From d3778f74cb9c4b59b6b7b2995b2b40ce9f19a62b Mon Sep 17 00:00:00 2001
From: Eduardo Nava Hernandez <naype888@gmail.com>
Date: Wed, 16 Sep 2026 11:54:00 -0600
Subject: [PATCH 01/10] feat(PhyslibAlpha): Fisher-Rao metric and Cramer-Rao
 bound for algebraic states

---
 PhyslibAlpha.lean                             |   1 +
 .../InformationGeometry/FisherRao.lean        | 216 ++++++++++++++++++
 2 files changed, 217 insertions(+)
 create mode 100644 PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean

diff --git a/PhyslibAlpha.lean b/PhyslibAlpha.lean
index 6102e0c8d..15b69d5e4 100644
--- a/PhyslibAlpha.lean
+++ b/PhyslibAlpha.lean
@@ -229,4 +229,5 @@ public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.JordanStatistics
 public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.JordanPositivity
 public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.JordanCFC
 public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.JordanSpecial
+public import PhyslibAlpha.AlgebraicFramework.InformationGeometry.FisherRao
 public import PhyslibAlpha.Relativity.General.Schwarzschild.IncompressibleSphere
diff --git a/PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean b/PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean
new file mode 100644
index 000000000..61fb7366c
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean
@@ -0,0 +1,216 @@
+/-
+Copyright (c) 2026 Eduardo Nava Hernandez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava Hernandez
+-/
+import Mathlib.Analysis.InnerProductSpace.Basic
+import Mathlib.Algebra.BigOperators.Finprod
+import Mathlib.Algebra.Order.Field.Basic
+import Mathlib.Algebra.BigOperators.Group.Finset.Basic
+import Mathlib.Topology.Algebra.InfiniteSum.Basic
+import PhyslibAlpha.Basic
+
+/-!
+# Fisher--Rao metric and Cramer--Rao bound (finite case)
+
+Defines the Fisher--Rao inner product on the open probability simplex
+and the classical Cramer--Rao lower bound on estimator variance,
+for finite sample spaces. All definitions reuse Mathlib primitives
+(`Finset.sum`, `div`, `sq`). The Cramer--Rao proof is Cauchy--Schwarz
+on the weighted inner product -- the same obstruction kernel as
+Robertson--Schrödinger in algebraic uncertainty frameworks.
+
+## Main definitions
+
+* `PhyslibAlpha.OpenSimplex` — a point on the open probability simplex.
+* `PhyslibAlpha.OpenSimplex.fisherRaoInner` — the Fisher--Rao inner product $g_p(u,v)$.
+* `PhyslibAlpha.fisherInfo` — classical Fisher information $I(θ)$ for a parametric family.
+* `PhyslibAlpha.cramerRao` — the Cramer--Rao lower bound $\mathrm{Var}(T) \ge 1/I(θ)$.
+
+## References
+
+* C. R. Rao, *Information and the accuracy attainable in the estimation
+  of statistical parameters*, Bull. Calcutta Math. Soc. 37, 81--91 (1945).
+* H. Cramer, *Mathematical Methods of Statistics*, Princeton (1946).
+-/
+
+noncomputable section
+
+open Finset BigOperators
+
+variable {α : Type*} [Fintype α]
+
+namespace PhyslibAlpha
+
+/-! ## I. Open probability simplex -/
+
+/-- A point on the open probability simplex: strictly positive weights summing to 1. -/
+structure OpenSimplex (α : Type*) [Fintype α] where
+  val : α → ℝ
+  pos : ∀ i, 0 < val i
+  sum_one : ∑ i : α, val i = 1
+
+namespace OpenSimplex
+
+variable (p : OpenSimplex α)
+
+/-- Every coordinate is nonzero (convenience lemma). -/
+theorem val_ne_zero (i : α) : p.val i ≠ 0 := ne_of_gt (p.pos i)
+
+/-- Every coordinate is nonneg (convenience lemma). -/
+theorem val_nonneg (i : α) : 0 ≤ p.val i := le_of_lt (p.pos i)
+
+/-! ## II. Fisher--Rao inner product -/
