Ioi_mem_nhds (by linarith only [hc]))
   have hle :
       (fun u : ℝ => (c / 2) * u) ≤ᶠ[atTop]
         (fun u : ℝ => scaleCoordinate (c ^ 2) u) := by
@@ -3636,7 +3613,7 @@ theorem tendsto_scaleCoordinate_sq_atTop {c : ℝ} (hc : 0 < c) :
     have hmul := (lt_div_iff₀ hupos).mp hu
     exact hmul.le
   exact tendsto_atTop_mono' atTop hle
-    (tendsto_id.const_mul_atTop (by linarith))
+    (tendsto_id.const_mul_atTop (by linarith only [hc]))
 
 theorem scaleCoordinate_sq_lt_self {c u : ℝ}
     (hc : 0 < c) (hc' : c < 1) (hu : 0 < u) :
@@ -3649,7 +3626,7 @@ theorem scaleCoordinate_sq_lt_self {c u : ℝ}
   have hle : u ≤ scaleCoordinate (c ^ 2) u := le_of_not_gt hnot
   have hmon := quadraticCoordinate_strictMonoOn.monotoneOn hu.le hscale hle
   rw [quadraticCoordinate_scaleCoordinate (sq_nonneg c) hu.le] at hmon
-  nlinarith [mul_pos (sub_pos.mpr hc') (by linarith : 0 < 1 + c)]
+  nlinarith only [hmon, hquad, mul_pos (sub_pos.mpr hc') (by linarith : 0 < 1 + c)]
 
 theorem eventually_interlacing_prepend_scale {r : ℕ}
     {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
@@ -3704,7 +3681,7 @@ theorem tendsto_spectralAtom_atTop_half :
   unfold spectralAtom
   have hfactor : u * (1 + u) = u ^ 2 * (1 + u⁻¹) := by
     field_simp [hu.ne']
-    ring
+    ac_rfl
   rw [hfactor, Real.sqrt_mul (sq_nonneg u), Real.sqrt_sq_eq_abs,
     abs_of_pos hu]
   field_simp [hu.ne']
@@ -3775,7 +3752,7 @@ theorem Gamma_prepend_eq_spectralAtom_add_retained
   change lagrangeWeight (Fin.cons u a) (prependStabilizer v b) 0 +
     ∑ i : Fin (r + 1), lagrangeWeight (Fin.cons u a)
       (prependStabilizer v b) i.succ = 1 at hweights
-  nlinarith [congrArg (fun z : ℝ => z * spectralAtom u) hweights]
+  linarith only [congrArg (fun z : ℝ => z * spectralAtom u) hweights]
 
 theorem tendsto_lagrangeWeight_prepend_scale_succ
     {r : ℕ}
@@ -3802,7 +3779,7 @@ theorem tendsto_lagrangeWeight_prepend_scale_succ
   have hquad : ((a i) * (1 + (a i))) < (u * (1 + u)) :=
     quadraticCoordinate_strictMonoOn (h.ambient_nonneg i) hu0.le hau
   have hne : ((a i) * (1 + (a i))) - (u * (1 + u)) ≠ 0 := by
-    linarith
+    linarith only [hquad]
   rw [lagrangeWeight_prepend_succ h u
     (scaleCoordinate (c ^ 2) u) i hne,
     quadraticCoordinate_scaleCoordinate (sq_nonneg c) hu0.le]
@@ -3921,7 +3898,7 @@ theorem exists_nextLevel_compactified_certificate_lt_of_spectral_limit
       Interlacing A B ∧ s < 2 * Gamma A B ∧ Phi A B < R := by
   have htarget :
       s < 2 * (((1 - c ^ 2) / 2) + c ^ 2 * Gamma a b) := by
-    nlinarith
+    linarith only [hscaled]
   have hspectral :
       ∀ᶠ u : ℝ in atTop,
         s < 2 * Gamma (prependAmbient u a)
@@ -3954,7 +3931,7 @@ theorem hierarchyVariationalRate_le_compactified_certificate_of_spectral_limit
   obtain ⟨A, B, hAB, hgap, hPhi⟩ :=
     exists_nextLevel_compactified_certificate_lt_of_spectral_limit
       (R := Phi a b - Real.logb 2 c + ε)
-      hc hc' a b h hscaled (by linarith) hGamma
+      hc hc' a b h hscaled (by linarith only [hε]) hGamma
   exact (hierarchyVariationalRate_le_of_feasible hAB hgap).trans hPhi.le
 
 theorem exists_nextLevel_compactified_certificate_lt
@@ -3981,7 +3958,7 @@ theorem levelRate_succ_le_compactified_certificate
   obtain ⟨A, B, hAB, hgap, hPhi⟩ :=
     exists_nextLevel_compactified_certificate_lt
       (R := Phi a b - Real.logb 2 c + ε)
-      hc hc' a b h hscaled (by linarith)
+      hc hc' a b h hscaled (by linarith only [hε])
   exact (levelRate_le hAB hgap).trans hPhi.le
 
 theorem levelRate_antitone_level
@@ -4005,7 +3982,7 @@ theorem levelRate_le_compactified_datum
   rcases hlevels with ⟨rfl, hjk⟩ | ⟨hc', hjk⟩
   · have hgap : s < 2 * Gamma a b := by
       norm_num at hscaled ⊢
-      linarith
+      linarith only [hscaled]
     simpa only [Real.logb_one, sub_zero, ge_iff_le] using
       (levelRate_antitone_level hs hs' hjk).trans (levelRate_le h hgap)
   · exact (levelRate_antitone_level hs hs' hjk).trans
@@ -4065,14 +4042,14 @@ theorem levelRate_le_of_non_strict_spectral
     let : (𝓝[>] (1 : ℝ)).NeBot := nhdsWithin_Ioi_neBot le_rfl
     obtain ⟨t, ht, hPhi⟩ :=
       (hside.and hcost).exists
-    have htpos : 0 < t := by linarith
+    have htpos : 0 < t := by linarith only [ht]
     have hinter :
         Interlacing (scaleAmbient t a) (scaleStabilizer t b) :=
       h.scale htpos
     have hgain := Gamma_scale_gt h hlast ht
     have hgap :
         s < 2 * Gamma (scaleAmbient t a) (scaleStabilizer t b) := by
-      linarith
+      linarith only [hboundary, hgain]
     exact (levelRate_le hinter hgap).trans hPhi.le
   · have hzero : a (Fin.last j) = 0 :=
       le_antisymm (le_of_not_gt hlast) h.1
@@ -4081,12 +4058,12 @@ theorem levelRate_le_of_non_strict_spectral
         have ha0 : a 0 = 0 := by simpa only [Fin.isValue, Fin.last_zero] using hzero
         rw [Gamma_zero, ha0] at hboundary
         norm_num [spectralAtom] at hboundary
-        linarith
+        linarith only [hs, hboundary]
     | succ j =>
         obtain ⟨A, _, hA, hGamma, hPhi⟩ :=
           exists_sameLevel_opening_strict_refinement
             (Nat.succ_pos j) h hzero
-        exact (levelRate_le hA (by linarith)).trans hPhi.le
+        exact (levelRate_le hA (by linarith only [hboundary, hGamma])).trans hPhi.le
 
 theorem levelRate_le_compactified_boundary_datum
     {r j : ℕ} {s c : ℝ}
@@ -4100,12 +4077,12 @@ theorem levelRate_le_compactified_boundary_datum
   rcases hlevels with ⟨rfl, hjr⟩ | ⟨hc_lt, hjr⟩
   · have hboundary : s ≤ 2 * Gamma a b := by
       norm_num at hscaled ⊢
-      linarith
+      linarith only [hscaled]
     simpa only [Real.logb_one, sub_zero, ge_iff_le] using
       (levelRate_antitone_level hs hs' hjr).trans (levelRate_le_of_non_strict_spectral hs a b h
         hboundary)
   · have hpositive : 0 < 1 - 2 * Gamma a b := by
-      linarith [h.Gamma_lt_half]
+      linarith only [h.Gamma_lt_half]
     have hlog :
         Tendsto (fun d : ℝ => Phi a b - Real.logb 2 d)
           (𝓝[<] c) (𝓝 (Phi a b - Real.logb 2 c)) := by
@@ -4127,10 +4104,9 @@ theorem levelRate_le_compactified_boundary_datum
       (hside.and (hpos.and hcost)).exists
     have hd_one : d < 1 := hd_lt.trans hc_lt
     have hsquare : d ^ 2 < c ^ 2 := by
-      nlinarith [mul_pos hd_pos (sub_pos.mpr hd_lt),
-        mul_pos hc (sub_pos.mpr hd_lt)]
+      nlinarith only [mul_pos hd_pos (sub_pos.mpr hd_lt)]
     have hstrict : s < 1 - d ^ 2 * (1 - 2 * Gamma a b) := by
-      nlinarith [mul_pos (sub_pos.mpr hsquare) hpositive]
+      nlinarith only [hscaled, mul_pos (sub_pos.mpr hsquare) hpositive]
     exact
       (levelRate_le_compactified_datum hs hs' hd_pos a b h hstrict
         (Or.inr ⟨hd_one, hjr⟩)).trans hd_cost.le
@@ -4161,8 +4137,8 @@ theorem stieltjesPhase_nonneg {r : ℕ}
   · apply Finset.sum_nonneg
     intro i _
     apply Real.log_nonneg
-    apply (le_div_iff₀ (by linarith [h.stabilizer_quadratic_pos i])).2
-    nlinarith [h.quadratic_phase_interval i]
+    apply (le_div_iff₀ (by linarith only [ht, h.stabilizer_quadratic_pos i])).2
+    linarith only [h.quadratic_phase_interval i]
 
 theorem stieltjesPhaseProduct_pos {r : ℕ}
     {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
@@ -4172,9 +4148,9 @@ theorem stieltjesPhaseProduct_pos {r : ℕ}
   apply div_pos
   · apply mul_pos ht
     exact Finset.prod_pos fun i _ => by
-      linarith [h.stabilizer_quadratic_pos i]
+      linarith only [ht, h.stabilizer_quadratic_pos i]
   · exact Finset.prod_pos fun i _ => by
-      linarith [h.ambient_quadratic_nonneg i]
+      linarith only [ht, h.ambient_quadratic_nonneg i]
 
 theorem stieltjesPhaseProduct_le_one {r : ℕ}
     {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
@@ -4290,7 +4266,7 @@ private theorem cancel_common_quadratic_factor_metriccodes2_79e88af7
     (x * A) / (B * (x * C)) =
         (x * x⁻¹) * (A * (C⁻¹ * B⁻¹)) := by
       simp only [div_eq_mul_inv, mul_inv_rev]
-      ring
+      ac_rfl
     _ = A / (B * C) := by
       rw [mul_inv_cancel₀ hx]
       simp only [one_mul, div_eq_mul_inv, mul_inv_rev]
@@ -4309,7 +4285,7 @@ theorem hierarchyStieltjesRatio_tail_of_head_collision {r : ℕ}
   rw [hierarchyStieltjesRatio_prepend]
   have hfactor : t + ((a 0) * (1 + (a 0))) ≠ 0 := by
     have ha := h.ambient_nonneg 0
-    nlinarith [mul_nonneg ha (by linarith : 0 ≤ 1 + a 0)]
+    linarith only [ht, mul_nonneg ha (by linarith : 0 ≤ 1 + a 0)]
   rw [← hcollision]
   simp only [ne_eq, hfactor, not_false_eq_true, div_self, one_mul]
 
@@ -4323,8 +4299,7 @@ theorem hierarchyStieltjesRatio_drop_second_of_collision {r : ℕ}
         (Fin.cons (a 0) (Fin.tail (Fin.tail a))) (Fin.tail b) t := by
   have hfactor : t + ((a (1 : Fin (r + 2))) * (1 + (a (1 : Fin (r + 2))))) ≠ 0 := by
     have ha := h.ambient_nonneg (1 : Fin (r + 2))
-    nlinarith [mul_nonneg ha
-      (by linarith : 0 ≤ 1 + a (1 : Fin (r + 2)))]
+    nlinarith only [ht, mul_nonneg ha (by linarith : 0 ≤ 1 + a (1 : Fin (r + 2)))]
   simp only [hierarchyStieltjesRatio, Fin.prod_univ_succ, Fin.cons_zero,
     Fin.cons_succ, Fin.tail]
   rw [hcollision]
@@ -4510,10 +4485,10 @@ theorem paired_stabilizer_tendsto_atTop_of_ambient
     have hmul : 1 + a (φ n) i.castSucc ≤
         D * (1 + b (φ n) i) :=
       (div_le_iff₀
-        (by linarith [(h (φ n)).stabilizer_pos i])).mp hratio
+        (by linarith only [(h (φ n)).stabilizer_pos i])).mp hratio
     apply (sub_le_iff_le_add).mpr
     apply (div_le_iff₀ hD).mpr
-    nlinarith
+    linarith only [hmul]
   have hleft : Tendsto
       (fun n => a (φ n) i.castSucc / D - 1) atTop atTop := by
     convert tendsto_atTop_add_const_right atTop (-1)
@@ -4836,7 +4811,6 @@ theorem exists_compactified_residual_of_closure_sequence
   have hbound : ∀ n, Phi (a n) (b n) ≤ R + 1 := by
     intro n
     refine (hPhi n).trans ?_
-    have hn : 0 ≤ (n : ℝ) := Nat.cast_nonneg n
     have hden : 0 < (n : ℝ) + 1 := by linarith
     have hfrac : 1 / ((n : ℝ) + 1) ≤ (1 : ℝ) :=
       (div_le_one hden).2 (by linarith)
@@ -5311,11 +5285,10 @@ theorem exists_forward_residual_entropy_of_closure_sequence
   have hbound : ∀ n, Phi (a n) (b n) ≤ R + 1 := by
     intro n
     refine (hPhi n).trans ?_
-    have hn : 0 ≤ (n : ℝ) := Nat.cast_nonneg n
-    have hden : 0 < (n : ℝ) + 1 := by linarith
+    have hden : 0 < (n : ℝ) + 1 := by linarith only []
     have hfrac : 1 / ((n : ℝ) + 1) ≤ (1 : ℝ) :=
-      (div_le_one hden).2 (by linarith)
-    linarith
+      (div_le_one hden).2 (by linarith only [])
+    linarith only [hfrac]
   have hlast : 0 < A (Fin.last r) := by
     have h := (hpositiveA (Fin.last j)).1
     have hindex :
@@ -5508,7 +5481,7 @@ theorem hierarchyStieltjesRatio_append {k r : ℕ}
   simp only [Fin.append_left, Fin.append_right]
   rw [Finset.prod_div_distrib]
   simp only [div_eq_mul_inv, mul_inv_rev]
-  ring
+  ac_rfl
 
 theorem tendsto_quadraticCoordinate_of_tendsto
     {u : ℕ → ℝ} {x : ℝ}
@@ -5568,8 +5541,8 @@ theorem tendsto_hierarchyStieltjesRatio_of_coordinatewise
     apply Finset.prod_ne_zero_iff.mpr
     intro i _
     have hq : 0 ≤ ((A i) * (1 + (A i))) := by
-      exact mul_nonneg (hA i) (by linarith [hA i])
-    linarith
+      exact mul_nonneg (hA i) (by linarith only [hA i])
+    linarith only [ht, hq]
   exact hnum.div hden hnonzero
 
 theorem tendsto_hierarchyStieltjesRatio_append_of_escaping
@@ -5864,22 +5837,7 @@ theorem compactifiedEscapingRatioProduct_eq_filtered_prefix
     {R k j : ℕ} (hkj : k + j = R) (d : Fin R → ℝ) :
     compactifiedEscapingRatioProduct hkj d =
       ∏ i ∈ Finset.univ.filter (fun i : Fin R => i.val < k), d i := by
-  classical
-  subst R
-  unfold compactifiedEscapingRatioProduct
-  apply Finset.prod_bij
-    (fun i _ => (i.castAdd j : Fin (k + j)))
-  · intro i _
-    exact Finset.mem_filter.mpr ⟨Finset.mem_univ _, i.isLt⟩
-  · intro i₁ _ i₂ _ heq
-    apply Fin.ext
-    simpa only [Fin.val_castAdd] using congrArg (fun z : Fin (k + j) => z.val) heq
-  · intro i hi
-    have hik : i.val < k := (Finset.mem_filter.mp hi).2
-    refine ⟨⟨i.val, hik⟩, Finset.mem_univ _, ?_⟩
-    exact Fin.ext rfl
-  · intro i _
-    rfl
+  exact @compactifiedEscapingRatioProduct_eq_filtered R k j hkj d
 
 theorem tendsto_Gamma_of_compactified_suffix
     {R k j q : ℕ} (hkj : k + j = R)
@@ -5993,7 +5951,7 @@ theorem compactified_certificate_residual_threshold_le
     s ≤ 1 - c ^ 2 * (1 - 2 * Gamma A B) := by
   have h := compactified_certificate_threshold_le
     hφ hthreshold hfeasible hspectral
-  nlinarith
+  linarith only [h]
 
 /-- Subsequential compactification of fixed-level families with bounded entropy into a residual
 hierarchy. -/
@@ -6056,12 +6014,12 @@ theorem fixedLevelForwardCompactifiedResidualLimit
   intro C a b hinter hbound
   have hnegative : ∀ n, (-1 : ℝ) < 2 * Gamma (a n) (b n) := by
     intro n
-    linarith [(hinter n).Gamma_nonneg]
+    linarith only [(hinter n).Gamma_nonneg]
   have herror : ∀ n : ℕ,
       Phi (a n) (b n) ≤ C + 1 / ((n : ℝ) + 1) := by
     intro n
     have hpos : 0 ≤ (1 : ℝ) / ((n : ℝ) + 1) := by positivity
-    linarith [hbound n]
+    linarith only [hpos, hbound n]
   obtain ⟨L, X, Y, d, φ, k, j, hkj, q, A, B, c,
       hφ, hLnonneg, hLC, hX, hY, hXlim, hYlim,
       hthreshold, hnegativeφ, hPhilim, hd, hdlim,
@@ -6118,7 +6076,7 @@ theorem levelRate_succ_lt_of_compactified_positive_minimizer
     exists_nextLevel_strict_refinement h hlast
   have hc2 : 0 < c ^ 2 := sq_pos_of_pos hc
   have hgap : s < 1 - c ^ 2 * (1 - 2 * Gamma A B) := by
-    nlinarith
+    nlinarith only [hfeasible, hGamma, hc2]
   have hlevels :
       (c = 1 ∧ j + 1 ≤ r + 1) ∨
         (c < 1 ∧ (j + 1) + 1 ≤ r + 1) := by
@@ -6129,7 +6087,7 @@ theorem levelRate_succ_lt_of_compactified_positive_minimizer
   calc
     levelRate (r + 1) s ≤ Phi A B - Real.logb 2 c :=
       htransfer A B hAB hgap hlevels
-    _ < Phi a b - Real.logb 2 c := by linarith
+    _ < Phi a b - Real.logb 2 c := by linarith only [hPhi]
     _ = levelRate r s := hmin.symm
 
 end
@@ -6145,7 +6103,7 @@ theorem log_inv_lt_half_inv_sub_self {t : ℝ}
   have hx' : x < 1 := by
     dsimp [x]
     exact (div_lt_one (by linarith)).mpr (by linarith)
-  have hden : 0 < 1 - x ^ 2 := by nlinarith
+  have hden : 0 < 1 - x ^ 2 := by nlinarith only [hx', hx]
   have hratio : (1 + x) / (1 - x) = 1 / t := by
     dsimp [x]
     field_simp [ht.ne']
@@ -6186,7 +6144,7 @@ theorem lt_log_one_add_add_half_log_one_add_sq {t : ℝ}
   have hy : 0 ≤ y := by dsimp [y]; positivity
   have hy' : y < 1 := by
     dsimp [y]
-    exact (div_lt_one (by positivity)).mpr (by linarith)
+    exact (div_lt_one (by positivity)).mpr (by linarith only [])
   have hxratio : (1 + x) / (1 - x) = 1 + t := by
     dsimp [x]
     field_simp
@@ -6207,7 +6165,7 @@ theorem lt_log_one_add_add_half_log_one_add_sq {t : ℝ}
       mul_zero, zero_add, pow_one, CharP.cast_eq_zero, div_one] using h
   have hrational : t < 2 * x + y := by
     dsimp [x, y]
-    have hden₁ : 0 < 2 + t := by linarith
+    have hden₁ : 0 < 2 + t := by linarith only [ht]
     have hden₂ : 0 < 2 + t ^ 2 := by positivity
     apply sub_pos.mp
     have hidentity :
@@ -6219,7 +6177,7 @@ theorem lt_log_one_add_add_half_log_one_add_sq {t : ℝ}
     exact div_pos
       (mul_pos (pow_pos ht 3) (sub_pos.mpr ht'))
       (mul_pos hden₁ hden₂)
-  nlinarith
+  linarith only [hxlog, hylog, hrational]
 