+
+/-- The Fisher--Rao inner product at $p \in \Delta^\circ(\alpha)$:
+    `g_p(u, v) = \sum_{i \in \alpha} \frac{u_i v_i}{p_i}` -/
+def fisherRaoInner (u v : α → ℝ) : ℝ :=
+  ∑ i : α, u i * v i / p.val i
+
+/-- The Fisher--Rao quadratic form (squared norm). -/
+def fisherRaoSq (u : α → ℝ) : ℝ := p.fisherRaoInner u u
+
+/-- $g_p(u, u) \ge 0$ for all $u$. -/
+theorem fisherRaoSq_nonneg (u : α → ℝ) : 0 ≤ p.fisherRaoSq u := by
+  apply Finset.sum_nonneg
+  intro i _
+  apply div_nonneg
+  · exact mul_self_nonneg (u i)
+  · exact p.val_nonneg i
+
+/-- $g_p(u, u) = 0 \iff u = 0$ (positive definiteness). -/
+theorem fisherRaoSq_eq_zero_iff (u : α → ℝ) :
+    p.fisherRaoSq u = 0 ↔ u = 0 := by
+  constructor
+  · intro h
+    have hnn : ∀ i ∈ Finset.univ, 0 ≤ u i * u i / p.val i := by
+      intro i _
+      exact div_nonneg (mul_self_nonneg _) (p.val_nonneg i)
+    have hall := Finset.sum_eq_zero_iff_of_nonneg hnn |>.mp h
+    ext i
+    simp only [Pi.zero_apply]
+    have hi := hall i (Finset.mem_univ i)
+    rcases div_eq_zero_iff.mp hi with hmul | habs
+    · exact mul_self_eq_zero.mp hmul
+    · exact absurd habs (p.val_ne_zero i)
+  · intro h
+    simp [fisherRaoSq, fisherRaoInner, h]
+
+/-- Symmetry: $g_p(u, v) = g_p(v, u)$. -/
+theorem fisherRaoInner_comm (u v : α → ℝ) :
+    p.fisherRaoInner u v = p.fisherRaoInner v u := by
+  simp only [fisherRaoInner]
+  congr 1; ext i; ring
+
+/-- Expansion of the Fisher--Rao norm under linear combinations. -/
+theorem fisherRaoInner_sub_smul (u v : α → ℝ) (t : ℝ) :
+    p.fisherRaoSq (fun i => u i - t * v i) =
+      p.fisherRaoSq u - 2 * t * p.fisherRaoInner u v +
+        t ^ 2 * p.fisherRaoSq v := by
+  simp only [fisherRaoSq, fisherRaoInner]
+  rw [show (∑ i : α, (u i - t * v i) * (u i - t * v i) / p.val i) =
+    (∑ i : α, u i * u i / p.val i) - 2 * t * (∑ i : α, u i * v i / p.val i) +
+      t ^ 2 * (∑ i : α, v i * v i / p.val i) from by
+    rw [Finset.mul_sum, Finset.mul_sum]
+    simp only [← Finset.sum_add_distrib, ← Finset.sum_sub_distrib]
+    apply Finset.sum_congr rfl; intro i _; ring]
+
+/-- Cauchy--Schwarz for the Fisher--Rao inner product:
+    $g_p(u, v)^2 \le g_p(u, u) \cdot g_p(v, v)$. -/
+theorem fisherRao_cauchy_schwarz (u v : α → ℝ) :
+    p.fisherRaoInner u v ^ 2 ≤ p.fisherRaoSq u * p.fisherRaoSq v := by
+  by_cases hv : p.fisherRaoSq v = 0
+  · rw [p.fisherRaoSq_eq_zero_iff] at hv
+    simp [fisherRaoInner, fisherRaoSq, hv]
+  · have hvpos : 0 < p.fisherRaoSq v :=
+      lt_of_le_of_ne (p.fisherRaoSq_nonneg v) (Ne.symm hv)
+    set t := p.fisherRaoInner u v / p.fisherRaoSq v with ht_def
+    have key := p.fisherRaoSq_nonneg (fun i => u i - t * v i)
+    rw [p.fisherRaoInner_sub_smul u v t] at key
+    have ht2 : t * p.fisherRaoInner u v =
+        p.fisherRaoInner u v ^ 2 / p.fisherRaoSq v := by
+      rw [ht_def]; field_simp
+    have ht3 : t ^ 2 * p.fisherRaoSq v =
+        p.fisherRaoInner u v ^ 2 / p.fisherRaoSq v := by
+      rw [ht_def]; field_simp
+    rw [show 2 * t * p.fisherRaoInner u v =
+        2 * (p.fisherRaoInner u v ^ 2 / p.fisherRaoSq v) from by
+      linarith [ht2]] at key
+    rw [ht3] at key
+    have h1 : p.fisherRaoInner u v ^ 2 / p.fisherRaoSq v ≤ p.fisherRaoSq u :=
+      by linarith
+    rwa [div_le_iff₀ hvpos] at h1
+
+end OpenSimplex
+
+/-! ## III. Classical Fisher information -/
+
+/-- Classical Fisher information for a 1-parameter discrete family
+    $\theta \mapsto p(\cdot|\theta)$, given as $I(\theta) = \sum_i \frac{(\partial p_i / \partial \theta)^2}{p_i(\theta)}$.
+    This equals the Fisher--Rao squared norm of the score tangent vector. -/
+def fisherInfo (p : α → ℝ) (dp : α → ℝ) : ℝ :=
+  ∑ i : α, dp i ^ 2 / p i
+
+/-- Fisher information is nonneg. -/
+theorem fisherInfo_nonneg (p dp : α → ℝ) (hpos : ∀ i, 0 < p i) :
+    0 ≤ fisherInfo p dp := by
+  apply Finset.sum_nonneg
+  intro i _
+  exact div_nonneg (sq_nonneg _) (le_of_lt (hpos i))
+
+/-- Fisher information equals the Fisher--Rao squared norm of the
+    derivative vector: $I(\theta) = \|dp/d\theta\|^2_{\mathrm{FR}}$. -/
+theorem fisherInfo_eq_fisherRaoSq (q : OpenSimplex α) (dp : α → ℝ) :
+    fisherInfo q.val dp = q.fisherRaoSq dp := by
+  simp only [fisherInfo, OpenSimplex.fisherRaoSq, OpenSimplex.fisherRaoInner]
+  congr 1; ext i; ring
+
+/-! ## IV. Cramer--Rao lower bound -/
+
+/-- Weighted expectation $\mathbb{E}_p[f] = \sum_i f(i) p(i)$. -/
+def expect (p : α → ℝ) (f : α → ℝ) : ℝ :=
+  ∑ i : α, f i * p i
+
+/-- Weighted variance $\mathrm{Var}_p(f) = \mathbb{E}_p[(f - \mathbb{E}_p[f])^2]$. -/
+def variance (p : α → ℝ) (f : α → ℝ) : ℝ :=
+  expect p (fun i => (f i - expect p f) ^ 2)
+
+/-- **Cramer--Rao lower bound.** For a parametric family with
+    all $p_i > 0$ and an estimator $T$ satisfying the unbiasedness
+    derivative condition $\sum_i T(i) \cdot (\partial p_i / \partial \theta) = 1$, we have
+
+      $\mathrm{Var}_p(T) \ge 1 / I(\theta)$
+
+    where $I(\theta) = \mathrm{fisherInfo}(p, dp)$. The proof is a single application
+    of Cauchy--Schwarz on the Fisher--Rao inner product to the pair
+    $(T - \mathbb{E}[T], dp/p)$. -/
+theorem cramerRao (p dp : α → ℝ) (T : α → ℝ)
+    (hpos : ∀ i, 0 < p i)
+    (hsum : ∑ i : α, p i = 1)
+    (hdsum : ∑ i : α, dp i = 0)
+    (hunbiased : ∑ i : α, T i * dp i = 1)
+    (hI : 0 < fisherInfo p dp) :
+    1 / fisherInfo p dp ≤ variance p T := by
+  let q : OpenSimplex α := ⟨p, hpos, hsum⟩
+  let a : α → ℝ := fun i => (T i - expect p T) * p i
+  have cs := q.fisherRao_cauchy_schwarz a dp
+  have hpne : ∀ i, p i ≠ 0 := fun i => ne_of_gt (hpos i)
+  have inner_eq : q.fisherRaoInner a dp = 1 := by
+    simp only [OpenSimplex.fisherRaoInner, a]
+    conv => lhs; arg 2; ext i; rw [show ((T i - expect p T) * p i) * dp i / p i =
+        (T i - expect p T) * dp i from by field_simp [hpne i]]
+    simp_rw [sub_mul]
+    rw [Finset.sum_sub_distrib, ← Finset.mul_sum, hdsum, mul_zero, sub_zero]
+    exact hunbiased
+  have sq_u_eq : q.fisherRaoSq a = variance p T := by
+    simp only [OpenSimplex.fisherRaoSq, OpenSimplex.fisherRaoInner, variance, expect, a]
+    apply Finset.sum_congr rfl; intro i _
+    simp only [q]; field_simp [hpne i]
+  have sq_v_eq : q.fisherRaoSq dp = fisherInfo p dp := by
+    simp only [OpenSimplex.fisherRaoSq, OpenSimplex.fisherRaoInner, fisherInfo]
+    apply Finset.sum_congr rfl; intro i _; ring
+  rw [inner_eq, sq_u_eq, sq_v_eq] at cs
+  rw [one_pow] at cs
+  rwa [div_le_iff₀ hI]
+
+end PhyslibAlpha

From 5f21d9867f8fbb06364d27d34fd701bcb28cc5bc Mon Sep 17 00:00:00 2001
From: Eduardo Nava Hernandez <naype888@gmail.com>
Date: Wed, 16 Sep 2026 12:05:40 -0600
Subject: [PATCH 02/10] style: wrap docstring line to satisfy 100-char linter
 rule

---
 .../AlgebraicFramework/InformationGeometry/FisherRao.lean      | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean b/PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean
index 61fb7366c..18995e088 100644
--- a/PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean
+++ b/PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean
@@ -146,7 +146,8 @@ end OpenSimplex
 /-! ## III. Classical Fisher information -/
 
 /-- Classical Fisher information for a 1-parameter discrete family
-    $\theta \mapsto p(\cdot|\theta)$, given as $I(\theta) = \sum_i \frac{(\partial p_i / \partial \theta)^2}{p_i(\theta)}$.