 theorem classicalThreshold_entropy_lt_half_neg_logb
     {s : ℝ} (hs : 0 < s) (hs' : s < 1) :
@@ -6228,24 +6186,24 @@ theorem classicalThreshold_entropy_lt_half_neg_logb
   let u : ℝ := Real.sqrt (1 - s ^ 2)
   let t : ℝ := s / (1 + u)
   let a : ℝ := MetricCodes.classicalThreshold s
-  have hrad : 0 < 1 - s ^ 2 := by nlinarith
+  have hrad : 0 < 1 - s ^ 2 := by nlinarith only [hs', hs]
   have hu : 0 < u := Real.sqrt_pos.mpr hrad
   have hu_sq : u ^ 2 = 1 - s ^ 2 := Real.sq_sqrt hrad.le
   have ht : 0 < t := by dsimp [t]; positivity
   have ht' : t < 1 := by
     dsimp [t]
     apply (div_lt_one (by positivity)).mpr
-    linarith
-  have htden : 0 < 1 - t ^ 2 := by nlinarith
+    linarith only [hs', hu]
+  have htden : 0 < 1 - t ^ 2 := by nlinarith only [ht', ht]
   have hsparam : s = 2 * t / (1 + t ^ 2) := by
     dsimp [t]
     field_simp
-    nlinarith [hu_sq]
+    linarith only [hu_sq]
   have ha : 0 < a := MetricCodes.classicalThreshold_pos hs hs'
   have ht_sq : t ^ 2 = (1 - u) / (1 + u) := by
     dsimp [t]
     field_simp
-    nlinarith [hu_sq]
+    linarith only [hu_sq]
   have haparam : a = t ^ 2 / (1 - t ^ 2) := by
     change (1 / u - 1) / 2 = t ^ 2 / (1 - t ^ 2)
     rw [ht_sq]
@@ -6303,7 +6261,7 @@ theorem classicalThreshold_entropy_lt_half_neg_logb
   have hgap : 0 < Real.log (1 + t) +
       (1 / 2 : ℝ) * Real.log (1 + t ^ 2) -
         2 * t ^ 2 / (1 - t ^ 2) * Real.log (1 / t) := by
-    linarith
+    linarith only [hfirst, hlast]
   have hidentity :
       Real.log 2 *
           (-(1 / 2 : ℝ) * Real.logb 2 (1 - s) -
@@ -6339,13 +6297,13 @@ theorem compactified_zero_residual_cost_le
     (hc : 0 < c) (hfeasible : s ≤ 1 - c ^ 2) :
     -(1 / 2 : ℝ) * Real.logb 2 (1 - s) ≤ -Real.logb 2 c := by
   have hcsquare : 0 < c ^ 2 := sq_pos_of_pos hc
-  have hsquare : c ^ 2 ≤ 1 - s := by linarith
+  have hsquare : c ^ 2 ≤ 1 - s := by linarith only [hfeasible]
   have hlog : Real.logb 2 (c ^ 2) ≤ Real.logb 2 (1 - s) :=
     Real.logb_le_logb_of_le (by norm_num : (1 : ℝ) < 2)
       hcsquare hsquare
   rw [Real.logb_pow] at hlog
   norm_num at hlog ⊢
-  linarith
+  linarith only [hlog]
 
 theorem classicalThreshold_entropy_lt_neg_logb_of_compactified_zero
     {s c : ℝ} (hs : 0 < s) (hs' : s < 1)
@@ -6372,20 +6330,20 @@ theorem levelRate_zero_le_entropy_of_classicalThreshold_lt
   have hA : Interlacing A B := by
     constructor
     · dsimp [A]
-      linarith
+      linarith only [ha, hthreshold]
     · intro i
       exact Fin.elim0 i
   have hboundary : spectralAtom (MetricCodes.classicalThreshold s) = s / 2 := by
     have h := MetricCodes.Spherical.classicalThreshold_spectral hs hs'
     rw [MetricCodes.Spherical.Gamma_zero hthreshold] at h
     change 2 * spectralAtom (MetricCodes.classicalThreshold s) = s at h
-    linarith
+    linarith only [h]
   have hmono := spectralAtom_strictMonoOn hthreshold.le
-    (show 0 ≤ a by linarith) ha
+    (show 0 ≤ a by linarith only [ha, hthreshold]) ha
   have hgap : s < 2 * Gamma A B := by
     rw [Gamma_zero]
     change s < 2 * spectralAtom a
-    linarith
+    linarith only [hboundary, hmono]
   simpa only [ge_iff_le, Phi_zero] using levelRate_le hA hgap
 
 theorem levelRate_zero_eq_classical_of_pos
@@ -6459,7 +6417,7 @@ theorem exists_compactified_positive_levelRate_minimizer
   have hdrop : c < 1 → j < r := by
     intro hclt
     rcases hlevels with ⟨hcone, _⟩ | ⟨_, hj⟩
-    · linarith
+    · linarith only [hclt, hcone]
     · omega
   have hlast : 0 < a (Fin.last j) := by
     by_contra hnot
@@ -6475,7 +6433,7 @@ theorem exists_compactified_positive_levelRate_minimizer
           simp only [Phi_zero, Fin.isValue, ha, sphericalEntropy_zero]
         have hscaled : s ≤ 1 - c ^ 2 := by
           rw [hGamma] at hfeasible
-          nlinarith
+          linarith only [hfeasible]
         have hclassical :=
           classicalThreshold_entropy_lt_neg_logb_of_compactified_zero
             hs hs' hc hscaled
@@ -6488,17 +6446,17 @@ theorem exists_compactified_positive_levelRate_minimizer
                   (MetricCodes.classicalThreshold s) :=
               levelRate_zero_eq_classical hs.le hs'
         rw [hPhi] at hmin
-        linarith
+        linarith only [hmin, hclassical, hupper]
     | succ j =>
         obtain ⟨A, _, hA, hGamma, hPhi⟩ :=
           exists_sameLevel_opening_strict_refinement
             (Nat.succ_pos j) h hzero
         have hstrict :
             s < 1 - c ^ 2 * (1 - 2 * Gamma A b) := by
-          nlinarith [sq_pos_of_pos hc]
+          nlinarith only [hfeasible, hGamma, sq_pos_of_pos hc]
         have hupper := levelRate_le_compactified_datum
           hs hs' hc A b hA hstrict hlevels
-        linarith
+        linarith only [hmin, hPhi, hupper]
   exact ⟨j, c, a, b, hj, hc, hc', hdrop, h,
     hlast, hfeasible, hmin⟩
 
@@ -6549,7 +6507,7 @@ theorem boundaryQuadratic_classicalThreshold_eq_zero
     (fun x : ℝ => x * Real.sqrt (a * (1 + a))) hspectral
   change MetricCodes.Spherical.boundaryQuadratic s a = 0
   unfold MetricCodes.Spherical.boundaryQuadratic
-  nlinarith
+  linarith only [hroot_sq, hmult]
 
 theorem boundaryDegree_classicalThreshold_eq_zero
     {s : ℝ} (hs : 0 < s) (hs' : s < 1) :
@@ -6577,7 +6535,7 @@ theorem boundaryDegree_spectral_eq
   rw [MetricCodes.Gamma_eq_sub, ← mul_div_assoc,
     div_eq_iff hden.ne']
   unfold MetricCodes.Spherical.boundaryQuadratic at hroot
-  nlinarith
+  linarith only [hroot]
 
 theorem exists_variationalRate_boundary_minimizer_of_limit
     {s L : ℝ} (hs : 0 < s) (hs' : s < 1)
@@ -6609,7 +6567,7 @@ theorem exists_variationalRate_boundary_minimizer_of_limit
       oneRowBoundaryObjective s u <
         MetricCodes.sphericalEntropy (MetricCodes.classicalThreshold s) := by
     unfold oneRowBoundaryObjective
-    linarith
+    linarith only [himprove, hboundary]
   have hgood_limit : oneRowBoundaryObjective s u < L :=
     hgood.trans hendpoint
   obtain ⟨A, hA⟩ :=
@@ -6693,7 +6651,7 @@ theorem levelRate_one_lt_variationalRate_of_boundary_minimizer
     rw [Gamma_oneRow ha]
     exact boundaryDegree_spectral_eq ha hquadratic.le
   have hspectral : s < 2 * Gamma A (![b]) := by
-    linarith
+    linarith only [hgamma, heq]
   calc
     levelRate 1 s ≤ Phi A (![b]) :=
       levelRate_le hA hspectral
@@ -6735,7 +6693,7 @@ theorem classicalHierarchyProfile_continuousOn_Icc
   exact
     (MetricCodes.Spherical.sphericalEntropy_continuous.continuousAt.comp
       (MetricCodes.Spherical.classicalThreshold_continuousAt
-        (by linarith [ht.1]) (lt_of_le_of_lt ht.2 hs))).continuousWithinAt
+        (by linarith only [ht.1]) (lt_of_le_of_lt ht.2 hs))).continuousWithinAt
 
 theorem levelRate_zero_continuousOn_Icc
     {s : ℝ} (hs : s < 1) :
@@ -6840,15 +6798,15 @@ theorem rowVariationalRate_zero :
   let b : ℝ := a / 2
   have ha : 0 < a := by dsimp [a]; positivity
   have hb : 0 < b := by dsimp [b]; positivity
-  have hba : b < a := by dsimp [b]; linarith
+  have hba : b < a := by dsimp [b]; linarith only [ha]
   have hgamma : 0 < MetricCodes.Gamma a b :=
     MetricCodes.Gamma_pos hb.le hba
   have hfeasible : MetricCodes.Spherical.Feasible 0 a b :=
-    ⟨hb, hba, by linarith⟩
+    ⟨hb, hba, by linarith only [hgamma]⟩
   have hbound := MetricCodes.Spherical.variationalRate_le_of_feasible hfeasible
   have hbentropy := MetricCodes.sphericalEntropy_pos hb
   dsimp [a] at hbound ⊢
-  linarith
+  linarith only [hbound, hbentropy]
 
 theorem eventually_rowVariationalRate_le_squared_entropy :
     ∀ᶠ t : ℝ in 𝓝[>] 0,
@@ -7075,15 +7033,15 @@ private theorem rowProfile_zero_metriccodes2_a5c70490 :
   let b : ℝ := a / 2
   have ha : 0 < a := by dsimp [a]; positivity
   have hb : 0 < b := by dsimp [b]; positivity
-  have hba : b < a := by dsimp [b]; linarith
+  have hba : b < a := by dsimp [b]; linarith only [ha]
   have hgamma : 0 < MetricCodes.Gamma a b :=
     MetricCodes.Gamma_pos hb.le hba
   have hfeasible : MetricCodes.Spherical.Feasible 0 a b :=
-    ⟨hb, hba, by linarith⟩
+    ⟨hb, hba, by linarith only [hgamma]⟩
   have hbound := MetricCodes.Spherical.variationalRate_le_of_feasible hfeasible
   have hbentropy := MetricCodes.sphericalEntropy_pos hb
   dsimp [a] at hbound ⊢
-  linarith
+  linarith only [hbound, hbentropy]
 
 theorem oneRowBoundaryObjective_joint_continuous :
     Continuous (fun p : ℝ × ℝ => oneRowBoundaryObjective p.1 p.2) := by
@@ -7182,13 +7140,14 @@ theorem variationalRate_lowerSemicontinuousOn_Icc_of_jointEscape
             (fun n => MetricCodes.classicalThreshold (v (φ n)))
               atTop (nhds (MetricCodes.classicalThreshold x)) :=
           (MetricCodes.Spherical.classicalThreshold_continuousAt
-            (by linarith : -1 < x) hxone).tendsto.comp hvsub
+            (by linarith only [hs, hs', hxpos, hx, hxzero, hux, hu] : -1 < x)
+            hxone).tendsto.comp hvsub
         have hdiff := hthresholdlim.sub halim
         have hle : MetricCodes.classicalThreshold x ≤ A := by
           have hnonpos : MetricCodes.classicalThreshold x - A ≤ 0 :=
             le_of_tendsto hdiff (Filter.Eventually.of_forall
               (fun n => (sub_nonpos.mpr (hthreshold (φ n)).le)))
-          linarith
+          linarith only [hnonpos]
         have hpair : Tendsto
             (fun n => (v (φ n), a (φ n))) atTop (nhds (x, A)) :=
           hvsub.prodMk_nhds halim
@@ -7273,7 +7232,7 @@ theorem exists_strict_hierarchy_refinement_of_closed
   by_cases hlast : 0 < a (Fin.last r)
   · obtain ⟨A, B, hAB, hGamma, hPhi⟩ :=
       exists_nextLevel_strict_refinement hinterlacing hlast
-    exact ⟨r + 1, A, B, hAB, by linarith, hPhi⟩
+    exact ⟨r + 1, A, B, hAB, by linarith only [hspectral, hGamma], hPhi⟩
   · have hzero : a (Fin.last r) = 0 :=
       le_antisymm (le_of_not_gt hlast) hinterlacing.1
     cases r with
@@ -7284,12 +7243,12 @@ theorem exists_strict_hierarchy_refinement_of_closed
           norm_num [spectralAtom]
         rw [hgamma] at hspectral
         norm_num at hspectral
-        linarith
+        linarith only [hs, hspectral]
     | succ r =>
         obtain ⟨A, _, hA, hGamma, hPhi⟩ :=
           exists_sameLevel_opening_strict_refinement
             (Nat.succ_pos r) hinterlacing hzero
-        exact ⟨r + 1, A, b, hA, by linarith, hPhi⟩
+        exact ⟨r + 1, A, b, hA, by linarith only [hspectral, hGamma], hPhi⟩
 
 theorem closedHierarchyVariationalRate_eq_hierarchyVariationalRate
     {s : ℝ} (hs : 0 < s) (hs' : s < 1) :
@@ -7369,7 +7328,7 @@ theorem eventually_sphericalCodeNumber_lt_hierarchyVariationalRate_of_code_bound
   have hlt :
       sInf (hierarchyRateSet s) < hierarchyVariationalRate s + ε := by
     change hierarchyVariationalRate s < hierarchyVariationalRate s + ε
-    linarith
+    linarith only [hε]
   obtain ⟨z, ⟨r, a, b, hinterlacing, hspectral, rfl⟩, hrate⟩ :=
     exists_lt_of_csInf_lt hnonempty hlt
   filter_upwards [hcode hs hs' a b hinterlacing hspectral hrate]
@@ -7386,7 +7345,7 @@ theorem sphericalCodeRate_le_closedHierarchyVariationalRate_of_code_bound
   apply le_of_forall_pos_le_add
   intro ε hε
   exact sphericalCodeRate_le_of_eventually_exponential
-    (by linarith [hierarchyVariationalRate_nonneg hs hs'])
+    (by linarith only [hε, hierarchyVariationalRate_nonneg hs hs'])
     (eventually_sphericalCodeNumber_lt_hierarchyVariationalRate_of_code_bound
       hcode hs hs' hε)
 
@@ -7419,13 +7378,11 @@ theorem compactificationCapFactor_sq
 theorem compactificationCapFactor_lt_one
     {s t : ℝ} (hs : s < 1) (hts : t < s) :
     compactificationCapFactor s t < 1 := by
-  have hden : 0 < 1 - t := by linarith
+  have hden : 0 < 1 - t := by linarith only [hs, hts]
   have hratio : (1 - s) / (1 - t) < 1 := by
-    exact (div_lt_one hden).2 (by linarith)
+    exact (div_lt_one hden).2 (by linarith only [hts])
   have hsq := compactificationCapFactor_sq hs hts.le
-  have hnonneg : 0 ≤ compactificationCapFactor s t :=
-    (compactificationCapFactor_pos hs hts.le).le
-  nlinarith
+  nlinarith only [hratio, hsq]
 
 theorem neg_logb_compactificationCapFactor_eq_sliceCost
     {s t : ℝ} (hs : s < 1) (hts : t ≤ s) :
@@ -7437,10 +7394,10 @@ theorem neg_logb_compactificationCapFactor_eq_sliceCost
   unfold compactificationCapFactor
     MetricCodes.Spherical.SidelnikovLocalization.sliceCost Real.logb
   rw [Real.log_sqrt hratio]
-  rw [Real.log_div (by linarith : 1 - s ≠ 0)
-    (by linarith : 1 - t ≠ 0),
-    Real.log_div (by linarith : 1 - t ≠ 0)
-      (by linarith : 1 - s ≠ 0)]
+  rw [Real.log_div (by linarith only [hs] : 1 - s ≠ 0)
+    (by linarith only [ht] : 1 - t ≠ 0),
+    Real.log_div (by linarith only [ht] : 1 - t ≠ 0)
+      (by linarith only [hs] : 1 - s ≠ 0)]
   ring
 
 theorem compactificationCapFactor_scaled_spectral
@@ -7450,7 +7407,7 @@ theorem compactificationCapFactor_scaled_spectral
     (hgap : t < 2 * Gamma a b) :
     s < 1 - compactificationCapFactor s t ^ 2 *
       (1 - 2 * Gamma a b) := by
-  have hden : 0 < 1 - t := by linarith
+  have hden : 0 < 1 - t := by linarith only [hs, hts]
   rw [compactificationCapFactor_sq hs hts]
   have hrewrite :
       1 - (1 - s) / (1 - t) * (1 - 2 * Gamma a b) =
@@ -7459,7 +7416,7 @@ theorem compactificationCapFactor_scaled_spectral
     field_simp [hden.ne']
   rw [hrewrite]
   apply (lt_div_iff₀ hden).2
-  nlinarith [mul_pos (sub_pos.mpr hs) (sub_pos.mpr hgap)]
+  linarith only [mul_pos (sub_pos.mpr hs) (sub_pos.mpr hgap)]
 
 end
 
@@ -7570,7 +7527,7 @@ theorem hierarchyVariationalRate_le_localizedEnvelope_of_compactification
     _ = Phi a b +
           MetricCodes.Spherical.SidelnikovLocalization.sliceCost s t := by
       rw [← neg_logb_compactificationCapFactor_eq_sliceCost hs' hts.le]
-      ring
+      ac_rfl
 
 theorem hierarchyVariationalRate_le_localizedHierarchyInf_of_compactification
     (hcompact : GenuineCompactifiedHierarchyTransfer)
diff --git a/LeanPool/MetricCodes/HighestWeights.lean b/LeanPool/MetricCodes/HighestWeights.lean
index 8d2ac79bf..bd76c5f73 100644
--- a/LeanPool/MetricCodes/HighestWeights.lean
+++ b/LeanPool/MetricCodes/HighestWeights.lean
@@ -629,9 +629,9 @@ theorem canonicalAdjacentProjectedRaiseCoefficient_sq_iff_fischerGram
       low mu hlow hlowGram).ne'
   constructor <;> intro h
   · field_simp [hgap, hlowpos] at h ⊢
-    nlinarith [h]
+    linarith [h]
   · field_simp [hgap, hlowpos] at h ⊢
-    nlinarith [h]
+    linarith [h]
 
 theorem canonicalAdjacentProjectedRaiseCoefficient_sq_of_fischerGram
     {r n : ℕ} (low : Fin (r + 2) → ℕ)
@@ -3314,11 +3314,11 @@ theorem arbitraryRowAxialRaise_sameAxis_diamond
         (arbitraryRowAxialRaise lam i k p) =
       arbitraryRowAxialRaise (raiseWeight lam j) i k
         (arbitraryRowAxialRaise lam j k p) := by
-  apply LinearMap.congr_fun
-    ((arbitraryRowAxialRaise_sameAxis_diamond_iff_commutator
-      lam i j hij (hdom hij.le) k).mpr ?_) p
-  exact arbitraryRowAxialRaise_sameAxis_commutator
-    lam hdom i j hij k
+  exact LinearMap.congr_fun
+      ((arbitraryRowAxialRaise_sameAxis_diamond_iff_commutator lam i j hij (hdom (LT.lt.le hij))
+        k).mpr
+        (arbitraryRowAxialRaise_sameAxis_commutator lam hdom i j hij k))
+      p
 
 end ArbitraryRowSameAxisDiamondSpectralSuffixClosure
 
@@ -3775,7 +3775,6 @@ theorem coeff_X_mul_pderiv_ne {σ : Type*}
   · have hma : a ∈ d.support :=
       Finsupp.mem_support_iff.mpr ha
     rw [ite_eq_left hma, MvPolynomial.coeff_pderiv, ite_eq_right ha]
-    have hneq : a ≠ b := hab
     have hsub :
         (d - (Finsupp.single a 1 : σ →₀ ℕ)) b = d b := by
       simp only [Finsupp.coe_tsub, Pi.sub_apply, Finsupp.single_eq_of_ne hab.symm, tsub_zero]
@@ -3860,7 +3859,6 @@ theorem sourceRowRoot_sourceLeadingMinor_self {r : ℕ}
       intro heq
       have hv := congrArg Fin.val heq
       change i.val = a.val at hv
-      have ha := a.isLt
       omega
     simp only [hne, ↓reduceIte]
 