+    $\theta \mapsto p(\cdot|\theta)$, given as
+    $I(\theta) = \sum_i \frac{(\partial p_i / \partial \theta)^2}{p_i(\theta)}$.
     This equals the Fisher--Rao squared norm of the score tangent vector. -/
 def fisherInfo (p : α → ℝ) (dp : α → ℝ) : ℝ :=
   ∑ i : α, dp i ^ 2 / p i

From 7db19c75101e0e2575017d0628aacf751a25e83c Mon Sep 17 00:00:00 2001
From: Eduardo Nava Hernandez <naype888@gmail.com>
Date: Wed, 16 Sep 2026 12:13:33 -0600
Subject: [PATCH 03/10] fix(PhyslibAlpha): add module header to FisherRao.lean
 for module system compliance

---
 .../InformationGeometry/FisherRao.lean             | 14 ++++++++------
 1 file changed, 8 insertions(+), 6 deletions(-)

diff --git a/PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean b/PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean
index 18995e088..396bad8d8 100644
--- a/PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean
+++ b/PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean
@@ -3,12 +3,14 @@ Copyright (c) 2026 Eduardo Nava Hernandez. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava Hernandez
 -/
-import Mathlib.Analysis.InnerProductSpace.Basic
-import Mathlib.Algebra.BigOperators.Finprod
-import Mathlib.Algebra.Order.Field.Basic
-import Mathlib.Algebra.BigOperators.Group.Finset.Basic
-import Mathlib.Topology.Algebra.InfiniteSum.Basic
-import PhyslibAlpha.Basic
+module
+
+public import Mathlib.Analysis.InnerProductSpace.Basic
+public import Mathlib.Algebra.BigOperators.Finprod
+public import Mathlib.Algebra.Order.Field.Basic
+public import Mathlib.Algebra.BigOperators.Group.Finset.Basic
+public import Mathlib.Topology.Algebra.InfiniteSum.Basic
+public import PhyslibAlpha.Basic
 
 /-!
 # Fisher--Rao metric and Cramer--Rao bound (finite case)

From f21aa8066adcac308289db646b980f997085b16c Mon Sep 17 00:00:00 2001
From: Eduardo Nava-Hernandez <naype888@gmail.com>
Date: Thu, 17 Sep 2026 07:42:37 -0600
Subject: [PATCH 04/10] feat(PhyslibAlpha): formalize the finite path spectral
 gap

Co-authored-by: Claude Opus 4.8 <no-reply+claude-opus-4-8@anthropic.com>
---
 .../HilbertSpace/PathSpectralGap.lean         |  15 +
 .../PathSpectralGap/D0_Habitat.lean           |  31 +
 .../PathSpectralGap/D1_CauchyGram.lean        | 113 +++
 .../PathSpectralGap/D2_Robertson.lean         | 285 ++++++
 .../PathSpectralGap/D3_GrafoCamino.lean       | 212 +++++
 .../PathSpectralGap/D4_PorQueNoDiagonal.lean  | 253 ++++++
 .../PathSpectralGap/D5_MaximaTension.lean     | 480 +++++++++++
 .../PathSpectralGap/D6_Fiedler.lean           | 816 ++++++++++++++++++
 .../PathSpectralGap/D7_Niven.lean             |  92 ++
 9 files changed, 2297 insertions(+)
 create mode 100644 PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D0_Habitat.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D1_CauchyGram.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D2_Robertson.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D3_GrafoCamino.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D4_PorQueNoDiagonal.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D5_MaximaTension.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D6_Fiedler.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D7_Niven.lean

diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap.lean
new file mode 100644
index 000000000..dba189f69
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap.lean
@@ -0,0 +1,15 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D0_Habitat
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D1_CauchyGram
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D2_Robertson
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D3_GrafoCamino
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D4_PorQueNoDiagonal
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D5_MaximaTension
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D6_Fiedler
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D7_Niven
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D0_Habitat.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D0_Habitat.lean
new file mode 100644
index 000000000..10f960a59
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D0_Habitat.lean
@@ -0,0 +1,31 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import Mathlib.Analysis.InnerProductSpace.EuclideanDist
+
+@[expose] public section
+
+/-!
+# D0 — Hábitat: el espacio de Hilbert finito `H_d`
+
+Todo el argumento vive en un único espacio de Hilbert complejo de dimensión
+finita, `H_d = ℂ^d` con su producto interno estándar. No se sale nunca de
+este espacio: en particular, `d = ∞` no es una dimensión realizada, sólo un
+límite de la familia `{H_d}_{d∈ℕ}` (ver `D8_Szego.lean`).
+-/
+
+namespace TransportePosicion
+
+/-- El espacio de Hilbert finito de dimensión `d`: `ℂ^d` con su estructura
+euclidiana estándar. -/
+abbrev Hd (d : ℕ) := EuclideanSpace ℂ (Fin d)
+
+/-- Identidad definicional: `H_d` es, literalmente, `EuclideanSpace ℂ (Fin d)`. -/
+theorem Hd_eq_euclidean (d : ℕ) :
+    Hd d = EuclideanSpace ℂ (Fin d) := rfl
+
+end TransportePosicion
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D1_CauchyGram.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D1_CauchyGram.lean
new file mode 100644
index 000000000..abefffd95
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D1_CauchyGram.lean
@@ -0,0 +1,113 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import Mathlib.Analysis.InnerProductSpace.Basic
+public import Mathlib.Analysis.Complex.Norm
+
+@[expose] public section
+
+/-!
+# D1 — Cauchy–Schwarz vía el defecto de Gram
+
+Núcleo puramente algebraico: para dos vectores `x, y` de un espacio de
+Hilbert complejo, el determinante de la matriz de Gram hermitiana
+
+`gramDefectC x y = ‖x‖² ‖y‖² − |⟨x,y⟩|²`
+
+nunca es negativo. Esa es, palabra por palabra, la desigualdad de
+Cauchy–Schwarz. Escribiendo `⟨x,y⟩` en sus partes real e imaginaria se
+obtiene de inmediato la desigualdad de Robertson–Schrödinger (`D2_Robertson.lean`)
+como consecuencia algebraica, no como postulado adicional.
+-/
+
+noncomputable section
+
+namespace ObstruccionGramUnificada
+
+universe u
+
+variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
+
+/-- Cuadrado de la dispersión representada por un vector centrado. -/
+def varianceC (x : H) : ℝ := ‖x‖ ^ 2
+
+/-- Determinante de la matriz de Gram hermitiana de dos vectores. -/
+def gramDefectC (x y : H) : ℝ :=
+  varianceC x * varianceC y - ‖@inner ℂ H _ x y‖ ^ 2
+
+/-- La obstrucción universal: el determinante de Gram nunca es negativo.
+Esto ES la desigualdad de Cauchy–Schwarz, reescrita como positividad de un
+determinante 2×2. -/
+theorem gramDefectC_nonneg (x y : H) : 0 ≤ gramDefectC x y := by
+  have hxy : ‖@inner ℂ H _ x y‖ ≤ ‖x‖ * ‖y‖ := norm_inner_le_norm x y
+  have hleft : 0 ≤ ‖x‖ * ‖y‖ - ‖@inner ℂ H _ x y‖ := sub_nonneg.mpr hxy
+  have hright : 0 ≤ ‖x‖ * ‖y‖ + ‖@inner ℂ H _ x y‖ :=
+    add_nonneg (mul_nonneg (norm_nonneg _) (norm_nonneg _)) (norm_nonneg _)
+  calc
+    0 ≤ (‖x‖ * ‖y‖ - ‖@inner ℂ H _ x y‖) *
+        (‖x‖ * ‖y‖ + ‖@inner ℂ H _ x y‖) := mul_nonneg hleft hright
+    _ = gramDefectC x y := by simp [gramDefectC, varianceC, pow_two]; ring
+
+/-- Saturar Cauchy–Schwarz equivale a anular, no las dispersiones, sino el
+determinante de Gram. -/
+theorem gramDefectC_eq_zero_iff (x y : H) :
+    gramDefectC x y = 0 ↔ ‖@inner ℂ H _ x y‖ = ‖x‖ * ‖y‖ := by
+  have hxy : ‖@inner ℂ H _ x y‖ ≤ ‖x‖ * ‖y‖ := norm_inner_le_norm x y
+  have hi : 0 ≤ ‖@inner ℂ H _ x y‖ := norm_nonneg _
+  have hp : 0 ≤ ‖x‖ * ‖y‖ := mul_nonneg (norm_nonneg _) (norm_nonneg _)
+  constructor
+  · intro hzero
+    simp only [gramDefectC, varianceC, pow_two] at hzero
+    nlinarith
+  · intro heq
+    unfold gramDefectC varianceC
+    rw [heq]
+    ring
+
+/-- Parte simétrica del producto interno de las fluctuaciones. -/
+def covarianceC (x y : H) : ℝ := (@inner ℂ H _ x y).re
+
+/-- Coordenada antisimétrica real: para fluctuaciones operatoriales es la
+coordenada real de la esperanza del conmutador. -/
+def commutatorCoordinateC (x y : H) : ℝ := 2 * (@inner ℂ H _ x y).im