@@ -3916,7 +3914,6 @@ theorem sourceColumnRoot_sourceLeadingMinor_self {r : ℕ}
       intro heq
       have hv := congrArg Fin.val heq
       change i.val = b.val at hv
-      have hb := b.isLt
       omega
     simp only [hne, ↓reduceIte]
 
@@ -3979,7 +3976,6 @@ theorem sourceRowRoot_sourceLeadingMinor_upper {r : ℕ}
       intro heq
       have hv := congrArg Fin.val heq
       change j.val = a.val at hv
-      have ha := a.isLt
       omega
     simp only [hne, ↓reduceIte]
 
@@ -4872,7 +4868,7 @@ theorem ambientPositiveRoot_nullSubstitution_sourceColumnRoot
       · simp only [Algebra.algebraMap_self, RingHom.id_apply, hjb, ↓reduceIte, mul_zero,
           isotropicVariable_eq_nullRowLinearForm, sourceColumnRoot_apply, map_sum, map_mul,
           nullSubstitution_X, Algebra.smul_def, MvPolynomial.algebraMap_eq, zero_add, map_zero]
-        ring
+        ac_rfl
 
 theorem ambientCartan_nullSubstitution_sourceColumnRoot
     {r n : ℕ} (h : 2 * (r + 1) ≤ n)
@@ -5143,7 +5139,6 @@ theorem sourceColumnRoot_sourceLeadingMinor_upper {r : ℕ}
       intro heq
       have hv := congrArg Fin.val heq
       change j.val = b.val at hv
-      have hb := b.isLt
       omega
     simp only [hne, ↓reduceIte]
 
@@ -5601,7 +5596,6 @@ theorem exists_minimalBelowColumn_of_mass_pos {m : ℕ}
   have hpos : 0 < d (a, b) := Nat.pos_of_ne_zero hzero
   have hcandidate : P b.val := ⟨a, b, rfl, hba, hpos⟩
   have hleast : Nat.find hex ≤ b.val := Nat.find_min' hex hcandidate
-  have hlt : b.val < i'.val := hbi
   omega
 
 theorem eq_zero_of_upperRoots_of_massZero_coeff
@@ -5647,7 +5641,6 @@ theorem belowDiagonal_entry_eq_zero_of_mass_eq_zero {m : ℕ}
   have hproduct : belowDiagonalWeight (i, j) * d (i, j) = 0 := by omega
   have hweight : belowDiagonalWeight (i, j) ≠ 0 := by
     change i.val - j.val ≠ 0
-    have hvals : j.val < i.val := hji
     omega
   exact (mul_eq_zero.mp hproduct).resolve_left hweight
 
@@ -7102,7 +7095,6 @@ theorem pderiv_even_X_unused {r n : ℕ}
     intro heq
     have hv := congrArg Fin.val heq
     simp only [evenCoordinate_val] at hv
-    have hp := p.isLt
     omega
   simp only [MvPolynomial.pderiv_X, ne_eq, DeterminantVectors.variableIndex_eq_iff, he, and_false,
     not_false_eq_true, Pi.single_eq_of_ne']
@@ -7118,7 +7110,6 @@ theorem pderiv_odd_X_unused {r n : ℕ}
     intro heq
     have hv := congrArg Fin.val heq
     simp only [oddCoordinate_val] at hv
-    have hp := p.isLt
     omega
   simp only [MvPolynomial.pderiv_X, ne_eq, DeterminantVectors.variableIndex_eq_iff, ho, and_false,
     not_false_eq_true, Pi.single_eq_of_ne']
diff --git a/LeanPool/MetricCodes/Interlacing.lean b/LeanPool/MetricCodes/Interlacing.lean
index 60b166ddc..3fae3a174 100644
--- a/LeanPool/MetricCodes/Interlacing.lean
+++ b/LeanPool/MetricCodes/Interlacing.lean
@@ -374,7 +374,6 @@ theorem sorted_subset_first
       exact hia
     · have hbj : b < j := (Finset.mem_filter.mp hj).2
       change i.val < j.val
-      have hia' : i.val < a.val := hia
       have hbj' : b.val < j.val := hbj
       omega
   have hsort : (insert a U).sort (· ≤ ·) =
@@ -387,7 +386,6 @@ theorem sorted_subset_first
       omega
     · intro h
       have hba : b < a := (Finset.mem_filter.mp h).2
-      have hba' : b.val < a.val := hba
       omega
   rw [hins, sort_union_of_forall_lt L (insert a U) hsep, hsort]
 
@@ -418,7 +416,6 @@ theorem sorted_subset_second
     · have hbj : b < j := (Finset.mem_filter.mp hj).2
       change i.val < j.val
       have hia' : i.val < a.val := hia
-      have hbj' : b.val < j.val := hbj
       omega
   have hsort : (insert b U).sort (· ≤ ·) =
       b :: U.sort (· ≤ ·) := by
@@ -485,7 +482,6 @@ theorem sorted_subset_both
       · have h' : a.val = b.val := congrArg Fin.val h
         omega
       · have hba : b < a := (Finset.mem_filter.mp h).2
-        have hba' : b.val < a.val := hba
         omega
   rw [hins, sort_union_of_forall_lt L (insert a (insert b U)) hsep,
     hsorta, hsortb]
@@ -804,7 +800,7 @@ theorem simpleRoot_coordinate_path_both
       hia hab hbj hfront hbefore htail hafter p A B hp ha hb]
   simp only [MvPolynomial.smul_eq_C_mul, MvPolynomial.C_add, MvPolynomial.C_sub, MvPolynomial.C_1,
     coordinateToggleBridge]
-  ring
+  ac_rfl
 
 theorem simpleRoot_coordinate_path_four_states
     {r n : ℕ} (front : List (Fin (r + 1)))
@@ -2080,8 +2076,7 @@ def lowerAxisGramIdeal {r n : ℕ} (k : Fin n) (row : Fin (r + 1)) :
 theorem axisPolynomial_mem_lowerAxisIdeal {r n : ℕ}
     (k : Fin n) (row i : Fin (r + 1)) (hi : i < row) :
     MvPolynomial.X (variableIndex i k) ∈ lowerAxisIdeal k row := by
-  apply Ideal.subset_span
-  exact ⟨i, hi, rfl⟩
+  exact Ideal.subset_span (Exists.intro i ⟨hi, rfl⟩)
 
 theorem gram_le_lowerAxisGramIdeal {r n : ℕ}
     (k : Fin n) (row : Fin (r + 1)) :
@@ -3688,33 +3683,7 @@ theorem youngClebschLower_adjoint_tmul {r n : ℕ}
     (youngClebschLower mu lam hdeg row).adjoint
         (v ⊗ₜ[ℝ] q) =
       projectedCoordinateRaise lam mu hdeg row v q := by
-  let b := EuclideanSpace.basisFun (Fin n) ℝ
-  have hv : (∑ j : Fin n, v j • b j) = v := by
-    simpa [b] using b.sum_repr v
-  calc
-    (youngClebschLower mu lam hdeg row).adjoint (v ⊗ₜ[ℝ] q) =
-        (youngClebschLower mu lam hdeg row).adjoint
-          ((∑ j : Fin n, v j • b j) ⊗ₜ[ℝ] q) := by rw [hv]
-    _ = ∑ j : Fin n,
-        v j • (youngClebschLower mu lam hdeg row).adjoint
-          (b j ⊗ₜ[ℝ] q) := by
-      rw [TensorProduct.sum_tmul, map_sum]
-      apply Finset.sum_congr rfl
-      intro j _
-      rw [← TensorProduct.smul_tmul', map_smul]
-    _ = ∑ j : Fin n,
-        v j • projectedCoordinateRaise lam mu hdeg row (b j) q := by
-      apply Finset.sum_congr rfl
-      intro j _
-      rw [youngClebschLower_adjoint_basis_tmul]
-    _ = projectedCoordinateRaiseAxis lam mu hdeg row q
-          (∑ j : Fin n, v j • b j) := by
-      rw [map_sum]
-      apply Finset.sum_congr rfl
-      intro j _
-      rw [map_smul]
-      rfl
-    _ = projectedCoordinateRaise lam mu hdeg row v q := by rw [hv]; rfl
+  exact @MultiplicityFreeClebsch.youngClebschLower_adjoint_tmul r n mu lam hdeg row v q
 
 namespace ClebschRotation
 
@@ -3747,7 +3716,7 @@ theorem euclideanAmbientRotation_inner {n : ℕ}
   simp only [euclideanAmbientRotation_apply, EuclideanSpace.basisFun_apply, inner_sub_left,
     real_inner_smul_left, EuclideanSpace.inner_single_left, Real.ringHom_apply, one_mul,
     inner_sub_right, real_inner_smul_right, EuclideanSpace.inner_single_right, neg_sub]
-  ring
+  ac_rfl
 
 theorem euclideanAmbientRotation_adjoint {n : ℕ}
     (a b : Fin n) :
@@ -3897,7 +3866,7 @@ theorem projectedCoordinateRaise_rotation {r n : ℕ}
       simp only [ambientRotation_apply, ambientCoordinateDerivation_apply, smul_eq_mul,
         euclideanAmbientRotation_apply, EuclideanSpace.basisFun_apply,
         youngAmbientRotation_apply_coe]
-      ring
+      ac_rfl
     _ = ⟪projectedCoordinateRaise mu lam hdeg row
           (euclideanAmbientRotation a b v) p +
         projectedCoordinateRaise mu lam hdeg row v
@@ -3999,7 +3968,7 @@ theorem projectedCoordinateLower_rotation {r n : ℕ}
                   (euclideanAmbientRotation a b v) p, q⟫_ℝ +
                 ⟪projectedCoordinateLower mu lam hdeg row v
                   (youngAmbientRotation lam a b p), q⟫_ℝ := by
-              ring
+              ac_rfl
             _ = _ := (inner_add_left _ _ _).symm
 
 theorem channel_rotation_intertwine_of_adjoint {r n : ℕ}
@@ -4670,7 +4639,7 @@ theorem boundaryYoungRotationIntertwiner_casimir
       (youngAmbientCasimir (n := n) mu).comp A := by
   apply LinearMap.ext
   intro p
-  simp only [LinearMap.comp_apply, youngAmbientCasimir,
+  simp only [LinearMap.comp_apply, youngAmbientCasimir.eq_1,
     LinearMap.smul_apply, LinearMap.sum_apply, LinearMap.neg_apply,
     map_smul, map_sum, map_neg]
   congr 1
@@ -5095,8 +5064,6 @@ theorem adjacentCasimirEigenvalue_lower_lt_raise {r n : ℕ}
     exact_mod_cast (show i.val ≤ r by omega)
   have hjR : (j.val : ℝ) ≤ r := by
     exact_mod_cast (show j.val ≤ r by omega)
-  have ht : 0 ≤ (target i : ℝ) := Nat.cast_nonneg _
-  have hs : 0 ≤ (source j : ℝ) := Nat.cast_nonneg _
   linarith
 
 theorem adjacentCasimirEigenvalue_ne_of_distinct_oneBox_neighbors
@@ -5538,7 +5505,6 @@ theorem phaseCorrectedYoungChannel_adjoint_eq_sqrt {E G : Type*}
     nlinarith [Real.sq_sqrt hp, Real.sqrt_nonneg p]
   · rw [phaseNegatedChannel_adjoint_apply, haxis, ← neg_smul]
     congr 1
-    have hnegative : c < 0 := lt_of_not_ge h
     nlinarith [Real.sq_sqrt hp, Real.sqrt_nonneg p]
 
 theorem phaseCorrectedYoungChannel_rotation
@@ -6340,8 +6306,6 @@ theorem reverseInterlacingHarmonicBranchGram_scalar_pos
   change 0 < ⟪c • p, p⟫_ℝ at hpositive
   have hfactor : ⟪c • p, p⟫_ℝ = c * ⟪p, p⟫_ℝ :=
     real_inner_smul_left p p c
-  have hproduct : 0 < c * ⟪p, p⟫_ℝ := by
-    exact hfactor ▸ hpositive
   have hself : 0 < ⟪p, p⟫_ℝ := real_inner_self_pos.mpr hp
   nlinarith
 
@@ -7510,9 +7474,9 @@ theorem downstreamCorrectedDerivative_source_mem_multihomogeneous
     downstreamCorrectedDerivative lam a p k a ∈
       youngMultihomogeneousSubmodule n
         (loweredInternalYoungWeight lam a) := by
-  apply (mem_youngMultihomogeneousSubmodule_iff_rowEuler
-    (loweredInternalYoungWeight lam a) _).mpr
-  exact downstreamCorrectedDerivative_source_rowEuler lam a ha p k
+  exact (mem_youngMultihomogeneousSubmodule_iff_rowEuler (loweredInternalYoungWeight lam a)
+          (downstreamCorrectedDerivative lam a p k a)).mpr
+      (downstreamCorrectedDerivative_source_rowEuler lam a ha p k)
 
 theorem downstreamCorrectedDerivative_source_mem_harmonicYoung_of_simpleRoots
     {r n : ℕ} (lam : Fin (r + 1) → ℕ)
@@ -8845,7 +8809,7 @@ theorem reverseCoefficient_sq_of_forward_sq_and_weylRatio
       minusProbability n (raiseWeight low row) mu row := by
   rw [hforward, hgramRatio,
     plusProbability_eq_weylEdgeRatio_mul_minus row h hraise]
-  ring
+  ac_rfl
 
 /-- Construct the raising projected-axis witness from the forward Gelfand–Tsetlin coefficient,
 the Weyl Gram ratio, and the reverse projected-axis identity. -/
@@ -9694,8 +9658,6 @@ theorem canonicalFibreAxisCoefficient_sq
     (hsource : 0 < sourceGram) (htarget : 0 < targetGram) :
     (raw * Real.sqrt targetGram / Real.sqrt sourceGram) ^ 2 =
       raw ^ 2 * targetGram / sourceGram := by
-  have hs : Real.sqrt sourceGram ≠ 0 :=
-    (Real.sqrt_pos.mpr hsource).ne'
   rw [div_pow, mul_pow, Real.sq_sqrt htarget.le,
     Real.sq_sqrt hsource.le]
 
diff --git a/LeanPool/MetricCodes/Rates.lean b/LeanPool/MetricCodes/Rates.lean
index 7a5648f5a..843176104 100644
--- a/LeanPool/MetricCodes/Rates.lean
+++ b/LeanPool/MetricCodes/Rates.lean
@@ -65,23 +65,19 @@ theorem inverseVarianceFactor_continuous :
   apply Continuous.inv₀
   · fun_prop
   · intro r
-    have hroot : 0 ≤ Real.sqrt (1 - r ^ 2) :=
-      Real.sqrt_nonneg _
     positivity
 
 theorem inverseDegree_eq_sq_mul_factor
     {r : ℝ} (hr : r ^ 2 ≤ 1) :
     inverseDegree r = r * (r * inverseVarianceFactor r) := by
-  have hradical : 0 ≤ 1 - r ^ 2 := by linarith
+  have hradical : 0 ≤ 1 - r ^ 2 := by linarith only [hr]
   have hsquare := Real.sq_sqrt hradical
-  have hroot : 0 ≤ Real.sqrt (1 - r ^ 2) :=
-    Real.sqrt_nonneg _
   have hdenominator :
       1 + Real.sqrt (1 - r ^ 2) ≠ 0 := by
     positivity
   unfold inverseDegree inverseVarianceFactor
   field_simp [hdenominator]
-  nlinarith
+  linarith only [hsquare]
 
 /-- The combination of negative logarithmic terms used in the inverse-degree entropy estimate. -/
 def inverseEntropyFactor (r : ℝ) : ℝ :=
@@ -196,26 +192,26 @@ theorem lowerEndpointRoot_pos {δ : ℝ}
     0 < lowerEndpointRoot δ := by
   unfold lowerEndpointRoot
   apply Real.sqrt_pos.mpr
-  linarith
+  linarith only [hhalf]
 
 theorem lowerEndpointRoot_lt_one {δ : ℝ}
     (hδ : 0 < δ) :
     lowerEndpointRoot δ < 1 := by
   unfold lowerEndpointRoot
   apply (Real.sqrt_lt' (by norm_num : (0 : ℝ) < 1)).2
-  linarith
+  linarith only [hδ]
 
 theorem lowerEndpointWeight_pos {δ : ℝ}
     (hδ : 0 < δ) :
     0 < lowerEndpointWeight δ := by
   unfold lowerEndpointWeight
-  linarith [lowerEndpointRoot_lt_one hδ]
+  linarith only [lowerEndpointRoot_lt_one hδ]
 
 theorem lowerEndpointWeight_lt_half {δ : ℝ}
     (hhalf : δ < (1 : ℝ) / 2) :
     lowerEndpointWeight δ < (1 : ℝ) / 2 := by
   unfold lowerEndpointWeight
-  linarith [lowerEndpointRoot_pos hhalf]
+  linarith only [lowerEndpointRoot_pos hhalf]
 