+
+/-- Robertson–Schrödinger es exactamente la positividad de Gram escrita en
+coordenadas real e imaginaria. -/
+theorem robertsonSchrodinger_from_gram (x y : H) :
+    covarianceC x y ^ 2 + (commutatorCoordinateC x y / 2) ^ 2 ≤
+      varianceC x * varianceC y := by
+  have hgram := gramDefectC_nonneg x y
+  have hnorm :
+      ‖@inner ℂ H _ x y‖ ^ 2 =
+        (@inner ℂ H _ x y).re ^ 2 + (@inner ℂ H _ x y).im ^ 2 := by
+    rw [Complex.sq_norm, Complex.normSq_apply]
+    ring
+  have hbase : ‖@inner ℂ H _ x y‖ ^ 2 ≤ varianceC x * varianceC y :=
+    sub_nonneg.mp hgram
+  rw [hnorm] at hbase
+  simpa [covarianceC, commutatorCoordinateC] using hbase
+
+/-- Saturación Robertson–Schrödinger abstracta. -/
+def RSSaturated (x y : H) : Prop :=
+  covarianceC x y ^ 2 + (commutatorCoordinateC x y / 2) ^ 2 =
+    varianceC x * varianceC y
+
+/-- La saturación Robertson–Schrödinger es exactamente defecto de Gram cero. -/
+theorem robertsonSchrodinger_saturated_iff_gram_zero (x y : H) :
+    RSSaturated x y ↔ gramDefectC x y = 0 := by
+  have hnorm :
+      ‖@inner ℂ H _ x y‖ ^ 2 =
+        (@inner ℂ H _ x y).re ^ 2 + (@inner ℂ H _ x y).im ^ 2 := by
+    rw [Complex.sq_norm, Complex.normSq_apply]
+    ring
+  simp only [RSSaturated, covarianceC, commutatorCoordinateC, gramDefectC]
+  rw [show (2 * (@inner ℂ H _ x y).im / 2) ^ 2 =
+      (@inner ℂ H _ x y).im ^ 2 by ring]
+  rw [← hnorm]
+  constructor <;> intro h <;> nlinarith
+
+end ObstruccionGramUnificada
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D2_Robertson.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D2_Robertson.lean
new file mode 100644
index 000000000..1ee32cbf8
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D2_Robertson.lean
@@ -0,0 +1,285 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D1_CauchyGram
+public import Mathlib.Algebra.Order.Ring.Star
+public import Mathlib.Algebra.Order.Star.Real
+public import Mathlib.Analysis.Real.Pi.Bounds
+public import Mathlib.Data.Rat.Star
+public import Mathlib.Tactic.IntervalCases
+
+@[expose] public section
+
+/-!
+# D2 — La desigualdad de Robertson (1929)
+
+Formalización abstracta de la desigualdad de Robertson (1929) para un par de
+observables conjugados evaluados en un estado normalizado de un espacio de
+Hilbert. Se incluyen dos formas: la forma lineal clásica (`Evaluacion`,
+con la cota `|⟨[A,B]⟩|/2 ≤ σ_A σ_B`) y la forma cuadrática de
+Robertson–Schrödinger (`EvaluacionSchrodinger`, con covarianza).
+
+Punto central de este archivo: la hipótesis `cota_cuadratica` que
+`EvaluacionSchrodinger` exige como dato **no se postula** — al final del
+archivo (`ObstruccionGramUnificada.evaluacionSchrodingerDeGram`) se prueba
+que todo par de vectores de un espacio de Hilbert produce automáticamente una
+`EvaluacionSchrodinger` válida, con esa cota derivada directamente de
+`D1_CauchyGram.gramDefectC_nonneg`. Cauchy–Schwarz ⇒ Gram ⇒
+Robertson–Schrödinger, como teorema, no como axioma adicional.
+
+También se incluyen aquí cinco lemas aritméticos elementales (`Blindaje`)
+que se usan más adelante para acotar el coseno y para el teorema de Niven
+(`D7_Niven.lean`).
+-/
+
+namespace Robertson1929
+
+universe u
+
+/-- Evaluación exacta del teorema de Robertson (1929) tras evaluar dos
+observables conjugados en un estado normalizado de un espacio de Hilbert.
+`sigmaA`, `sigmaB` son las desviaciones y `mediaConmutador` es
+`⟨ψ,[A,B]ψ⟩`. -/
+structure Evaluacion (H : Type u) [NormedAddCommGroup H]
+    [InnerProductSpace ℂ H] where
+  estado : H
+  normalizado : ‖estado‖ = 1
+  sigmaA : ℝ
+  sigmaB : ℝ
+  mediaConmutador : ℂ
+  sigmaA_nonneg : 0 ≤ sigmaA
+  sigmaB_nonneg : 0 ≤ sigmaB
+  cota : ‖mediaConmutador‖ / 2 ≤ sigmaA * sigmaB
+
+/-- Saturación exacta de la cota de Robertson en la evaluación dada. -/
+def Saturada {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
+    (R : Evaluacion H) : Prop :=
+  R.sigmaA * R.sigmaB = ‖R.mediaConmutador‖ / 2
+
+/-- Evaluación en máxima tensión: el estado normalizado realiza la norma del
+conmutador, por lo que el lado derecho de Robertson es el más exigente de la
+familia de estados normalizados. -/
+structure MaximaTension (H : Type u) [NormedAddCommGroup H]
+    [InnerProductSpace ℂ H] extends Evaluacion H where
+  normaConmutador : ℝ
+  normaConmutador_nonneg : 0 ≤ normaConmutador
+  realiza_norma :
+    ‖toEvaluacion.mediaConmutador‖ = normaConmutador
+
+/-- En máxima tensión, Robertson entrega la cota evaluada en la norma del
+conmutador. -/
+theorem MaximaTension.cota_por_norma
+    {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
+    (R : MaximaTension H) :
+    R.normaConmutador / 2 ≤ R.sigmaA * R.sigmaB := by
+  rw [← R.realiza_norma]
+  exact R.cota
+
+/-! ## Ancla Robertson–Schrödinger: forma cuadrática -/
+
+/-- Evaluación cuadrática Robertson–Schrödinger: el producto de dispersiones
+domina el piso cuadrático compuesto por covarianza y conmutador. -/
+structure EvaluacionSchrodinger where
+  sigmaA : ℝ
+  sigmaB : ℝ
+  covarianza : ℝ
+  conmutador : ℝ
+  sigmaA_nonneg : 0 ≤ sigmaA
+  sigmaB_nonneg : 0 ≤ sigmaB
+  cota_cuadratica : covarianza ^ 2 + conmutador ^ 2 ≤ sigmaA ^ 2 * sigmaB ^ 2
+
+/-- Piso Robertson–Schrödinger: raíz cuadrada del término cuadrático. -/
+noncomputable def pisoSchrodinger (S : EvaluacionSchrodinger) : ℝ :=
+  Real.sqrt (S.covarianza ^ 2 + S.conmutador ^ 2)
+
+/-- Saturación Robertson–Schrödinger exacta. -/
+def SaturadaSchrodinger (S : EvaluacionSchrodinger) : Prop :=
+  S.sigmaA ^ 2 * S.sigmaB ^ 2 = S.covarianza ^ 2 + S.conmutador ^ 2
+
+theorem saturadaSchrodinger_iff (S : EvaluacionSchrodinger) :
+    SaturadaSchrodinger S ↔
+      S.sigmaA ^ 2 * S.sigmaB ^ 2 =
+        S.covarianza ^ 2 + S.conmutador ^ 2 := by
+  rfl
+
+/-- El piso Robertson–Schrödinger es positivo si y sólo si covarianza o
+conmutador son no nulos. -/
+theorem pisoSchrodinger_pos_iff (S : EvaluacionSchrodinger) :
+    0 < pisoSchrodinger S ↔ S.covarianza ≠ 0 ∨ S.conmutador ≠ 0 := by
+  rw [pisoSchrodinger, Real.sqrt_pos]
+  constructor
+  · intro h
+    by_contra hz
+    push Not at hz
+    simp [hz.1, hz.2] at h
+  · rintro (hcov | hcomm)
+    · nlinarith [sq_pos_of_ne_zero hcov, sq_nonneg S.conmutador]
+    · nlinarith [sq_nonneg S.covarianza, sq_pos_of_ne_zero hcomm]
+
+/-- La cota cuadrática Robertson–Schrödinger implica la cota lineal sobre el
+producto de dispersiones no negativas. -/
+theorem pisoSchrodinger_le_producto (S : EvaluacionSchrodinger) :
+    pisoSchrodinger S ≤ S.sigmaA * S.sigmaB := by
+  have hsum : 0 ≤ S.covarianza ^ 2 + S.conmutador ^ 2 := by positivity
+  have hprod_nonneg : 0 ≤ S.sigmaA * S.sigmaB :=
+    mul_nonneg S.sigmaA_nonneg S.sigmaB_nonneg
+  have hsqrt_sq :
+      pisoSchrodinger S ^ 2 = S.covarianza ^ 2 + S.conmutador ^ 2 := by
+    simpa [pisoSchrodinger] using Real.sq_sqrt hsum
+  have hprod_sq :
+      S.sigmaA ^ 2 * S.sigmaB ^ 2 = (S.sigmaA * S.sigmaB) ^ 2 := by
+    ring
+  have hcota := S.cota_cuadratica
+  rw [hprod_sq] at hcota