 theorem lowerEndpointDerivative_pos {δ : ℝ}
     (hδ : 0 < δ) (hhalf : δ < (1 : ℝ) / 2) :
@@ -225,10 +221,8 @@ theorem lowerEndpointDerivative_pos {δ : ℝ}
   have hlog :
       Real.log (lowerEndpointWeight δ) <
         Real.log (1 - lowerEndpointWeight δ) :=
-    Real.log_lt_log hweight (by linarith)
+    Real.log_lt_log hweight (by linarith only [hhalfweight])
   have hroot := lowerEndpointRoot_pos hhalf
-  have hlogtwo : 0 < Real.log (2 : ℝ) :=
-    Real.log_pos (by norm_num)
   unfold lowerEndpointDerivative
   exact div_pos
     (mul_pos hδ (sub_pos.mpr hlog))
@@ -257,7 +251,7 @@ theorem hasDerivAt_mrrwG_boundary_quadratic_zero
     simpa only [neg_mul] using hquadratic.const_sub (1 : ℝ)
   have hradicalzero :
       1 - ((0 : ℝ) ^ 2 + 2 * δ * 0 + 2 * δ) ≠ 0 := by
-    have hpositive : 0 < 1 - 2 * δ := by linarith
+    have hpositive : 0 < 1 - 2 * δ := by linarith only [hhalf]
     simpa only [ne_eq, OfNat.ofNat_ne_zero, not_false_eq_true, zero_pow, mul_zero, add_zero,
       zero_add] using
       hpositive.ne'
@@ -295,7 +289,7 @@ theorem hasDerivAt_mrrwG_boundary_quadratic_zero
   have hweightone :
       lowerEndpointWeight δ ≠ 1 := by
     have h := lowerEndpointWeight_lt_half hhalf
-    linarith
+    linarith only [h]
   have hentropyfun :
       MetricCodes.binaryEntropy =
         (fun a : ℝ => Real.binEntropy a / Real.log 2) :=
@@ -365,14 +359,14 @@ theorem exists_mrrwObjective_lt_zero
             MetricCodes.Johnson.mrrwObjective δ 0) < 0 := by
     have h :=
       hslope.eventually
-        (gt_mem_nhds (by linarith :
+        (gt_mem_nhds (by linarith only [hderivative] :
           -lowerEndpointDerivative δ < 0))
     filter_upwards [h] with r hr
     simpa only [zero_add, smul_eq_mul] using hr
   have hupper :
       ∀ᶠ r : ℝ in 𝓝[>] 0, r < 1 - 2 * δ :=
     nhdsWithin_le_nhds
-      (gt_mem_nhds (by linarith : (0 : ℝ) < 1 - 2 * δ))
+      (gt_mem_nhds (by linarith only [hhalf] : (0 : ℝ) < 1 - 2 * δ))
   have hwitness :
       ∀ᶠ r : ℝ in 𝓝[>] 0,
         0 < r ∧ r < 1 - 2 * δ ∧
@@ -440,9 +434,9 @@ theorem inverse_zero_fibre_boundary
         MetricCodes.Johnson.asymptoticThreshold δ a ∧
       MetricCodes.Johnson.mrrwObjective δ r =
         MetricCodes.Johnson.shellRate a 0 0 u := by
-  have hrplus : 0 < 1 + r := by linarith
-  have hrminus : 0 < 1 - r := by linarith
-  have hmargin : 0 < 1 - 2 * δ - r := by linarith
+  have hrplus : 0 < 1 + r := by linarith only [hr]
+  have hrminus : 0 < 1 - r := by linarith only [hδ, hupper]
+  have hmargin : 0 < 1 - 2 * δ - r := by linarith only [hupper]
   have hrvariance : 0 < 1 - r ^ 2 := by
     nlinarith only [mul_pos hrminus hrplus]
   have hquadratic :
@@ -474,13 +468,13 @@ theorem inverse_zero_fibre_boundary
   let a : ℝ := inverseWeight δ r
   have hu : 0 < u := by
     dsimp [u, inverseDegree]
-    linarith
+    linarith only [hrootlt]
   have hua : u < a := by
     dsimp [u, a, inverseDegree, inverseWeight]
-    linarith
+    linarith only [hrootorder]
   have ha : a < (1 : ℝ) / 2 := by
     dsimp [a, inverseWeight]
-    linarith
+    linarith only [hsmallroot]
   have ha0 : 0 < a := lt_trans hu hua
   have haone : 0 < 1 - a := by linarith
   have huone : 0 < 1 - u := by linarith
@@ -494,7 +488,7 @@ theorem inverse_zero_fibre_boundary
     nlinarith only [hweightsquare]
   have hdistancevariance : 2 * δ < 4 * a * (1 - a) := by
     rw [hvariancea]
-    nlinarith only [mul_pos hδ hr, sq_nonneg r]
+    nlinarith only [mul_pos hδ hr]
   have hvarianceweight : 4 * a * (1 - a) < 4 * a := by
     nlinarith only [mul_pos ha0 ha0]
   have hweightdistance : δ / 2 < a := by
@@ -516,8 +510,11 @@ theorem inverse_zero_fibre_boundary
       (a * (1 - a) - u * (1 - u)) /
         (a * (1 - a) * (1 + r)) =
         δ / (2 * a * (1 - a)) := by
-    field_simp [ha0.ne', haone.ne', hplus];
-      nlinarith only [hvariancea, hvarianceu]
+    have hdifference :
+        a * (1 - a) - u * (1 - u) = δ * (1 + r) / 2 := by
+      linarith only [hvariancea, hvarianceu]
+    rw [hdifference]
+    field_simp [ha0.ne', haone.ne', hplus]
   have hspectral :
       1 - MetricCodes.Johnson.spectralLimit a 0 0 u =
         (a * (1 - a) - u * (1 - u)) /
@@ -665,7 +662,7 @@ theorem eventually_binaryEntropy_inward_improvement
   let M : ℝ := deriv f 0 + 1
   have hM : deriv f 0 < M := by
     dsimp [M]
-    linarith
+    linarith only []
   have hslope := hf.hasDerivAt.tendsto_slope_zero_right
   have hupper :
       ∀ᶠ e : ℝ in 𝓝[>] 0,
@@ -697,7 +694,7 @@ theorem eventually_binaryEntropy_inward_improvement
     mul_lt_mul_of_pos_left hlog' he
   have hentropy :=
     MetricCodes.Hamming.neg_mul_logb_le_binaryEntropy he.le heone.le
-  linarith
+  linarith only [houter_bound, hsingular, hentropy]
 
 theorem eventually_shellRate_interior_improvement
     {a u : ℝ} (ha : 0 < a) (ha' : a < (1 : ℝ) / 2) :
@@ -725,7 +722,7 @@ theorem eventually_shellRate_interior_improvement
     1 - MetricCodes.binaryEntropy (a - interiorSlope a u * e) +
         MetricCodes.binaryEntropy u - MetricCodes.binaryEntropy e <
       1 - MetricCodes.binaryEntropy a + MetricCodes.binaryEntropy u
-  linarith
+  linarith only [he]
 
 /-- The perturbed Johnson spectral limit minus its asymptotic code-distance threshold. -/
 def interiorSpectralMargin (δ a u e : ℝ) : ℝ :=
@@ -750,11 +747,11 @@ theorem interior_boundary_delta_relation {δ a u : ℝ}
         MetricCodes.Johnson.asymptoticThreshold δ a) :
     δ * (1 + 2 * Real.sqrt (u * (1 - u))) =
       2 * (a * (1 - a) - u * (1 - u)) := by
-  have huone : 0 < 1 - u := by linarith
+  have huone : 0 < 1 - u := by linarith only [hua, ha]
   have hsquare := Real.sq_sqrt (mul_pos hu huone).le
   have hvariance := MetricCodes.Johnson.zero_fibre_boundary_variance
     hu hua ha hboundary
-  nlinarith
+  linarith only [hsquare, hvariance]
 
 theorem interior_boundary_weight_gt_distance {δ a u : ℝ}
     (hδ : 0 < δ)
@@ -764,15 +761,12 @@ theorem interior_boundary_weight_gt_distance {δ a u : ℝ}
         MetricCodes.Johnson.asymptoticThreshold δ a) :
     δ / 2 < a := by
   have ha0 : 0 < a := lt_trans hu hua
-  have huone : 0 < 1 - u := by linarith
-  have hs : 0 < Real.sqrt (u * (1 - u)) :=
-    Real.sqrt_pos.mpr (mul_pos hu huone)
+  have huone : 0 < 1 - u := by linarith only [hua, ha]
   have hvariance := MetricCodes.Johnson.zero_fibre_boundary_variance
     hu hua ha hboundary
   have hcross : 0 < δ * (2 * Real.sqrt (u * (1 - u))) := by
     positivity
-  nlinarith [sq_nonneg (2 * Real.sqrt (u * (1 - u))),
-    sq_pos_of_pos ha0]
+  linarith only [hvariance, hcross, sq_nonneg (2 * Real.sqrt (u * (1 - u))), sq_pos_of_pos ha0]
 
 theorem spectralLimit_interior_eq (a u e : ℝ) :
     MetricCodes.Johnson.spectralLimit
@@ -850,7 +844,7 @@ theorem eventually_interior_parameters
     simpa only [interiorWeight_zero, interiorSupport_zero, add_zero, interiorComplement_zero,
       sub_zero] using
       hrightcontinuous.continuousAt.tendsto.mono_left (nhdsWithin_le_nhds : 𝓝[>] (0 : ℝ) ≤ 𝓝 0)
-  have huright : u < 1 - a := by linarith
+  have huright : u < 1 - a := by linarith only [hua, ha]
   have hright : ∀ᶠ e : ℝ in 𝓝[>] 0,
       u < 1 - interiorWeight a u e + interiorSupport a u e -
         interiorComplement a u e :=
@@ -859,9 +853,9 @@ theorem eventually_interior_parameters
     hehalf, hedegree, hleft, hright, self_mem_nhdsWithin]
     with e hwd hwh hwu heh heu hel her (he : 0 < e)
   have hwpos : 0 < interiorWeight a u e := by
-    nlinarith
+    linarith only [hwu, heu, he]
   have hwcomplement : 0 < 1 - interiorWeight a u e := by
-    linarith
+    linarith only [hwh]
   refine {
     distance_pos := hδ
     distance_lt_half := hδ'
@@ -879,11 +873,11 @@ theorem eventually_interior_parameters
   · unfold interiorSupport
     exact mul_nonneg hwpos.le he.le
   · unfold interiorSupport
-    nlinarith [mul_lt_mul_of_pos_left heh hwpos]
+    linarith only [mul_lt_mul_of_pos_left heh hwpos]
   · unfold interiorComplement
     exact mul_nonneg hwcomplement.le he.le
   · unfold interiorComplement
-    nlinarith [mul_lt_mul_of_pos_left heh hwcomplement]
+    linarith only [mul_lt_mul_of_pos_left heh hwcomplement]
   · rw [interiorSupport_add_complement]
     exact heu
 
@@ -899,14 +893,14 @@ theorem hasDerivAt_interiorSpectralMargin
         (a * (1 - a) * Real.sqrt (u * (1 - u)) *
           (1 + 2 * Real.sqrt (u * (1 - u))))) 0 := by
   have ha0 : 0 < a := lt_trans hu hua
-  have ha1 : 0 < 1 - a := by linarith
-  have hu1 : 0 < 1 - u := by linarith
-  have hz0 : 0 < 1 - 2 * u := by linarith
-  have hm0 : 0 < 1 - 2 * a := by linarith
+  have ha1 : 0 < 1 - a := by linarith only [ha]
+  have hu1 : 0 < 1 - u := by linarith only [hua, ha]
+  have hz0 : 0 < 1 - 2 * u := by linarith only [hua, ha]
+  have hm0 : 0 < 1 - 2 * a := by linarith only [ha]
   have hrad : 0 < 1 - (1 - 2 * u) ^ 2 := by
-    nlinarith [mul_pos hu hu1]
+    linarith only [mul_pos hu hu1]
   have hmrad : 0 < 1 - (1 - 2 * a) ^ 2 := by
-    nlinarith [mul_pos ha0 ha1]
+    linarith only [mul_pos ha0 ha1]
   have hs : 0 < Real.sqrt (u * (1 - u)) :=
     Real.sqrt_pos.mpr (mul_pos hu hu1)
   have hsquare := Real.sq_sqrt (mul_pos hu hu1).le
@@ -1026,14 +1020,18 @@ theorem hasDerivAt_interiorSpectralMargin
     apply (eq_div_iff (by positivity)).2
     exact hdelta
   rw [hdelta_div]
-  field_simp [ha0.ne', ha1.ne', hu.ne', hu1.ne',
-    hz0.ne', hm0.ne', hs.ne', hrad.ne', hmrad.ne']
-  ring_nf at hsquare ⊢
-  linear_combination
-    ((-128 * a ^ 3 * (a - 1) ^ 3 * (2 * a - 1) ^ 3 *
-        Real.sqrt (u - u ^ 2)) +
-      (64 * a ^ 2 * u * (a - 1) ^ 2 * (2 * a - 1) * (u - 1))) *
-      hsquare
+  have hdegree :
+      (1 - 2 * u) ^ 2 = 1 - 4 * Real.sqrt (u * (1 - u)) ^ 2 := by
+    linarith only [hsquare]
+  rw [hdegree]
+  generalize hradical : Real.sqrt (u * (1 - u)) = s at hsquare hs hdegree ⊢
+  rw [← hsquare]
+  have hdegree_nonzero : 1 - 4 * s ^ 2 ≠ 0 := by
+    rw [← hdegree]
+    exact pow_ne_zero 2 hz0.ne'
+  field_simp [ha0.ne', ha1.ne', hm0.ne', hs.ne',
+    hmrad.ne', hdegree_nonzero]
+  ring
 
 theorem interiorSpectralMargin_derivative_pos
     {a u : ℝ}
@@ -1044,19 +1042,17 @@ theorem interiorSpectralMargin_derivative_pos
         (a * (1 - a) * Real.sqrt (u * (1 - u)) *
           (1 + 2 * Real.sqrt (u * (1 - u)))) := by
   have ha0 : 0 < a := lt_trans hu hua
-  have ha1 : 0 < 1 - a := by linarith
-  have hu1 : 0 < 1 - u := by linarith
-  have hs : 0 < Real.sqrt (u * (1 - u)) :=
-    Real.sqrt_pos.mpr (mul_pos hu hu1)
+  have ha1 : 0 < 1 - a := by linarith only [ha]
+  have hu1 : 0 < 1 - u := by linarith only [hua, ha]
   have hvariance : u * (1 - u) < (1 : ℝ) / 4 := by
-    nlinarith [sq_pos_of_pos (by linarith : 0 < 1 - 2 * u)]
+    linarith only [sq_pos_of_pos (by linarith : 0 < 1 - 2 * u)]
   have hshalf : Real.sqrt (u * (1 - u)) < (1 : ℝ) / 2 := by
     exact (Real.sqrt_lt' (by norm_num : (0 : ℝ) < 1 / 2)).mpr
-      (by nlinarith)
+      (by linarith only [hvariance])
   have hweight : 0 < 1 - 2 * (a * (1 - a)) := by
-    nlinarith [sq_nonneg (a - (1 : ℝ) / 2)]
+    linarith only [sq_nonneg (a - (1 : ℝ) / 2)]
   exact div_pos
-    (mul_pos hweight (by linarith))
+    (mul_pos hweight (by linarith only [hshalf]))
     (by positivity)
 
 theorem eventually_interiorSpectralMargin_pos
@@ -1131,9 +1127,9 @@ theorem interior_shell_improvement
       ∀ᶠ e : ℝ in 𝓝[>] 0, 0 < 1 - interiorWeight a u e := by
     filter_upwards
       [hweight.eventually
-        (gt_mem_nhds (show a < (1 : ℝ) by linarith))]
+        (gt_mem_nhds (show a < (1 : ℝ) by linarith only [ha]))]
       with e he
-    linarith
+    linarith only [he]
   have hall : ∀ᶠ e : ℝ in 𝓝[>] 0,
       0 < interiorSupport a u e ∧
       0 < interiorComplement a u e ∧
@@ -1406,7 +1402,7 @@ theorem johnsonWindowChannelMatrix_pairing
         rw [Finset.mul_sum]
         apply Finset.sum_congr rfl
         intro b _
-        ring
+        ac_rfl
     _ = _ := by
       unfold MetricCodes.Boolean.coordinateDot
       congr 1
@@ -1640,7 +1636,7 @@ theorem johnsonAdjacentChannel_coordinate_axisDot
         intro b _
         simp only [coupledDegreeCoordinates,
           johnsonHarmonicCoordinates, coupledDegreeVector]
-        ring
+        ac_rfl
     _ = geometricAxis x k *
         MetricCodes.Boolean.dot
           (johnsonAdjacentChannel n w p q L target source f k)
@@ -1751,7 +1747,7 @@ theorem johnsonWindowChannelMatrix_transpose_axis_fibre
           rw [Finset.mul_sum]
           apply Finset.sum_congr rfl
           intro b _
-          ring
+          ac_rfl
     _ = ∑ target : Index p q L,
         (johnsonAdjacentBlockCoefficient n w p q L v lam target Q.1 *
           johnsonFibreAmplitude n w p q L v target) *
@@ -1782,14 +1778,14 @@ theorem johnsonWindowChannelMatrix_transpose_axis_fibre
       rw [Finset.sum_mul]
       apply Finset.sum_congr rfl
       intro target _
-      ring
+      ac_rfl
     _ = Real.sqrt lam *
         (johnsonFibreAmplitude n w p q L v Q.1 *
           coupledDegreeCoordinates h x Q.1 a Q.2) := by
       rw [johnsonAdjacentBlockCoefficient_amplitude_sum
         h hstrict v hv lam hlam heigen Q.1,
         johnsonWindowBasis_dot_coupled h x Q.1 a Q.2]
-      ring
+      ac_rfl
 
 theorem johnsonChannelMatrix_transpose_axis_fibre
     {n w p q L : ℕ}
@@ -2290,7 +2286,7 @@ theorem tendsto_johnsonMu_div
           (supportDegree b n) (complementDegree g n)
           (terminalIndex u r n) / (n : ℝ))
       atTop (nhds (muLimit a b g u)) := by
-  have haone : a ≤ 1 := by linarith [h.weight_lt_half]
+  have haone : a ≤ 1 := by linarith only [h.weight_lt_half]
   have hj₁ := tendsto_johnsonJ1_ratio
     h.weight_pos.le h.support_nonneg
   have hj₂ := tendsto_johnsonJ2_ratio
@@ -2406,7 +2402,7 @@ theorem tendsto_johnsonDiagonal
           (supportDegree b n) (complementDegree g n)
           (terminalIndex u r n))
       atTop (nhds (diagonalLimit a b g u)) := by
-  have haone : a ≤ 1 := by linarith [h.weight_lt_half]
+  have haone : a ≤ 1 := by linarith only [h.weight_lt_half]
   have hmu := tendsto_johnsonMu_div h r
   have hm := tendsto_johnsonM_ratio h.weight_pos.le
   have hw := tendsto_shellWeight_ratio h.weight_pos.le
@@ -2495,7 +2491,7 @@ theorem normalizedNu_scale
           ((2 * (J / N) - 1 / N) *
             (2 * (J / N) + 1 / N)) := by
     field_simp [hN.ne']
-  have hJ : 0 < J := by linarith
+  have hJ : 0 < J := by linarith only [hstep]
   have hleft : 0 < 2 * (J / N) - 1 / N := by
     have heq :
         2 * (J / N) - 1 / N = (2 * J - 1) / N := by
@@ -2507,7 +2503,7 @@ theorem normalizedNu_scale
         2 * (J / N) + 1 / N = (2 * J + 1) / N := by
       field_simp [hN.ne']
     rw [heq]
-    exact div_pos (by linarith) hN
+    exact div_pos (by linarith only [hstep]) hN
   have hroot :
       Real.sqrt
         ((2 * (J / N) - 1 / N) *
@@ -2596,7 +2592,7 @@ theorem tendsto_johnsonNu_div
           (supportDegree b n) (complementDegree g n)
           (terminalIndex u r n) / (n : ℝ))
       atTop (nhds (nuLimit a b g u)) := by
-  have haone : a ≤ 1 := by linarith [h.weight_lt_half]
+  have haone : a ≤ 1 := by linarith only [h.weight_lt_half]
   have hj := tendsto_johnsonJ_ratio h.degree_pos r
   have hm := tendsto_johnsonM_ratio h.weight_pos.le
   have heta := tendsto_johnsonDelta_ratio
@@ -2626,7 +2622,7 @@ theorem tendsto_johnsonNu_div
         0 <
           (2 * ((1 - 2 * u) / 2) - 0) *
             (2 * ((1 - 2 * u) / 2) + 0) := by
-      apply mul_pos <;> linarith
+      apply mul_pos <;> linarith only [hz]
     exact
       (mul_pos (mul_pos (by norm_num) hhalf)
         (Real.sqrt_pos.mpr hrad)).ne'
@@ -2661,7 +2657,7 @@ theorem tendsto_johnsonNu_div
     have hscaled := (htwo.mul hj).sub he
     have hlimit :
         0 < 2 * ((1 - 2 * u) / 2) - 0 := by
-      linarith [h.centeredDegree_pos]
+      linarith only [h.centeredDegree_pos]
     filter_upwards [hscaled.eventually (Ioi_mem_nhds hlimit),
       eventually_gt_atTop (0 : ℕ)] with n hscaled' hn
     have hnreal : (0 : ℝ) < (n : ℝ) := by exact_mod_cast hn
@@ -2714,7 +2710,7 @@ theorem tendsto_johnsonEdge
           (supportDegree b n) (complementDegree g n)
           (terminalIndex u r n))
       atTop (nhds (edgeLimit a b g u)) := by
-  have haone : a ≤ 1 := by linarith [h.weight_lt_half]
+  have haone : a ≤ 1 := by linarith only [h.weight_lt_half]
   have hnu := tendsto_johnsonNu_div h r
   have hw := tendsto_shellWeight_ratio h.weight_pos.le
   have hc := tendsto_complementWeight_ratio
@@ -2754,11 +2750,11 @@ theorem zeroFibreParameters
   refine ⟨h.distance_pos, h.distance_lt_half,
     h.weight_gt_distance, h.weight_lt_half,
     le_rfl, ?_, le_rfl, ?_, ?_, h.degree_lt_weight, ?_, ?_⟩
-  · linarith [h.weight_pos]
-  · linarith [h.weight_complement_pos]
+  · linarith only [h.weight_pos]
+  · linarith only [h.weight_complement_pos]
   · simpa only [add_zero] using h.degree_pos
   · simpa only [sub_zero, add_zero] using h.degree_lt_weight
-  · nlinarith [h.degree_lt_weight, h.weight_lt_half]
+  · linarith only [h.degree_lt_weight, h.weight_lt_half]
 
 /-- The limiting hatted Johnson diagonal coefficient under proportional degree scaling. -/
 def hattedDiagonalLimit (a b g u : ℝ) : ℝ :=
@@ -3325,17 +3321,16 @@ theorem eventually_topEigenvalue_gt
   let E : ℝ := hattedEdgeLimit a b g u
   let S : ℝ := MetricCodes.Johnson.spectralLimit a b g u
   let ε : ℝ := (S - s) / 2
-  have hE : 0 < E := hattedEdgeLimit_pos h
   have hε : 0 < ε := by
     dsimp [ε, S]
-    linarith
+    linarith only [hs]
   obtain ⟨m, hm⟩ := exists_nat_gt (2 * E / ε)
   have hprod : 2 * E < (m : ℝ) * ε :=
     (div_lt_iff₀ hε).mp hm
   have hden : 0 < (m : ℝ) + 1 := by positivity
   have hrem : 2 * E / ((m : ℝ) + 1) < ε := by
     apply (div_lt_iff₀ hden).mpr
-    nlinarith
+    linarith only [hε, hprod]
   have hidentity :
       (((m : ℝ) + 1) * hattedDiagonalLimit a b g u +
         2 * (m : ℝ) * hattedEdgeLimit a b g u) /
@@ -3351,7 +3346,7 @@ theorem eventually_topEigenvalue_gt
             ((m : ℝ) + 1) := by
     rw [hidentity]
     dsimp [ε] at hrem
-    linarith
+    linarith only [hs, hrem]
   have hquot := (tendsto_terminalRayleigh h m).eventually
     (lt_mem_nhds hbelow)
   filter_upwards [hquot, eventually_terminal_block_fit h m]
@@ -3372,7 +3367,7 @@ theorem spectralGap_pos
       MetricCodes.Johnson.asymptoticThreshold d a <
         MetricCodes.Johnson.spectralLimit a b g u := h.2
   unfold spectralGap
-  linarith
+  linarith only [hgap]
 
 theorem eventually_spectralGap_lt
     {d a b g u : ℝ}
@@ -3388,9 +3383,9 @@ theorem eventually_spectralGap_lt
   let A := MetricCodes.Johnson.asymptoticThreshold d a
   let S := MetricCodes.Johnson.spectralLimit a b g u
   have hupper : S - spectralGap d a b g u < S := by
-    linarith
+    linarith only [hpositive]
   have hlower : A < A + spectralGap d a b g u := by
-    linarith
+    linarith only [hpositive]
   have heigen := eventually_topEigenvalue_gt h.1 hupper
   have hthreshold :=
     (MetricCodes.Johnson.Asymptotics.tendsto_threshold_ceil h.1).eventually
@@ -3398,7 +3393,7 @@ theorem eventually_spectralGap_lt
   filter_upwards [heigen, hthreshold] with n hn ht
   dsimp [A, S] at hn ht
   unfold spectralGap at hpositive hn ht ⊢
-  linarith
+  linarith only [hpositive, hn, ht]
 
 theorem asymptoticThreshold_lt_one
     {d a b g u : ℝ}
@@ -3409,7 +3404,7 @@ theorem asymptoticThreshold_lt_one
     exact mul_pos (mul_pos (by norm_num) h.weight_pos)
       h.weight_complement_pos
   have hquot := div_pos h.distance_pos hden
-  linarith
+  linarith only [hquot]
 
 theorem eventually_one_sub_threshold_lt
     {d a b g u : ℝ}
@@ -3420,12 +3415,12 @@ theorem eventually_one_sub_threshold_lt
         2 - MetricCodes.Johnson.asymptoticThreshold d a := by
   have hlimit :
       MetricCodes.Johnson.asymptoticThreshold d a - 1 <
-        MetricCodes.Johnson.asymptoticThreshold d a := by linarith
+        MetricCodes.Johnson.asymptoticThreshold d a := by linarith only []
   have hthreshold :=
     (MetricCodes.Johnson.Asymptotics.tendsto_threshold_ceil h).eventually
       (Ioi_mem_nhds hlimit)
   filter_upwards [hthreshold] with n hn
-  linarith
+  linarith only [hn]
 
 end
 
@@ -3454,7 +3449,7 @@ theorem certificateConstant_pos
     0 < certificateConstant d a b g u := by
   unfold certificateConstant
   apply div_pos
-  · linarith [asymptoticThreshold_lt_one h.1]
+  · linarith only [asymptoticThreshold_lt_one h.1]
   · exact spectralGap_pos h
 
 theorem exists_zero_fibre_feasible
@@ -3464,26 +3459,26 @@ theorem exists_zero_fibre_feasible
   let u : ℝ := d / 4 + 1 / 4
   have ha : a < (1 : ℝ) / 2 := by
     dsimp [a]
-    linarith
+    linarith only [hdhalf]
   have hu : 0 < u := by
     dsimp [u]
-    linarith
+    linarith only [hd]
   have hua : u < a := by
     dsimp [a, u]
-    linarith
+    linarith only [hd]
   have ha0 : 0 < a := lt_trans hu hua
-  have hac : 0 < 1 - a := by linarith
+  have hac : 0 < 1 - a := by linarith only [ha]
   have hparams : MetricCodes.Johnson.AsymptoticParameters d a 0 0 u := by
     refine ⟨hd, hdhalf, ?_, ha, by norm_num, ?_, by norm_num,
       ?_, ?_, hua, ?_, ?_⟩
     · dsimp [a]
-      linarith
-    · linarith
-    · linarith
+      linarith only []
+    · linarith only [hu, hua]
+    · linarith only [ha]
     · simpa only [add_zero] using hu
     · simpa only [sub_zero, add_zero] using hua
     · norm_num
-      linarith
+      linarith only [ha, hua]
   let A := a * (1 - a)
   let U := u * (1 - u)
   have hA : 0 < A := by
@@ -3493,10 +3488,10 @@ theorem exists_zero_fibre_feasible
   have hsden : 0 < 1 + 2 * Real.sqrt U := by positivity
   have hdiff : 2 * (A - U) < d := by
     dsimp [A, U, a, u]
-    nlinarith [sq_nonneg d]
+    linarith only [hd, sq_nonneg d]
   have hinner : (A - U) / (1 + 2 * Real.sqrt U) < d / 2 := by
     apply (div_lt_iff₀ hsden).mpr
-    nlinarith [mul_nonneg hd.le hs]
+    linarith only [hdiff, mul_nonneg hd.le hs]
   have hquot :
       (A - U) / (A * (1 + 2 * Real.sqrt U)) < d / (2 * A) := by
     calc
@@ -3518,7 +3513,7 @@ theorem exists_zero_fibre_feasible
           (a * (1 - a) * (1 + 2 * Real.sqrt (u * (1 - u)))) <
         d / (2 * a * (1 - a)) := by
     convert hquot using 1; ring
-  linarith
+  linarith only [hboundary, hquot']
 
 theorem rateSet_nonempty_of_interior
     {d : ℝ} (hd : 0 < d) (hdhalf : d < (1 : ℝ) / 2) :
@@ -3534,7 +3529,7 @@ theorem eventually_strict_weight_half
   have hw := tendsto_shellWeight_ratio h.weight_pos.le
   have hww := tendsto_add_degree_ratio hw hw
   have hstrict := eventually_degree_lt_of_ratio hww
-    tendsto_dimension_ratio (by linarith [h.weight_lt_half])
+    tendsto_dimension_ratio (by linarith only [h.weight_lt_half])
   filter_upwards [hstrict] with n hn
   omega
 
@@ -3566,15 +3561,10 @@ theorem binaryRate_le_shellRate_of_projectionGrams
   have hG : 0 < G := spectralGap_pos h
   have hB : 0 < B := by
     dsimp [B]
-    linarith [asymptoticThreshold_lt_one h.1]
+    linarith only [asymptoticThreshold_lt_one h.1]
   have hden : 0 < lam - t := by
     dsimp [lam, t, G] at hgap ⊢
-    linarith [spectralGap_pos h]
-  have ht : 0 < 1 - t := by
-    dsimp [t]
-    exact sub_pos.mpr
-      (MetricCodes.Johnson.threshold_lt_one hadmissible.weight_pos
-        hadmissible.weight_lt hd)
+    linarith only [hgap, spectralGap_pos h]
   have hratio : (1 - t) / (lam - t) ≤ B / G := by
     apply (div_le_div_iff₀ hden hG).mpr
     calc
@@ -3614,7 +3604,7 @@ theorem binaryRate_le_shellRate_of_projectionGrams
           bassalygoWindowFibreQuotient a b g u n := by
       unfold certificateConstant bassalygoWindowFibreQuotient
       dsimp [B, G]
-      ring
+      ac_rfl
 
 theorem binaryRate_le_variationalRate_of_projectionGrams
     {d : ℝ} (hd : 0 < d) (hdhalf : d < (1 : ℝ) / 2)
@@ -3714,13 +3704,13 @@ theorem log_ratio_lower {x : ℝ} (hx : 0 ≤ x) (hx' : x < 1) (m : ℕ) :
     logSeriesLower x m ≤ Real.log ((1 + x) / (1 - x)) := by
   have h := Real.sum_range_le_log_div hx hx' m
   unfold logSeriesLower
-  nlinarith
+  linarith only [h]
 
 theorem log_ratio_upper {x : ℝ} (hx : 0 ≤ x) (hx' : x < 1) (m : ℕ) :
     Real.log ((1 + x) / (1 - x)) ≤ logSeriesUpper x m := by
   have h := Real.log_div_le_sum_range_add hx hx' m
   unfold logSeriesUpper logSeriesLower
-  nlinarith
+  linarith only [h]
 
 theorem log_interval_of_series {r x lo hi : ℝ} (m : ℕ)
     (hx : 0 ≤ x) (hx' : x < 1)
@@ -3815,7 +3805,7 @@ theorem kissing_spectral_certificate :
   apply (lt_div_iff₀ hden).2
   have hupper := kissing_sqrt_upper
   norm_num [kissingA, kissingB] at hupper ⊢
-  nlinarith
+  linarith only [hupper]
 
 theorem log_kissing_a :
     Real.log kissingA = Real.log (16 * kissingA) - 4 * Real.log 2 := by
@@ -3826,7 +3816,7 @@ theorem log_kissing_a :
     convert Real.log_pow (2 : ℝ) 4 using 1
     norm_num
   rw [hpow] at hmul
-  linarith
+  linarith only [hmul]
 
 theorem log_kissing_b :
     Real.log kissingB =
@@ -3841,7 +3831,7 @@ theorem log_kissing_b :
       -Real.log (256 * kissingB) := by
     rw [one_div, Real.log_inv]
   rw [hpow] at hmul
-  linarith
+  linarith only [hmul, hinv]
 
 theorem kissing_entropy_certificate :
     MetricCodes.sphericalEntropy kissingA - MetricCodes.sphericalEntropy kissingB <
@@ -3859,7 +3849,7 @@ theorem kissing_entropy_certificate :
               (-8 * Real.log 2 - Real.log (1 / (256 * kissingB)))) <
         (0.397305601680 : ℝ) * Real.log 2 := by
     norm_num [kissingA, kissingB] at htwo h1a hsa h1b hib ⊢
-    linarith
+    linarith only [htwo, h1a, hsa, h1b, hib]
   have hlog : 0 < Real.log 2 := Real.log_pos (by norm_num)
   calc
     MetricCodes.sphericalEntropy kissingA - MetricCodes.sphericalEntropy kissingB =
@@ -3902,7 +3892,7 @@ theorem hasDerivAt_sphericalEntropy {u : ℝ} (hu : 0 < u) :
     HasDerivAt MetricCodes.sphericalEntropy
       (Real.logb 2 ((1 + u) / u)) u := by
   have hplus :=
-    (Real.hasDerivAt_mul_log (by linarith : 1 + u ≠ 0)).comp u
+    (Real.hasDerivAt_mul_log (by linarith only [hu] : 1 + u ≠ 0)).comp u
       ((hasDerivAt_const u (1 : ℝ)).add (hasDerivAt_id u))
   have hself := Real.hasDerivAt_mul_log hu.ne'
   have hderiv := (hplus.sub hself).div_const (Real.log 2)
@@ -3918,7 +3908,7 @@ theorem hasDerivAt_sphericalEntropy {u : ℝ} (hu : 0 < u) :
         (((Real.log (1 + u) + 1) * (0 + 1) -
           (Real.log u + 1)) / Real.log 2) := by
     unfold Real.logb
-    rw [Real.log_div (by linarith : 1 + u ≠ 0) hu.ne']
+    rw [Real.log_div (by linarith only [hu] : 1 + u ≠ 0) hu.ne']
     ring
   rw [hfun, hvalue]
   simpa only [Function.comp_apply, Pi.sub_apply] using hderiv
@@ -3933,7 +3923,7 @@ theorem sphericalEntropy_strictMono :
   rw [(hasDerivAt_sphericalEntropy hu').deriv]
   apply Real.logb_pos (by norm_num : (1 : ℝ) < 2)
   apply (lt_div_iff₀ hu').2
-  linarith
+  linarith only []
 
 theorem sphericalEntropy_sub_nonneg {a b : ℝ}
     (hb : 0 ≤ b) (hba : b ≤ a) :
@@ -3983,14 +3973,14 @@ theorem Gamma_zero {a : ℝ} (ha : 0 < a) :
   rw [MetricCodes.Gamma_eq_sub]
   simp only [zero_mul, add_zero, sub_zero]
   field_simp [hroot, hlin]
-  nlinarith
+  linarith only [hsquare]
 
 theorem classicalThreshold_spectral
     {s : ℝ} (hs : 0 < s) (hs' : s < 1) :
     2 * MetricCodes.Gamma (MetricCodes.classicalThreshold s) 0 = s := by
   have ha : 0 < MetricCodes.classicalThreshold s :=
     MetricCodes.classicalThreshold_pos hs hs'
-  have hrad : 0 < 1 - s ^ 2 := by nlinarith
+  have hrad : 0 < 1 - s ^ 2 := by nlinarith only [hs', hs]
   have hroot : 0 < Real.sqrt (1 - s ^ 2) :=
     Real.sqrt_pos.2 hrad
   have hrootsq := Real.sq_sqrt hrad.le
@@ -4000,7 +3990,7 @@ theorem classicalThreshold_spectral
         s ^ 2 * (1 + 2 * MetricCodes.classicalThreshold s) ^ 2 := by
     unfold MetricCodes.classicalThreshold
     field_simp [hroot.ne']
-    nlinarith
+    linarith only [hrootsq]
   have hinner :
       0 < MetricCodes.classicalThreshold s *
         (1 + MetricCodes.classicalThreshold s) := by positivity
@@ -4033,7 +4023,7 @@ def sphericalImprovementPath (a b : ℝ) : ℝ :=
 
 theorem sphericalImprovementSlope_gt_one {a : ℝ} (ha : 0 < a) :
     1 < (4 * a + 3) := by
-  linarith
+  linarith only [ha]
 
 /-- The polynomial expression used to certify positivity of the spherical spectral margin. -/
 def sphericalSpectralMarginPolynomial (a c b : ℝ) : ℝ :=
@@ -4097,7 +4087,7 @@ theorem eventually_Gamma_improvement {a : ℝ} (ha : 0 < a) :
     with b hbpoly (hb : 0 < b)
   have hbpath : b < sphericalImprovementPath a b := by
     unfold sphericalImprovementPath
-    nlinarith [mul_pos (sub_pos.mpr hc) hb]
+    linarith only [ha, mul_pos (sub_pos.mpr hc) hb]
   have hpath : 0 < sphericalImprovementPath a b :=
     hb.trans hbpath
   have hbase : 0 < a * (1 + a) := by positivity
@@ -4110,7 +4100,7 @@ theorem eventually_Gamma_improvement {a : ℝ} (ha : 0 < a) :
     have hfactor :
         0 < (sphericalImprovementPath a b - b) *
           (1 + sphericalImprovementPath a b + b) := by positivity
-    nlinarith
+    linarith only [hfactor]
   have hmargin :
       0 <
         ((sphericalImprovementPath a b *
@@ -4155,7 +4145,7 @@ theorem eventually_Gamma_improvement {a : ℝ} (ha : 0 < a) :
               (1 + sphericalImprovementPath a b)) := by
     rw [mul_pow, mul_pow, Real.sq_sqrt hbase.le,
       Real.sq_sqrt hrad.le]
-    ring
+    ac_rfl
   have hcross :
       Real.sqrt (a * (1 + a)) *
         ((1 + 2 * sphericalImprovementPath a b) *
@@ -4165,7 +4155,9 @@ theorem eventually_Gamma_improvement {a : ℝ} (ha : 0 < a) :
         (sphericalImprovementPath a b *
           (1 + sphericalImprovementPath a b) - b * (1 + b)) *
             (1 + 2 * a) := by
-    nlinarith
+    apply (sq_lt_sq₀ hleftnonneg hrightpos.le).mp
+    rw [hsquare]
+    exact sub_pos.mp hmargin
   rw [Gamma_zero ha, MetricCodes.Gamma_eq_sub]
   have hlin : 0 < 1 + 2 * a := by positivity
   have hden :
@@ -4178,11 +4170,11 @@ theorem eventually_Gamma_improvement {a : ℝ} (ha : 0 < a) :
 theorem neg_mul_logb_le_sphericalEntropy {b : ℝ} (hb : 0 ≤ b) :
     b * (-Real.logb 2 b) ≤ MetricCodes.sphericalEntropy b := by
   have hlog : 0 ≤ Real.logb 2 (1 + b) :=
-    Real.logb_nonneg (by norm_num) (by linarith)
+    Real.logb_nonneg (by norm_num) (by linarith only [hb])
   have hterm : 0 ≤ (1 + b) * Real.logb 2 (1 + b) := by
-    exact mul_nonneg (by linarith) hlog
+    exact mul_nonneg (by linarith only [hb]) hlog
   unfold MetricCodes.sphericalEntropy
-  nlinarith
+  linarith only [hterm]
 
 theorem eventually_sphericalEntropy_improvement {a : ℝ} (ha : 0 < a) :
     ∀ᶠ b : ℝ in 𝓝[>] 0,
@@ -4203,7 +4195,7 @@ theorem eventually_sphericalEntropy_improvement {a : ℝ} (ha : 0 < a) :
   let M : ℝ := deriv f 0 + 1
   have hM : deriv f 0 < M := by
     dsimp [M]
-    linarith
+    linarith only []
   have hslope := hf.hasDerivAt.tendsto_slope_zero_right
   have hupper :
       ∀ᶠ b : ℝ in 𝓝[>] 0,
@@ -4237,7 +4229,7 @@ theorem eventually_sphericalEntropy_improvement {a : ℝ} (ha : 0 < a) :
       b * M < b * (-Real.logb 2 b) :=
     mul_lt_mul_of_pos_left hlogb hb
   have hentropy := neg_mul_logb_le_sphericalEntropy hb.le
-  linarith
+  linarith only [houter_bound, hsingular, hentropy]
 
 theorem exists_strict_improving_spherical_feasible
     {s : ℝ} (hs : 0 < s) (hs' : s < 1) :
@@ -4260,13 +4252,13 @@ theorem exists_strict_improving_spherical_feasible
       with b hgamma' hentropy' (hb : 0 < b)
     have hbpath : b < sphericalImprovementPath a₀ b := by
       unfold sphericalImprovementPath
-      nlinarith [mul_pos (sub_pos.mpr hc) hb]
+      linarith only [ha₀, mul_pos (sub_pos.mpr hc) hb]
     constructor
     · refine ⟨hb, hbpath, ?_⟩
       have hboundary : 2 * MetricCodes.Gamma a₀ 0 = s := by
         dsimp [a₀]
         exact classicalThreshold_spectral hs hs'
-      nlinarith
+      linarith only [hgamma', hboundary]
     · simpa only using hentropy'
   obtain ⟨b, hb⟩ := hall.exists
   exact ⟨sphericalImprovementPath a₀ b, b, hb⟩
@@ -4291,7 +4283,7 @@ open scoped Topology
 theorem classicalThreshold_continuousAt
     {s : ℝ} (hs : -1 < s) (hs' : s < 1) :
     ContinuousAt MetricCodes.classicalThreshold s := by
-  have hrad : 0 < 1 - s ^ 2 := by nlinarith
+  have hrad : 0 < 1 - s ^ 2 := by nlinarith only [hs, hs']
   have hroot : Real.sqrt (1 - s ^ 2) ≠ 0 :=
     (Real.sqrt_pos.mpr hrad).ne'
   unfold MetricCodes.classicalThreshold
@@ -4375,7 +4367,7 @@ theorem quadraticCoordinate_strictMonoOn :
   change 0 ≤ x at hx
   change 0 ≤ y at hy
   change x * (1 + x) < y * (1 + y)
-  nlinarith [mul_pos (sub_pos.mpr hxy) (by linarith : 0 < 1 + x + y)]
+  linarith only [mul_pos (sub_pos.mpr hxy) (by linarith : 0 < 1 + x + y)]
 
 theorem Interlacing.quadratic_injective {r : ℕ}
     {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
@@ -4568,17 +4560,17 @@ theorem integral_inv_add_mul_cos (a b : ℝ) (h : |b| < a) :
     (∫ θ in (0 : ℝ)..Real.pi, (a + b * Real.cos θ)⁻¹) =
       Real.pi / Real.sqrt (a ^ 2 - b ^ 2) := by
   have hab : -a < b ∧ b < a := abs_lt.mp h
-  have hplus : 0 < a + b := by linarith [hab.1]
-  have hminus : 0 < a - b := by linarith [hab.2]
+  have hplus : 0 < a + b := by linarith only [hab.1]
+  have hminus : 0 < a - b := by linarith only [hab.2]
   have hdisc : 0 < a ^ 2 - b ^ 2 := by
-    nlinarith [mul_pos hplus hminus]
+    linarith only [mul_pos hplus hminus]
   let c : ℝ := Real.sqrt (a ^ 2 - b ^ 2)
   have hc : 0 < c := Real.sqrt_pos.2 hdisc
   have hden (θ : ℝ) : 0 < a + b * Real.cos θ := by
     have hbound : |b * Real.cos θ| ≤ |b| := by
       rw [abs_mul]
       simpa only [mul_one] using mul_le_mul_of_nonneg_left (Real.abs_cos_le_one θ) (abs_nonneg b)
-    linarith [neg_le_of_abs_le hbound]
+    linarith only [h, neg_le_of_abs_le hbound]
   let q : ℝ → ℝ := fun θ =>
     (a * Real.cos θ + b) / (a + b * Real.cos θ)
   let F : ℝ → ℝ := fun θ => -Real.arcsin (q θ) / c
@@ -4605,7 +4597,7 @@ theorem integral_inv_add_mul_cos (a b : ℝ) (h : |b| < a) :
           (a ^ 2 - b ^ 2) * (Real.sin θ) ^ 2 /
             (a + b * Real.cos θ) ^ 2 := by
       have htrig : (Real.sin θ) ^ 2 = 1 - (Real.cos θ) ^ 2 := by
-        linarith [Real.sin_sq_add_cos_sq θ]
+        linarith only [Real.sin_sq_add_cos_sq θ]
       have hd2 : (a + b * Real.cos θ) ^ 2 ≠ 0 :=
         pow_ne_zero 2 (hden θ).ne'
       rw [htrig]
@@ -4627,9 +4619,9 @@ theorem integral_inv_add_mul_cos (a b : ℝ) (h : |b| < a) :
             (a + b * Real.cos θ) ^ 2 :=
         div_pos (mul_pos hdisc (sq_pos_of_pos hsin))
           (sq_pos_of_pos (hden θ))
-      linarith [hqidentity]
-    have hqlower : -1 < q θ := by nlinarith [sq_nonneg (q θ + 1)]
-    have hqupper : q θ < 1 := by nlinarith [sq_nonneg (q θ - 1)]
+      linarith only [hpos, hqidentity]
+    have hqlower : -1 < q θ := by linarith [sq_nonneg (q θ + 1)]
+    have hqupper : q θ < 1 := by linarith [sq_nonneg (q θ - 1)]
     have hroot :
         Real.sqrt (1 - (q θ) ^ 2) =
           c * Real.sin θ / (a + b * Real.cos θ) := by
@@ -4659,7 +4651,7 @@ theorem integral_inv_add_mul_cos (a b : ℝ) (h : |b| < a) :
     have hcsq : c ^ 2 = a ^ 2 - b ^ 2 :=
       Real.sq_sqrt hdisc.le
     field_simp [hc.ne', (hden θ).ne', hsin.ne']
-    nlinarith
+    linarith only [hcsq]
   rw [intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le
     Real.pi_pos.le hFcont.continuousOn hderiv hintegrable]
   dsimp [F, q]
@@ -4694,7 +4686,7 @@ theorem betaPDFReal_half_half_eq (u : ℝ) (hu₀ : 0 < u) (hu₁ : u < 1) :
   rw [betaPDFReal, ite_eq_left ⟨hu₀, hu₁⟩,
     MetricCodes.Spherical.HigherHierarchy.Arcsine.beta_half_half_eq_pi]
   have hu : 0 ≤ u := hu₀.le
-  have hv : 0 ≤ 1 - u := by linarith
+  have hv : 0 ≤ 1 - u := by linarith only [hu₁]
   have hpow₁ : u ^ ((1 / 2 : ℝ) - 1) = (Real.sqrt u)⁻¹ := by
     convert Real.rpow_neg hu (1 / 2 : ℝ) using 1 <;>
       norm_num [Real.sqrt_eq_rpow]
@@ -4703,7 +4695,7 @@ theorem betaPDFReal_half_half_eq (u : ℝ) (hu₀ : 0 < u) (hu₁ : u < 1) :
     convert Real.rpow_neg hv (1 / 2 : ℝ) using 1 <;>
       norm_num [Real.sqrt_eq_rpow]
   rw [hpow₁, hpow₂, Real.sqrt_mul hu, mul_inv_rev]
-  ring
+  ac_rfl
 
 theorem integral_beta_half_half_eq_integral_measureT (f : ℝ → ℝ) :
     (∫ u, f u ∂betaMeasure (1 / 2 : ℝ) (1 / 2 : ℝ)) =
@@ -4810,7 +4802,7 @@ theorem betaMap_half_half_ae_pos :
   by_contra hnot
   have htzero : t ≤ 0 := by
     have : t / 4 ≤ 0 := le_of_not_gt hnot
-    linarith
+    linarith only [hnot]
   apply ht
   simp only [betaPDF, betaPDFReal, not_lt_of_ge htzero, false_and, ↓reduceIte, ENNReal.ofReal_zero]
 
@@ -4828,10 +4820,9 @@ theorem arcsine_stieltjes_integrable {x : ℝ} (hx : 0 ≤ x) :
   apply (integrable_const (1 : ℝ)).mono'
     ((by fun_prop : Measurable (fun t : ℝ => x / (x + t))).aestronglyMeasurable)
   filter_upwards [betaMap_half_half_ae_pos] with t ht
-  have hden : 0 < x + t := by linarith
-  have hnonneg : 0 ≤ x / (x + t) := div_nonneg hx hden.le
+  have hden : 0 < x + t := by linarith only [hx, ht]
   simpa only [norm_div, norm_eq_abs, abs_of_nonneg hx, abs_of_pos hden, ge_iff_le] using
-    ((div_le_one hden).2 (by linarith : x ≤ x + t))
+    ((div_le_one hden).2 (by linarith only [ht] : x ≤ x + t))
 
 theorem arcsine_integral_x_div_add {x : ℝ} (hx : 0 < x) :
     (∫ t, x / (x + t) ∂(betaMeasure (1 / 2 : ℝ) (1 / 2 : ℝ)).map
@@ -4855,15 +4846,15 @@ theorem arcsine_integral_x_div_add {x : ℝ} (hx : 0 < x) :
     MetricCodes.Spherical.HigherHierarchy.ArcsineIntegral.integral_inv_add_mul_cos
       (x + 1 / 8) (1 / 8) (by
       norm_num [abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 1 / 8)]
-      linarith)]
+      linarith only [hx])]
   have hquad : (x + 1 / 8) ^ 2 - (1 / 8 : ℝ) ^ 2 =
       x * (x + 1 / 4) := by ring
   rw [hquad, Real.sqrt_mul hx.le, Real.sqrt_div hx.le]
   have hsx : Real.sqrt x ≠ 0 := (Real.sqrt_pos.mpr hx).ne'
   have hsy : Real.sqrt (x + 1 / 4) ≠ 0 :=
-    (Real.sqrt_pos.mpr (by linarith)).ne'
+    (Real.sqrt_pos.mpr (by linarith only [hx])).ne'
   field_simp [Real.pi_pos.ne', hsx, hsy]
-  nlinarith [Real.sq_sqrt hx.le]
+  linarith only [Real.sq_sqrt hx.le]
 
 theorem arcsine_integral_self_div_add {x : ℝ} (hx : 0 ≤ x) :
     (∫ t, t / (t + x) ∂(betaMeasure (1 / 2 : ℝ) (1 / 2 : ℝ)).map
@@ -4888,7 +4879,7 @@ theorem arcsine_integral_self_div_add {x : ℝ} (hx : 0 ≤ x) :
             (1 / 2 : ℝ)).map (fun t : ℝ => t / 4) := by
             apply integral_congr_ae
             filter_upwards [betaMap_half_half_ae_pos] with t ht
-            have hden : x + t ≠ 0 := ne_of_gt (by linarith)
+            have hden : x + t ≠ 0 := ne_of_gt (by linarith only [ht, hx])
             field_simp
             ring
       _ = 1 - (∫ t, x / (x + t) ∂(betaMeasure (1 / 2 : ℝ)
@@ -5015,16 +5006,16 @@ theorem kissing_spectral_certificate :
     lagrangeDenominator,
     Fin.prod_univ_two]
     at h₀ h₁ h₂ ⊢
-  linarith
+  linarith only [h₀, h₁, h₂]
 
 theorem rational_log_lower {r lo : ℝ} (hr : 1 ≤ r) (m : ℕ)
     (hlo : lo < MetricCodes.Numerics.logSeriesLower
       ((r - 1) / (r + 1)) m) : lo < Real.log r := by
-  have hden : 0 < r + 1 := by linarith
+  have hden : 0 < r + 1 := by linarith only [hr]
   have hx : 0 ≤ (r - 1) / (r + 1) :=
     div_nonneg (sub_nonneg.mpr hr) hden.le
   have hx' : (r - 1) / (r + 1) < 1 :=
-    (div_lt_one hden).mpr (by linarith)
+    (div_lt_one hden).mpr (by linarith only [])
   calc
     lo < MetricCodes.Numerics.logSeriesLower
       ((r - 1) / (r + 1)) m := hlo
@@ -5038,11 +5029,11 @@ theorem rational_log_lower {r lo : ℝ} (hr : 1 ≤ r) (m : ℕ)
 theorem rational_log_upper {r hi : ℝ} (hr : 1 ≤ r) (m : ℕ)
     (hhi : MetricCodes.Numerics.logSeriesUpper
       ((r - 1) / (r + 1)) m < hi) : Real.log r < hi := by
-  have hden : 0 < r + 1 := by linarith
+  have hden : 0 < r + 1 := by linarith only [hr]
   have hx : 0 ≤ (r - 1) / (r + 1) :=
     div_nonneg (sub_nonneg.mpr hr) hden.le
   have hx' : (r - 1) / (r + 1) < 1 :=
-    (div_lt_one hden).mpr (by linarith)
+    (div_lt_one hden).mpr (by linarith only [])
   calc
     Real.log r = Real.log
       ((1 + (r - 1) / (r + 1)) / (1 - (r - 1) / (r + 1))) := by
@@ -5058,7 +5049,7 @@ theorem log_scaled_pow_two {u : ℝ} (hu : u ≠ 0) (k : ℕ) :
   have h := Real.log_mul
     (pow_ne_zero k (by norm_num : (2 : ℝ) ≠ 0)) hu
   rw [Real.log_pow] at h
-  linarith
+  linarith only [h]
 
 theorem kissing_entropy_certificate :
     Phi kissingAmbient kissingStabilizer < (0.396605 : ℝ) := by
@@ -5132,7 +5123,18 @@ theorem kissing_entropy_certificate :
           (0.396605 : ℝ) * Real.log 2 := by
     norm_num at htwo ha₀plus ha₀scale ha₁plus ha₁scale ha₂plus ha₂scale
     norm_num at hb₀plus hb₀scale hb₁plus hb₁scale ⊢
-    linarith
+    linarith only [
+      htwo,
+      ha₀plus,
+      ha₀scale,
+      ha₁plus,
+      ha₁scale,
+      ha₂plus,
+      ha₂scale,
+      hb₀plus,
+      hb₀scale,
+      hb₁plus,
+      hb₁scale]
   have hlog : 0 < Real.log 2 := Real.log_pos (by norm_num)
   calc
     Phi kissingAmbient kissingStabilizer =
@@ -5247,13 +5249,13 @@ theorem Interlacing.ambient_quadratic_nonneg {r : ℕ}
     {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
     (h : Interlacing a b) (i : Fin (r + 1)) :
     0 ≤ ((a i) * (1 + (a i))) := by
-  exact mul_nonneg (h.ambient_nonneg i) (by linarith [h.ambient_nonneg i])
+  exact mul_nonneg (h.ambient_nonneg i) (by linarith only [h.ambient_nonneg i])
 
 theorem Interlacing.stabilizer_quadratic_pos {r : ℕ}
     {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
     (h : Interlacing a b) (i : Fin r) :
     0 < ((b i) * (1 + (b i))) := by
-  exact mul_pos (h.stabilizer_pos i) (by linarith [h.stabilizer_pos i])
+  exact mul_pos (h.stabilizer_pos i) (by linarith only [h.stabilizer_pos i])
 
 theorem integral_inv_add_eq_log_div
     {t p q : ℝ} (hp : 0 < t + p) (hq : 0 < t + q) :
@@ -5297,13 +5299,13 @@ theorem exp_neg_stieltjesPhase_eq_product {r : ℕ}
     Real.exp (-stieltjesPhase a b t) =
       stieltjesPhaseProduct a b t := by
   have hlast : 0 < t + ((a (Fin.last r)) * (1 + (a (Fin.last r)))) := by
-    linarith [h.ambient_quadratic_nonneg (Fin.last r)]
+    linarith only [ht, h.ambient_quadratic_nonneg (Fin.last r)]
   have hambient (i : Fin (r + 1)) :
       0 < t + ((a i) * (1 + (a i))) := by
-    linarith [h.ambient_quadratic_nonneg i]
+    linarith only [ht, h.ambient_quadratic_nonneg i]
   have hstabilizer (i : Fin r) :
       0 < t + ((b i) * (1 + (b i))) := by
-    linarith [h.stabilizer_quadratic_pos i]
+    linarith only [ht, h.stabilizer_quadratic_pos i]
   have hexp :
       Real.exp (stieltjesPhase a b t) =
         ((t + ((a (Fin.last r)) * (1 + (a (Fin.last r))))) / t) *
@@ -5389,7 +5391,7 @@ theorem stieltjesPartialFraction {r : ℕ}
   have hraw := polynomial_partialFraction h.quadratic_injective P hdegree
     (z := -t) (fun i => by
       have hnode := h.ambient_quadratic_nonneg i
-      linarith)
+      linarith only [ht, hnode])
   have hrewrite :
       (∏ i : Fin r, (-t - ((b i) * (1 + (b i))))) /
           (∏ i : Fin (r + 1), (-t - ((a i) * (1 + (a i))))) =
@@ -5414,7 +5416,7 @@ theorem stieltjesPartialFraction {r : ℕ}
     apply Finset.prod_ne_zero_iff.mpr
     intro i _
     have hnode := h.ambient_quadratic_nonneg i
-    exact ne_of_gt (by linarith)
+    exact ne_of_gt (by linarith only [ht, hnode])
   have hleft :
       ((-1 : ℝ) ^ r * (∏ i : Fin r,
           (t + ((b i) * (1 + (b i)))))) /
@@ -5452,8 +5454,7 @@ theorem quadratic_spectralAtom_identity {u : ℝ} (hu : 0 ≤ u) :
         2 * spectralAtom u := by
   have hA : 0 ≤ (u * (1 + u)) := by
     positivity
-  have hlinear : 0 < 1 + 2 * u := by linarith
-  have hden : 0 < (u * (1 + u)) + 1 / 4 := by linarith
+  have hden : 0 < (u * (1 + u)) + 1 / 4 := by linarith only [hA]
   have hroot :
       Real.sqrt ((u * (1 + u)) + 1 / 4) =
         (1 + 2 * u) / 2 := by
@@ -5463,7 +5464,8 @@ theorem quadratic_spectralAtom_identity {u : ℝ} (hu : 0 ≤ u) :
     have htarget :
         ((1 + 2 * u) / 2) ^ 2 = (u * (1 + u)) + 1 / 4 := by
       ring
-    nlinarith
+    apply (sq_eq_sq₀ hnonneg (by positivity)).mp
+    exact hsquare.trans htarget.symm
   rw [Real.sqrt_div hA, hroot]
   unfold spectralAtom
   field_simp
@@ -5476,11 +5478,11 @@ theorem stieltjesKernel_integrable
   apply (integrable_const (α := ℝ) (μ := μ) (c := (1 : ℝ))).mono'
     (measurable_id.div (measurable_id.add measurable_const)).aestronglyMeasurable
   filter_upwards [hpositive] with t ht
-  have hden : 0 < t + x := by linarith
+  have hden : 0 < t + x := by linarith only [hx, ht]
   have hnonneg : 0 ≤ t / (t + x) := div_nonneg ht.le hden.le
   simp only [Pi.div_apply, Pi.add_apply, id_eq]
   rw [Real.norm_eq_abs, abs_of_nonneg hnonneg]
-  simpa only [ge_iff_le] using (div_le_one hden).mpr (by linarith)
+  simpa only [ge_iff_le] using (div_le_one hden).mpr (by linarith only [hx])
 
 theorem integral_exp_neg_stieltjesPhase_eq_one_sub_two_Gamma
     {r : ℕ} {a : Fin (r + 1) → ℝ} {b : Fin r → ℝ}
@@ -5527,7 +5529,7 @@ theorem integral_exp_neg_stieltjesPhase_eq_one_sub_two_Gamma
           rw [Finset.mul_sum]
           apply Finset.sum_congr rfl
           intro i _
-          ring
+          ac_rfl
 
 end
 
diff --git a/LeanPool/MetricCodes/Representation.lean b/LeanPool/MetricCodes/Representation.lean
index fc46f3a82..d0e8849ee 100644
--- a/LeanPool/MetricCodes/Representation.lean
+++ b/LeanPool/MetricCodes/Representation.lean
@@ -55,9 +55,8 @@ def normalizedSequence (n : ℕ) (hn : 2 ≤ n) : Polynomial.Sequence ℝ where
 theorem normalizedSequence_leadingCoeff_isUnit
     (n : ℕ) (hn : 2 ≤ n) (i : ℕ) :
     IsUnit ((normalizedSequence n hn i).leadingCoeff) := by
-  apply isUnit_iff_ne_zero.mpr
-  exact Polynomial.leadingCoeff_ne_zero.mpr
-    (Polynomial.Sequence.ne_zero (normalizedSequence n hn) i)
+  exact isUnit_iff_ne_zero.mpr (leadingCoeff_ne_zero.mpr (Sequence.ne_zero (normalizedSequence n hn)
+    i))
 
 /-- The basis of real polynomials given by the normalized Gegenbauer sequence. -/
 def normalizedBasis (n : ℕ) (hn : 2 ≤ n) :
@@ -1575,7 +1574,6 @@ theorem rowFactor_pos {r n : ℕ}
   · apply mul_pos
     · apply div_pos
       · have h := linearDenominator_pos hn i
-        have hlam : 0 ≤ (lam i : ℝ) := Nat.cast_nonneg _
         linarith
       · exact linearDenominator_pos hn i
     · positivity
@@ -1614,8 +1612,6 @@ theorem pairFactor_pos {r n : ℕ}
     linarith
   · apply div_pos
     · have hden := sumDenominator_pos hn i j
-      have hi : 0 ≤ (lam i : ℝ) := Nat.cast_nonneg _
-      have hj : 0 ≤ (lam j : ℝ) := Nat.cast_nonneg _
       linarith
     · exact sumDenominator_pos hn i j
 
@@ -1680,7 +1676,6 @@ theorem rowFactor_update_succ_self {r n : ℕ}
   have hsmall : ((lam ell + rowTail ell).factorial : ℝ) ≠ 0 := by positivity
   have hsmalllin : (lam ell : ℝ) + (rowTail ell : ℝ) + 1 ≠ 0 := by
     have hl : 0 ≤ (lam ell : ℝ) := Nat.cast_nonneg _
-    have hr : 0 ≤ (rowTail ell : ℝ) := Nat.cast_nonneg _
     linarith
   have hargbig :
       lam ell + 1 + tailLength n r ell =
@@ -2697,7 +2692,7 @@ theorem weighted_eigenvector_equation (source : I) :
               A.eigenvalue *
                 (Real.sqrt (A.dimension source : ℝ) *
                   A.eigenvector source)
-          ring
+          ac_rfl
 
 /-- The square root of the vertex weight divided by the total weight. -/
 def amplitude (i : I) : ℝ :=
@@ -2925,7 +2920,7 @@ theorem edgeCoefficient_axis_sum [Nonempty I] (source : I) :
   have ha : A.amplitude source ≠ 0 :=
     (A.amplitude_pos source).ne'
   field_simp
-  nlinarith [Real.sq_sqrt A.eigenvalue_pos.le]
+  linarith [Real.sq_sqrt A.eigenvalue_pos.le]
 
 theorem assembledChannel_axis [Nonempty I] (x : X) :
     (A.assembledChannel)ᵀ * A.axis x * A.combinedFibre x =
@@ -3697,10 +3692,10 @@ theorem rowEuler_mul {r n : ℕ} (i : Fin (r + 1))
   congr 1
   · apply Finset.sum_congr rfl
     intro j _
-    ring
+    ac_rfl
   · apply Finset.sum_congr rfl
     intro j _
-    ring
+    ac_rfl
 
 theorem polarization_mul {r n : ℕ} (i j : Fin (r + 1))
     (p q : PolynomialSpace r n) :
@@ -3712,10 +3707,10 @@ theorem polarization_mul {r n : ℕ} (i j : Fin (r + 1))
   congr 1
   · apply Finset.sum_congr rfl
     intro k _
-    ring
+    ac_rfl
   · apply Finset.sum_congr rfl
     intro k _
-    ring
+    ac_rfl
 
 end HigherHarmonicYoung.GelfandTsetlin
 
@@ -4350,15 +4345,7 @@ theorem polynomialInner_sub_left_harmonicLift {N : ℕ}
     SpherePacking.Fischer.polynomialInner N (p - q) s =
       SpherePacking.Fischer.polynomialInner N p s -
         SpherePacking.Fischer.polynomialInner N q s := by
-  rw [sub_eq_add_neg, SpherePacking.Fischer.polynomialInner_add_left]
-  have hneg :
-      SpherePacking.Fischer.polynomialInner N (-q) s =
-        -SpherePacking.Fischer.polynomialInner N q s := by
-    rw [← neg_one_smul ℝ q,
-      SpherePacking.Fischer.polynomialInner_smul_left]
-    ring
-  rw [hneg]
-  rfl
+  exact @SpherePacking.fischer_polynomialInner_sub_left N p q s
 
 theorem simultaneousHarmonicProjection_polarization
     {r n m : ℕ} (i j : Fin (r + 1))
@@ -4615,16 +4602,7 @@ theorem polarization_mul_euler {r n : ℕ}
     (i j : Fin (r + 1)) (p q : PolynomialSpace r n) :
     polarization r n i j (p * q) =
       polarization r n i j p * q + p * polarization r n i j q := by
-  classical
-  simp only [polarization_apply, MvPolynomial.pderiv_mul,
-    mul_add, Finset.sum_add_distrib, Finset.sum_mul, Finset.mul_sum]
-  apply congrArg₂ (· + ·)
-  · apply Finset.sum_congr rfl
-    intro k _
-    ring
-  · apply Finset.sum_congr rfl
-    intro k _
-    ring
+  exact @GelfandTsetlin.polarization_mul r n i j p q
 
 theorem polarization_X_euler {r n : ℕ}
     (i j a : Fin (r + 1)) (k : Fin n) :
@@ -4714,7 +4692,6 @@ theorem polarization_mem_youngMultihomogeneous_transfer
       simp only [Function.update, ↓reduceDIte, Nat.cast_add, Nat.cast_one, polarization_apply,
         add_smul, one_smul, hji, ↓reduceIte, add_zero, add_sub_cancel_right]
     · have hki' : i ≠ k := Ne.symm hki
-      have hkj' : j ≠ k := Ne.symm hkj
       simp only [Function.update, hkj, ↓reduceDIte, polarization_apply, hki, ↓reduceIte, add_zero,
         sub_zero]
 
@@ -5220,13 +5197,7 @@ theorem sum_nullCoordinateCoefficient_mul {m n : ℕ}
 theorem variableIndex_eq_iff {r n : ℕ}
     (i j : Fin (r + 1)) (k l : Fin n) :
     variableIndex i k = variableIndex j l ↔ i = j ∧ k = l := by
-  constructor
-  · intro h
-    have hp : (i, k) = (j, l) :=
-      variableIndex_injective (r := r) (n := n) h
-    exact ⟨congrArg Prod.fst hp, congrArg Prod.snd hp⟩
-  · rintro ⟨rfl, rfl⟩
-    rfl
+  exact @variableIndex_eq_iff_harmonicLift r n i j k l
 
 /-- The null row linear form used in the spherical-code argument. -/
 def nullRowLinearForm {r m n : ℕ} (hn : 2 * m ≤ n)
@@ -5300,7 +5271,7 @@ theorem pderiv_nullSubstitution {r m n : ℕ}
         · rw [Finset.sum_mul]
           apply Finset.sum_congr rfl
           intro x _
-          ring
+          ac_rfl
         · simp only [↓reduceIte, mul_comm, true_and, MonoidWithZeroHom.map_ite_one_zero, ite_mul,
             one_mul, zero_mul, mul_ite, Finset.sum_ite_eq, Finset.mem_univ]
       · have hrow' : h ≠ i := Ne.symm hrow
@@ -5309,7 +5280,7 @@ theorem pderiv_nullSubstitution {r m n : ℕ}
         rw [Finset.sum_mul]
         apply Finset.sum_congr rfl
         intro x _
-        ring
+        ac_rfl
 
 theorem complexTraceOperator_nullSubstitution {r m n : ℕ}
     (hn : 2 * m ≤ n) (i j : Fin (r + 1))
@@ -5360,7 +5331,7 @@ theorem complexPolarization_nullSubstitution {r m n : ℕ}
   rw [nullRowLinearForm, Finset.sum_mul]
   apply Finset.sum_congr rfl
   intro k _
-  ring
+  ac_rfl
 
 theorem nullRowLinearForm_eq_even_add_I_odd {r m n : ℕ}
     (hn : 2 * m ≤ n) (i : Fin (r + 1)) (j : Fin m) :
@@ -5514,12 +5485,7 @@ theorem evenCoordinate_ne_oddCoordinate {r n : ℕ}
 @[simp] theorem variableIndex_eq_iff {r n : ℕ}
     (i j : Fin (r + 1)) (a b : Fin n) :
     variableIndex i a = variableIndex j b ↔ i = j ∧ a = b := by
-  constructor
-  · intro heq
-    have hpair : (i, a) = (j, b) := variableIndex_injective heq
-    exact ⟨congrArg Prod.fst hpair, congrArg Prod.snd hpair⟩
-  · rintro ⟨rfl, rfl⟩
-    rfl
+  exact @variableIndex_eq_iff_harmonicLift r n i j a b
 
 /-- The isotropic variable used in the spherical-code argument. -/
 def isotropicVariable {r n : ℕ} (h : 2 * (r + 1) ≤ n)
@@ -5753,7 +5719,6 @@ theorem rowDerivation_leadingMinor_upper {r n : ℕ}
       intro heq
       have hv := congrArg Fin.val heq
       change j.val = a.val at hv
-      have ha := a.isLt
       omega
     simp only [hne, ↓reduceIte]
 
@@ -5809,7 +5774,6 @@ theorem rowDerivation_leadingMinor_self {r n : ℕ}
       intro heq
       have hv := congrArg Fin.val heq
       change i.val = a.val at hv
-      have ha := a.isLt
       omega
     simp only [hne, ↓reduceIte]
 
@@ -6693,7 +6657,7 @@ theorem ambientCartan_highestWeightPolynomial {r n : ℕ}
                 mul_zero, CharP.cast_eq_zero, zero_smul]
     | succ m =>
         simp [Algebra.smul_def, pow_succ, smul_eq_mul]
-        ring
+        ac_rfl
   · simp only [hpk, ↓reduceIte, smul_eq_mul, mul_zero, nsmul_zero, CharP.cast_eq_zero, zero_smul]
 
 theorem ambientCartan_dominantHighestWeightWitness {r n : ℕ}
diff --git a/LeanPool/MetricCodes/Rigidity.lean b/LeanPool/MetricCodes/Rigidity.lean
index 7349ec86e..9a5b4b54b 100644
--- a/LeanPool/MetricCodes/Rigidity.lean
+++ b/LeanPool/MetricCodes/Rigidity.lean
@@ -191,7 +191,7 @@ theorem actualExteriorRootCreation_contraction_contraction_cubic_anticommute
   split_ifs <;>
     simp only [mul_one, mul_zero, zero_mul, sub_zero, zero_sub, add_zero,
       zero_add, sub_self] <;>
-    noncomm_ring
+    ac_rfl
 
 end
 
@@ -280,14 +280,6 @@ theorem fullRootExteriorBracketUnordered_eq_two_smul
     apply Finset.sum_congr rfl
     intro γ hγ
     exact fullRootExteriorBracket_weightedAtom_swap α β γ
-  have hdiag : ∀ α : PositiveRoot r, F α α = 0 := by
-    intro α
-    dsimp [F]
-    apply Finset.sum_eq_zero
-    intro γ hγ
-    have h := rootStructureConstant_swap α α γ
-    have hz : rootStructureConstant α α γ = 0 := by linarith
-    simp only [hz, zero_smul]
   have hincomparable : ∀ α β : PositiveRoot r,
       (¬ α < β) → (¬ β < α) → F α β = 0 := by
     intro α β hαβ hβα
@@ -941,7 +933,7 @@ theorem fullRootExteriorBracketAtom_single_apply
           rw [hSα, ← hW]
           simp only [smul_smul]
           congr 1
-          ring
+          ac_rfl
         · have hnot : ¬ (α ∈ S.val ∧ β ∈ S.val.erase α ∧
               γ ∉ (S.val.erase α).erase β ∧
               T.val = insert γ ((S.val.erase α).erase β)) := by
@@ -1096,7 +1088,7 @@ theorem smul_operator_anticommutator
   intro x
   simp only [LinearMap.add_apply, Module.End.mul_apply,
     LinearMap.smul_apply, map_smul, smul_smul, smul_add]
-  module
+  ac_rfl
 
 end
 
@@ -1179,7 +1171,7 @@ theorem fullRootExteriorBracketUnordered_square_eq_cubicIncidences
   simp only [I, Fintype.sum_prod_type] at *
   dsimp [A]
   simp_rw [smul_operator_anticommutator]
-  simp only [fullRootExteriorBracketAtom]
+  simp only [fullRootExteriorBracketAtom.eq_1]
   simp_rw [actualExteriorRootCreation_contraction_contraction_cubic_anticommute]
   simp only [smul_add, smul_sub, Finset.sum_add_distrib,
     Finset.sum_sub_distrib, smul_ite, smul_zero]
@@ -2402,26 +2394,7 @@ theorem rootJointHarmonicActionUpperRootStructureCross_apply_coe
                       PolynomialSpace r n)
             else 0
         else 0 := by
-  classical
-  rw [rootJointHarmonicActionUpperRootStructureCross_apply]
-  simp only [Submodule.coe_sum]
-  apply Finset.sum_congr rfl
-  intro α _
-  split_ifs with hα
-  · simp only [Submodule.coe_sum]
-    apply Finset.sum_congr rfl
-    intro β _
-    split_ifs with hβ hadm
-    · simp only [ZeroMemClass.coe_zero]
-    · simp only [Submodule.coe_sum]
-      apply Finset.sum_congr rfl
-      intro γ _
-      split_ifs with hγ
-      · simp only [ZeroMemClass.coe_zero]
-      · simp only [Submodule.coe_smul]
-        rw [rootSwapUpperStructureEdge_coe]
-    · simp only [ZeroMemClass.coe_zero]
-  · simp only [ZeroMemClass.coe_zero]
+  exact @rootJointHarmonicActionUpperRootStructureCross_polynomial_apply r n k lam f S
 
 end
 
@@ -2875,43 +2848,14 @@ theorem weightedChevalleyEilenbergDifferential_comp_self_of_unweighted
       (rootChevalleyEilenbergBoundary r n (k + 1)) = 0) :
     (weightedChevalleyEilenbergDifferential n lam k).comp
       (weightedChevalleyEilenbergDifferential n lam (k + 1)) = 0 := by
-  apply LinearMap.ext
-  intro f
-  apply (rootJointHarmonicPolynomialInclusion_injective lam)
-  change
-    rootJointHarmonicPolynomialInclusion n lam k
-        (weightedChevalleyEilenbergDifferential n lam k
-          (weightedChevalleyEilenbergDifferential n lam (k + 1) f)) =
-      rootJointHarmonicPolynomialInclusion n lam k 0
-  rw [map_zero]
-  have hfirst := LinearMap.congr_fun
-    (rootJointHarmonicPolynomialInclusion_chevalley_intertwines
-      (n := n) (k := k) lam)
-      (weightedChevalleyEilenbergDifferential n lam (k + 1) f)
-  change
-    rootJointHarmonicPolynomialInclusion n lam k
-        (weightedChevalleyEilenbergDifferential n lam k
-          (weightedChevalleyEilenbergDifferential n lam (k + 1) f)) =
-      rootChevalleyEilenbergBoundary r n k
-        (rootJointHarmonicPolynomialInclusion n lam (k + 1)
-          (weightedChevalleyEilenbergDifferential n lam (k + 1) f))
-    at hfirst
-  have hsecond := LinearMap.congr_fun
-    (rootJointHarmonicPolynomialInclusion_chevalley_intertwines
-      (n := n) (k := k + 1) lam) f
-  change
-    rootJointHarmonicPolynomialInclusion n lam (k + 1)
-        (weightedChevalleyEilenbergDifferential n lam (k + 1) f) =
-      rootChevalleyEilenbergBoundary r n (k + 1)
-        (rootJointHarmonicPolynomialInclusion n lam (k + 2) f)
-    at hsecond
-  change
-    rootJointHarmonicPolynomialInclusion n lam k
-        (weightedChevalleyEilenbergDifferential n lam k
-          (weightedChevalleyEilenbergDifferential n lam (k + 1) f)) = 0
-  rw [hfirst, hsecond]
-  exact LinearMap.congr_fun h
-    (rootJointHarmonicPolynomialInclusion n lam (k + 2) f)
+  apply (LinearMap.cancel_left
+    (f := rootJointHarmonicPolynomialInclusion n lam k)
+    (rootJointHarmonicPolynomialInclusion_injective lam)).mp
+  rw [LinearMap.comp_zero, ← LinearMap.comp_assoc,
+    rootJointHarmonicPolynomialInclusion_chevalley_intertwines,
+    LinearMap.comp_assoc,
+    rootJointHarmonicPolynomialInclusion_chevalley_intertwines,
+    ← LinearMap.comp_assoc, h, LinearMap.zero_comp]
 
 end
 
@@ -3389,12 +3333,8 @@ open MetricCodes.Spherical.HigherYoungPenultimateRowProjectedLower
 theorem loweredInternalYoungWeight_raiseWeight
     {r : ℕ} (low : Fin (r + 1) → ℕ) (row : Fin (r + 1)) :
     loweredInternalYoungWeight (raiseWeight low row) row = low := by
-  funext j
-  by_cases hj : j = row
-  · subst j
-    simp only [loweredInternalYoungWeight, raiseWeight, Function.update_self, add_tsub_cancel_right]
-  · simp only [loweredInternalYoungWeight, raiseWeight, Function.update_self, add_tsub_cancel_right,
-      ne_eq, hj, not_false_eq_true, Function.update_of_ne]
+  exact @HigherYoungArbitraryRowLoweringProjectedAxisWitness.loweredInternalYoungWeight_raiseWeight
+    r low row
 
 theorem finrank_raiseWeight_eq_finrank_mul_weylEdgeRatio
     {r n : ℕ} (low : Fin (r + 1) → ℕ) (mu : Fin r → ℕ)
@@ -5386,9 +5326,8 @@ theorem unusedMixedHessian_of_mem_fullYoungComplexPolynomialSpan
             (antiholomorphicDerivative h b p f) +
             antiholomorphicDerivative h a p
               (holomorphicDerivative h b p f)))) := by
-  apply unusedMixedHessian_eq_neg_isotropicDerivative_of_traceFree
-  exact complexTraceOperator_of_mem_fullYoungComplexPolynomialSpan
-    lam hf a b
+  exact unusedMixedHessian_eq_neg_isotropicDerivative_of_traceFree h a b f
+      (complexTraceOperator_of_mem_fullYoungComplexPolynomialSpan lam hf a b)
 
 end
 
@@ -5949,7 +5888,7 @@ theorem ambientPositiveRoot_antiholomorphicDerivative_commutator_apply
           MvPolynomial.algebraMap_eq]
         rw [map_ofNat (MvPolynomial.C :
           ℂ →+* MvPolynomial (Fin ((r + 1) * n)) ℂ) 2]
-        ring
+        ac_rfl
       · simp only [HigherYoungIsotropicComplexTraceDecomposition.antiholomorphicDerivative_apply,
           map_add, Derivation.leibniz, smul_eq_mul, MvPolynomial.derivation_C, mul_zero, add_zero,
           hsq, and_false, ↓reduceIte, sub_self]
@@ -6240,7 +6179,7 @@ theorem sum_ambientSumPositiveRoot_holomorphicDerivative_of_all_positive_root_hi
           rw [hcount]
           congr 1
           push_cast
-          ring
+          ac_rfl
 
 end HigherYoungAmbientSumRootAllHighestHolomorphicEvaluation
 
@@ -7482,7 +7421,7 @@ theorem coeff_signedDiagonalDerivation
   rw [Finset.mul_sum, Finset.sum_mul]
   apply Finset.sum_congr rfl
   intro i _
-  ring
+  ac_rfl
 
 theorem signedDiagonal_weightedHomogeneousComponent
     {ι : Type*} [Finite ι] {m : ℕ}
diff --git a/LeanPool/MetricCodes/RootComplex.lean b/LeanPool/MetricCodes/RootComplex.lean
index 4e29cdb7c..c4857d0d7 100644
--- a/LeanPool/MetricCodes/RootComplex.lean
+++ b/LeanPool/MetricCodes/RootComplex.lean
@@ -845,7 +845,7 @@ theorem naturalDiagonalDerivation_coeff {ι : Type*} [Fintype ι]
   rw [Finset.sum_mul]
   apply Finset.sum_congr rfl
   intro i _
-  ring
+  ac_rfl
 
 theorem conjugatedTotalAmbientCartan_X {r n : ℕ}
     (h : 2 * (r + 1) ≤ n)
@@ -1167,7 +1167,7 @@ theorem complex_smul_youngComplexPair
     Complex.I • polynomialComplexification
       ((q : PolynomialSpace r n) + (q' : PolynomialSpace r n)) = _
   rw [map_add, map_add, smul_add]
-  module
+  ac_rfl
 
 theorem complexRotationCombination_youngComplexPair
     {r n k : ℕ} (lam : Fin (r + 1) → ℕ)
@@ -4590,7 +4590,7 @@ theorem det_orthogonal_denominator
       det_chebyshev_geometric_sum,
       det_quadratic_projective_vandermonde,
       ← upper_gram_pair_product]
-    ring
+    ac_rfl
 
 end OrthogonalDenominator
 
@@ -5495,8 +5495,6 @@ theorem rootJointHarmonicChainDifferential_top_eq_zero
   rw [weightedRootBracketBoundary_apply]
   apply Finset.sum_eq_zero
   intro S _
-  have hzero : rootBracketBoundaryCoefficient S.val T.val = 0 :=
-    rootBracketBoundaryCoefficient_degree_zero S.val T.val
   simp only [Nat.reduceAdd, rootBracketBoundaryCoefficient_degree_zero, ↓reduceDIte]
 
 @[simp] theorem weightedChevalleyEilenbergDifferential_degree_zero
@@ -5540,12 +5538,6 @@ theorem finrank_harmonicYoung_eq_rootExteriorEulerCharacteristic_of_rootExact
   change rootExteriorEulerCharacteristic n lam =
     (Module.finrank ℝ (RootJointHarmonicChain n lam 0) : ℤ) -
       (Module.finrank ℝ (LinearMap.range (d 0)) : ℤ) at h
-  have hzero' :
-      (Module.finrank ℝ
-          (LinearMap.ker (rootDegreeZeroPositiveCochain n lam)) : ℤ) +
-        (Module.finrank ℝ (LinearMap.range (d 0)) : ℤ) =
-          (Module.finrank ℝ (RootJointHarmonicChain n lam 0) : ℤ) := by
-    exact_mod_cast hzero
   have hker := finrank_rootDegreeZeroPositiveCochain_kernel
     (n := n) lam
   omega
diff --git a/LeanPool/MetricCodes/SpectralDecomposition.lean b/LeanPool/MetricCodes/SpectralDecomposition.lean
index 81e0f93a4..39f9a7151 100644
--- a/LeanPool/MetricCodes/SpectralDecomposition.lean
+++ b/LeanPool/MetricCodes/SpectralDecomposition.lean
@@ -28,6 +28,43 @@ namespace Spherical
 
 section
 
+open scoped InnerProductSpace
+
+/-- Spanning ranges detect vanishing of an adjoint composition. -/
+theorem adjoint_composition_eq_zero_of_spanning_ranges
+    {ι E F G : Type*}
+    [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
+    [NormedAddCommGroup F] [InnerProductSpace ℝ F] [FiniteDimensional ℝ F]
+    [AddCommGroup G] [Module ℝ G]
+    (V : ι → Type*) [∀ i, AddCommGroup (V i)] [∀ i, Module ℝ (V i)]
+    (L : (i : ι) → V i →ₗ[ℝ] E)
+    (hspan : (⨆ i, LinearMap.range (L i)) = ⊤)
+    (C : E →ₗ[ℝ] F) (A : G →ₗ[ℝ] F)
+    (hzero : ∀ i x p, ⟪C (L i x), A p⟫_ℝ = 0) :
+    C.adjoint.comp A = 0 := by
+  apply LinearMap.ext
+  intro p
+  change C.adjoint (A p) = 0
+  apply ext_inner_left ℝ
+  intro q
+  rw [inner_zero_right]
+  have hq : q ∈ ⨆ i, LinearMap.range (L i) := by
+    rw [hspan]
+    trivial
+  refine Submodule.iSup_induction
+    (motive := fun z => ⟪z, C.adjoint (A p)⟫_ℝ = 0) _ hq ?_ ?_ ?_
+  · intro i z hz
+    obtain ⟨x, rfl⟩ := hz
+    exact (LinearMap.adjoint_inner_right C _ _).trans (hzero i x p)
+  · exact inner_zero_left _
+  · intro u v hu hv
+    exact (inner_add_left (𝕜 := ℝ) u v (C.adjoint (A p))).trans
+      (by rw [hu, hv, add_zero])
+
+end
+
+section
+
 
 namespace HigherYoungAllRankInvariantPositiveRootKernel
 
@@ -376,11 +413,11 @@ theorem dominantHighestRotationWordSpan_eq_top_of_orthogonalPositiveRootRigidity
         orthogonalPositiveRootDerivation hn α z = 0) →
       z ∈ ambientIsotropicHighestSubmodule hn lam) :
     dominantHighestRotationWordSpan hn lam hdom = ⊤ := by
-  apply (allRankRotationIrreducible_of_orthogonalPositiveRootRigidity
-    hn lam hdom N hnil hhighest
-    (dominantHighestRotationWordSpan hn lam hdom)
-    (dominantHighestRotationWordSpan_rotationInvariant hn lam hdom)).resolve_left
-  exact dominantHighestRotationWordSpan_ne_bot hn lam hdom
+  exact Or.resolve_left
+      (allRankRotationIrreducible_of_orthogonalPositiveRootRigidity hn lam hdom N hnil hhighest
+        (dominantHighestRotationWordSpan hn lam hdom)
+          (dominantHighestRotationWordSpan_rotationInvariant hn lam hdom))
+      (dominantHighestRotationWordSpan_ne_bot hn lam hdom)
 
 theorem orthogonalPositiveRootWord_eq_zero_of_mem_fullYoungComplexPolynomialSpan
     {r n : ℕ} (hn : 2 * (r + 1) ≤ n)
@@ -472,12 +509,8 @@ theorem finiteInterlacing_two_mul_row_count_le
 theorem canonicalEdge_loweredInternalYoungWeight_raiseWeight
     {r : ℕ} (low : Fin (r + 1) → ℕ) (row : Fin (r + 1)) :
     loweredInternalYoungWeight (raiseWeight low row) row = low := by
-  funext j
-  by_cases hj : j = row
-  · subst j
-    simp only [loweredInternalYoungWeight, raiseWeight, Function.update_self, add_tsub_cancel_right]
-  · simp only [loweredInternalYoungWeight, raiseWeight, Function.update_self, add_tsub_cancel_right,
-      ne_eq, hj, not_false_eq_true, Function.update_of_ne]
+  exact @HigherYoungArbitraryRowLoweringProjectedAxisWitness.loweredInternalYoungWeight_raiseWeight
+    r low row
 
 theorem canonicalEdge_raiseWeight_strictly_removable
     {r : ℕ} (low : Fin (r + 1) → ℕ) (hdom : Antitone low)
@@ -3190,7 +3223,7 @@ theorem cartanCharacteristicInterpolationBasis_mul_eq_nodal
   rw [Lagrange.basis_eq_prod_sub_inv_mul_nodal_div (Finset.mem_univ i),
     ← Lagrange.nodal_erase_eq_nodal_div (Finset.mem_univ i),
     Lagrange.nodal_eq_mul_nodal_erase (Finset.mem_univ i)]
-  ring
+  ac_rfl
 
 theorem signedAmbientCharacteristicNodes_injective
     {r n : ℕ} {lam : Fin (r + 1) → ℕ} {mu : Fin r → ℕ}
@@ -3667,9 +3700,9 @@ theorem allRankCartanCharacteristicProjector_raise_channel
     (p : HarmonicYoungSpace (n := n) (raiseWeight target row)) :
     allRankCartanCharacteristicProjector target selected (A p) =
       if selected = (row, true) then A p else 0 := by
-  apply allRankCartanCharacteristicProjector_apply_eigenvector
-    target mu h selected (row, true) (A p)
-  exact gtRelativeCasimir_raise_channel target row A hA p
+  exact allRankCartanCharacteristicProjector_apply_eigenvector target mu h selected (row, true) (A
+    p)
+      (gtRelativeCasimir_raise_channel target row A hA p)
 
 theorem allRankCartanCharacteristicProjector_lower_channel
     {r n : ℕ} (target : Fin (r + 1) → ℕ)
@@ -3686,9 +3719,9 @@ theorem allRankCartanCharacteristicProjector_lower_channel
     (p : HarmonicYoungSpace (n := n) source) :
     allRankCartanCharacteristicProjector target selected (A p) =
       if selected = (row, false) then A p else 0 := by
-  apply allRankCartanCharacteristicProjector_apply_eigenvector
-    target mu h selected (row, false) (A p)
-  exact gtRelativeCasimir_lower_channel target source row hsource A hA p
+  exact allRankCartanCharacteristicProjector_apply_eigenvector target mu h selected (row, false) (A
+    p)
+      (gtRelativeCasimir_lower_channel target source row hsource A hA p)
 
 theorem allRankCartanCharacteristicProjector_youngClebschLower
     {r n : ℕ} (target : Fin (r + 1) → ℕ)
@@ -4232,11 +4265,17 @@ theorem canonicalGelfandTsetlinFibre_selectedClebschCompression_iff_reverse
               (reverseInterlacingHarmonicBranch low mu hlow p))) =
       plusProbability (n + 1) low mu row •
         reverseInterlacingHarmonicBranch low mu hlow p := by
-  have hscalar := (canonicalGelfandTsetlinFibre_phase_pos
-    low mu hlow hgram).ne'
-  simp only [canonicalGelfandTsetlinFibre_apply, map_smul, smul_smul]
-  rw [mul_comm (plusProbability (n + 1) low mu row), mul_smul,
-    smul_right_inj hscalar]
+  let A := gtYoungAxisTensor low
+    (EuclideanSpace.basisFun (Fin (n + 1)) ℝ (Fin.last n))
+  let T := A.adjoint.comp ((gtSelectedRowClebschRangeProjector low row).comp A)
+  let c : ℝ := (Real.sqrt (canonicalGelfandTsetlinFischerGram
+    low mu hlow hgram))⁻¹
+  let d := plusProbability (n + 1) low mu row
+  let x := reverseInterlacingHarmonicBranch low mu hlow p
+  have hc : c ≠ 0 := (canonicalGelfandTsetlinFibre_phase_pos low mu hlow hgram).ne'
+  rw [canonicalGelfandTsetlinFibre_apply]
+  change T (c • x) = d • (c • x) ↔ T x = d • x
+  rw [map_smul, smul_comm d c, smul_right_inj hc]
 
 theorem gtSelectedRowClebschRangeProjector_axisCompression_mem_reverseBranch_range
     {r n : ℕ} (low : Fin (r + 2) → ℕ)
@@ -4434,7 +4473,7 @@ theorem gtSelectedProjectorCompression_eq_plusProbability_of_minor
             (plusProbability (n + 1) lam mu row •
               (q : PolynomialSpace r n))
       rw [SpherePacking.Fischer.polynomialInner_smul_right]
-      ring
+      ac_rfl
 
 /-- The selected Clebsch range projector compressed along tensoring with the last coordinate
 axis. -/
@@ -4913,7 +4952,6 @@ theorem orthogonalTensorPieriCoefficient_padded_zero_lt
     (n r j : ℕ) (hj : j < r) :
     orthogonalTensorPieriCoefficient n (-(r : ℤ)) (j : ℤ) = 0 := by
   unfold orthogonalTensorPieriCoefficient
-  have hj' : (j : ℤ) < (r : ℤ) := by exact_mod_cast hj
   have hfirst : -(r : ℤ) + (j : ℤ) < 0 := by omega
   have hsecond : -(r : ℤ) - (j : ℤ) - 2 < 0 := by omega
   rw [orthogonalCompleteSymmetricCoefficient_of_neg n hfirst,
@@ -6102,7 +6140,7 @@ theorem orthogonalTensorPieriNextColumn_last
   rw [harg]
   convert orthogonalTensorPieriCoefficient_padded_zero_succ n r using 1
   push_cast
-  ring
+  ac_rfl
 
 theorem sum_det_updateRow_orthogonalTensorPieriSignedShiftMatrix
     {r : ℕ} (n : ℕ) (lam : Fin (r + 1) → ℕ)
@@ -6537,7 +6575,7 @@ theorem zeroRowTensorIsometryEquiv_symm_tensorCasimir_intertwine
           ((zeroRowTensorIsometryEquiv (n := n) lam).symm.toLinearMap y) :=
     LinearMap.congr_fun
       (zeroRowTensorIsometryEquiv_symm_rotation_intertwine lam a b) y
-  simp only [LinearMap.comp_apply, gtTensorCasimir, LinearMap.smul_apply,
+  simp only [LinearMap.comp_apply, gtTensorCasimir.eq_1, LinearMap.smul_apply,
     LinearMap.sum_apply, LinearMap.neg_apply, map_smul, map_sum, map_neg]
   simp_rw [hrot]
 
@@ -6552,7 +6590,7 @@ theorem zeroRowTensorIsometryEquiv_symm_relativeCasimir_intertwine
   have h := LinearMap.congr_fun
     (zeroRowTensorIsometryEquiv_symm_tensorCasimir_intertwine lam) x
   simp only [LinearMap.comp_apply] at h
-  simp only [gtRelativeCasimir, LinearMap.comp_apply, LinearMap.smul_apply,
+  simp only [gtRelativeCasimir.eq_1, LinearMap.comp_apply, LinearMap.smul_apply,
     LinearMap.sub_apply, LinearMap.id_apply, map_smul, map_sub]
   rw [allRankCasimirEigenvalue_appendZeroWeight, h]
 
@@ -8529,27 +8567,15 @@ theorem appendedPaddedPieriLower_adjoint_originalPaddedSelectedAxisTensor_eq_zer
   let C := (normalizedPaddedPieriLower (n := n + 1)
     (appendZeroWeight lam) hdominant (Fin.last (r + 2)) hsource).toLinearMap
   let A := originalPaddedSelectedAxisTensor lam mu h hgram
-  apply LinearMap.ext
-  intro p
-  change C.adjoint (A p) = 0
-  apply ext_inner_left ℝ
-  intro q
-  rw [inner_zero_right]
-  have hsum := canonicalFullBranch_sum_projection source hn hsource q
-  rw [← hsum, sum_inner]
-  apply Finset.sum_eq_zero
-  intro nu _
-  let F := (canonicalFullBranchFibre source hn nu).toLinearMap
-  obtain ⟨z, hz⟩ := Submodule.starProjection_apply_mem
-    (LinearMap.range F) q
-  change F z = (LinearMap.range F).starProjection q at hz
-  rw [← hz]
-  calc
-    ⟪F z, C.adjoint (A p)⟫_ℝ = ⟪C (F z), A p⟫_ℝ :=
-      LinearMap.adjoint_inner_right C (F z) (A p)
-    _ = ⟪A p, C (F z)⟫_ℝ := real_inner_comm (A p) (C (F z))
-    _ = 0 := originalAppendedFullBranchAxis_inner_eq_zero
-      lam mu h hgram hdominant hsource hn nu p z
+  apply adjoint_composition_eq_zero_of_spanning_ranges
+    (fun nu : FullBranchWeight source =>
+      HarmonicYoungSpace (n := n) (fullBranchSignature nu))
+    (fun nu => (canonicalFullBranchFibre source hn nu).toLinearMap)
+    (canonicalFullBranch_iSup_range_eq_top_unconditional source hn hsource) C A
+  intro nu x p
+  exact (real_inner_comm _ _).trans
+    (originalAppendedFullBranchAxis_inner_eq_zero
+      lam mu h hgram hdominant hsource hn nu p x)
 
 theorem appendedPaddedPieriLower_originalCanonicalAxis_orthogonal
     {r n : ℕ} (lam : Fin (r + 2) → ℕ)
diff --git a/LeanPool/MetricCodes/Weyl.lean b/LeanPool/MetricCodes/Weyl.lean
index c8bca76e3..a56b7b0f5 100644
--- a/LeanPool/MetricCodes/Weyl.lean
+++ b/LeanPool/MetricCodes/Weyl.lean
@@ -1353,7 +1353,6 @@ theorem weylTargetExponent_sub_reversePermutationExponent_apply
     (weylTargetExponent lam - reversePermutationExponent σ) i =
       (weylShift lam σ i).toNat := by
   have h := hσ i
-  have hi := i.isLt
   have hσi := (σ i).isLt
   simp only [reversePermutationExponent_apply, weylTargetExponent_apply, tsub_le_iff_right,
     Finsupp.coe_tsub, Pi.sub_apply, weylShift] at h ⊢
@@ -1652,7 +1651,7 @@ theorem alternatingGramKoszulCoefficient_eq_orthogonalJacobiTrudiDimension
                 (r + j.val + 2))).coeff (weylTargetExponent lam) := by
       rw [hdenominator]
       congr 2
-      ring
+      ac_rfl
     _ = orthogonalJacobiTrudiDimension n lam :=
       coeff_ambientBinomialPolynomial_mul_orthogonalDenominator lam hB
 
@@ -1673,7 +1672,7 @@ theorem risingFactorProduct_eq_ascFactorial (M q : ℕ) :
   apply Finset.prod_congr rfl
   intro a _
   simp only [Nat.cast_add, Nat.cast_one]
-  ring
+  ac_rfl
 
 theorem risingFactorProduct_pos (M q : ℕ) :
     0 < risingFactorProduct M q := by
@@ -1989,7 +1988,7 @@ def centeredProduct (x : ℝ) (j : ℕ) : ℝ :=
 @[simp] theorem centeredProduct_zero (x : ℝ) :
     centeredProduct x 0 = (x - 1) * x := by
   norm_num [centeredProduct, Finset.prod_range_succ]
-  ring
+  ac_rfl
 
 theorem centeredProduct_succ (x : ℝ) (j : ℕ) :
     centeredProduct x (j + 1) =
@@ -2920,7 +2919,7 @@ theorem orthogonalJacobiTrudiMatrix_entry_of_factored_transport
     (risingFactorProduct_pos _ _).ne'
   unfold orthogonalRowScale
   field_simp [hD]
-  nlinarith [hfact]
+  linarith [hfact]
 
 /-- The difference of the two reflected factorial products in an orthogonal determinant entry. -/
 def orthogonalReflectedBracket (n z : ℝ) (r j : ℕ) : ℝ :=
@@ -4500,7 +4499,9 @@ def rootJointHarmonicIncludedDescendingFischerLaplacian {r : ℕ}
     (f : RootJointHarmonicChain n lam k)
     (S : AdmissibleRootWedge lam k) :
     rootJointHarmonicIncludedDescendingFischerLaplacian n lam k f S =
-      includedDescendingPositiveRootFischerLaplacian n lam S (f S) := rfl
+      includedDescendingPositiveRootFischerLaplacian n lam S (f S) := by
+  simp only [rootJointHarmonicIncludedDescendingFischerLaplacian,
+    LinearMap.pi_apply, LinearMap.comp_apply, LinearMap.proj_apply]
 
 theorem rootJointHarmonicIncludedDescendingFischerLaplacian_energy_ge_self
     {r n k : ℕ} (lam : Fin (r + 1) → ℕ)
@@ -4698,7 +4699,9 @@ def rootJointHarmonicHodgeDiagonal {r : ℕ}
     rootJointHarmonicHodgeDiagonal n lam k f S =
       includedDescendingPositiveRootFischerLaplacian n lam S (f S) +
         includedNondescendingPositiveRootFischerLaplacian n lam S (f S) +
-        excludedActivePositiveRootFischerLaplacian n lam S (f S) := rfl
+        excludedActivePositiveRootFischerLaplacian n lam S (f S) := by
+  simp only [rootJointHarmonicHodgeDiagonal, LinearMap.pi_apply,
+    LinearMap.comp_apply, LinearMap.proj_apply, LinearMap.add_apply]
 
 theorem includedNondescendingPositiveRootFischerLaplacian_energy
     {r n k : ℕ} (lam : Fin (r + 1) → ℕ)
@@ -7534,10 +7537,6 @@ theorem rootActionCoboundary_eq_exteriorCreation_zeroExtension
   · have hcard : (S.val.erase α).card = k := by
       rw [Finset.card_erase_of_mem hα, S.property]
       omega
-    have hwedge : rootWedgeErase S α hα =
-        (⟨S.val.erase α, hcard⟩ : RootWedge r k) := by
-      apply Subtype.ext
-      rfl
     simp only [hα, ↓reduceDIte, actualExteriorRootCreation_apply,
       ↓reduceIte, rootPolynomialChainZeroExtension_apply,
       map_smul]
@@ -8070,7 +8069,7 @@ theorem fullRootExteriorPolynomialAction_orderedBracketCoboundary_commute
           actualExteriorRootCreation (PolynomialSpace r n) α) *
         (actualExteriorRootCreation (PolynomialSpace r n) β *
           actualExteriorRootContraction (PolynomialSpace r n) δ) := by
-          noncomm_ring
+          ac_rfl
     _ = (actualExteriorRootCreation (PolynomialSpace r n) α *
           fullRootExteriorPolynomialAction r n γ) *
         (actualExteriorRootCreation (PolynomialSpace r n) β *
@@ -8080,7 +8079,7 @@ theorem fullRootExteriorPolynomialAction_orderedBracketCoboundary_commute
         (fullRootExteriorPolynomialAction r n γ *
           actualExteriorRootCreation (PolynomialSpace r n) β) *
           actualExteriorRootContraction (PolynomialSpace r n) δ := by
-          noncomm_ring
+          ac_rfl
     _ = actualExteriorRootCreation (PolynomialSpace r n) α *
         (actualExteriorRootCreation (PolynomialSpace r n) β *
           fullRootExteriorPolynomialAction r n γ) *
@@ -8090,7 +8089,7 @@ theorem fullRootExteriorPolynomialAction_orderedBracketCoboundary_commute
         actualExteriorRootCreation (PolynomialSpace r n) β *
         (fullRootExteriorPolynomialAction r n γ *
           actualExteriorRootContraction (PolynomialSpace r n) δ) := by
-          noncomm_ring
+          ac_rfl
     _ = actualExteriorRootCreation (PolynomialSpace r n) α *
         actualExteriorRootCreation (PolynomialSpace r n) β *
         (actualExteriorRootContraction (PolynomialSpace r n) δ *
@@ -8208,26 +8207,7 @@ theorem rootJointHarmonicActionLowerRootStructureCross_apply_coe
                       PolynomialSpace r n)
             else 0
         else 0 := by
-  classical
-  rw [rootJointHarmonicActionLowerRootStructureCross_apply]
-  simp only [Submodule.coe_sum]
-  apply Finset.sum_congr rfl
-  intro α _
-  split_ifs with hα
-  · simp only [Submodule.coe_sum]
-    apply Finset.sum_congr rfl
-    intro β _
-    split_ifs with hβ hadm
-    · simp only [ZeroMemClass.coe_zero]
-    · simp only [Submodule.coe_sum]
-      apply Finset.sum_congr rfl
-      intro γ _
-      split_ifs with hγ
-      · simp only [ZeroMemClass.coe_zero]
-      · simp only [Submodule.coe_smul]
-        rw [rootSwapLowerStructureEdge_coe]
-    · simp only [ZeroMemClass.coe_zero]
-  · simp only [ZeroMemClass.coe_zero]
+  exact @rootJointHarmonicActionLowerRootStructureCross_polynomial_apply r n k lam f S
 
 end