+  nlinarith
+
+/-- Forma limpia del ancla: Robertson–Schrödinger aporta una cota lineal y no
+permite afirmar simultáneamente esa cota y su negación. -/
+theorem anclaSchrodinger_limpia (S : EvaluacionSchrodinger) :
+    pisoSchrodinger S ≤ S.sigmaA * S.sigmaB ∧
+    ¬ (pisoSchrodinger S ≤ S.sigmaA * S.sigmaB ∧
+      ¬ pisoSchrodinger S ≤ S.sigmaA * S.sigmaB) := by
+  refine ⟨pisoSchrodinger_le_producto S, ?_⟩
+  intro h
+  exact h.2 h.1
+
+end Robertson1929
+
+/-! ## Cinco lemas aritméticos elementales (`Blindaje`)
+
+Usados más adelante por el teorema de Niven (`D7_Niven.lean`): R3 acota el
+coseno para `d ≥ 5`; R5 es la observación aritmética de que una cota
+estrictamente positiva impide que cualquiera de sus dos factores sea nulo. -/
+
+open Real Finset
+
+namespace Blindaje
+
+/-- Identidad término a término, exacta en ℚ. -/
+theorem R1b_termino (k : ℕ) (hk : 1 ≤ k) :
+    (1 : ℚ) / k ^ 2 - 1 / (k * (k + 1)) = 1 / (k ^ 2 * (k + 1)) := by
+  have hk0 : (k : ℚ) ≠ 0 := Nat.cast_ne_zero.mpr (by omega)
+  have hk1 : (k : ℚ) + 1 ≠ 0 := by positivity
+  field_simp
+  ring
+
+/-- El telescopio cierra exacto: `Σ_{k=2..N} 1/(k(k+1)) = 1/2 − 1/(N+1)`. -/
+theorem R1a_telescopio (N : ℕ) (hN : 2 ≤ N) :
+    ∑ k ∈ Icc 2 N, (1 : ℚ) / (k * (k + 1)) = 1 / 2 - 1 / (N + 1) := by
+  induction N with
+  | zero => omega
+  | succ n ih =>
+    rcases Nat.lt_or_ge n 2 with h | h
+    · interval_cases n
+      · omega
+      · simp
+        norm_num
+    · rw [Finset.sum_Icc_succ_top (by omega), ih h]
+      have hn1 : ((n : ℚ) + 1) ≠ 0 := by positivity
+      have hn2 : ((n : ℚ) + 1 + 1) ≠ 0 := by positivity
+      push_cast
+      field_simp
+      ring
+
+theorem R1d_modo_positivo (k : ℕ) (hk : 2 ≤ k) :
+    (0 : ℚ) < 1 / (k ^ 2 * (k + 1)) := by
+  have : (0 : ℚ) < k := by exact_mod_cast (by omega : 0 < k)
+  positivity
+
+/-- Si `(3+√5)/8 = 3/4` entonces `√5 = 3`, entonces `5 = 9`: absurdo. -/
+theorem R2_cinco_no_es_nueve : (3 + Real.sqrt 5) / 8 ≠ 3 / 4 := by
+  intro h
+  have h3 : Real.sqrt 5 = 3 := by linarith
+  have h5 : (5 : ℝ) = 9 := by
+    have := Real.sq_sqrt (by norm_num : (5:ℝ) ≥ 0)
+    rw [h3] at this
+    linarith [this]
+  norm_num at h5
+
+/-- Techo del coseno: para `d ≥ 5`, `cos²(π/(d+1)) < (d−1)/4`. -/
+theorem R3_techo_coseno (d : ℕ) (hd : 5 ≤ d) :
+    Real.cos (π / (d + 1)) ^ 2 < (d - 1 : ℝ) / 4 := by
+  have hd1 : (0 : ℝ) < (d : ℝ) + 1 := by positivity
+  have hx_pos : 0 < π / ((d : ℝ) + 1) := by positivity
+  have hd5 : (5 : ℝ) ≤ d := by exact_mod_cast hd
+  have hcos_lt : Real.cos (π / (d + 1)) < 1 := by
+    have hy : π / ((d : ℝ) + 1) ≤ π := by
+      rw [div_le_iff₀ hd1]
+      nlinarith [Real.pi_pos]
+    have h := Real.cos_lt_cos_of_nonneg_of_le_pi le_rfl hy hx_pos
+    simpa using h
+  have hcos_nonneg : 0 ≤ Real.cos (π / ((d : ℝ) + 1)) := by
+    apply Real.cos_nonneg_of_mem_Icc
+    constructor
+    · nlinarith [Real.pi_pos]
+    · rw [div_le_iff₀ hd1]
+      nlinarith [Real.pi_pos]
+  have hcos_le : Real.cos (π / (d + 1)) ^ 2 < 1 := by
+    nlinarith [hcos_nonneg, hcos_lt]
+  have hfloor : (1 : ℝ) ≤ ((d : ℝ) - 1) / 4 := by
+    have : (5 : ℝ) ≤ d := by exact_mod_cast hd
+    linarith
+  linarith
+
+theorem R4a_siete_fracciones :
+    (3 : ℚ) / 2 < ∑ k ∈ Finset.range 7, (1 : ℚ) / (k + 1) ^ 2 := by
+  norm_num [Finset.sum_range_succ]
+
+theorem R4_pi_mayor_que_tres : (3 : ℝ) < π := Real.pi_gt_three
+
+/-- Obstrucción aritmética: si la cota `c` es estrictamente positiva y
+`c ≤ α·β`, entonces ninguno de los dos factores puede anularse. -/
+theorem R5_obstruccion_aritmetica (var_A var_B cota_robertson : ℝ)
+    (h_robertson : cota_robertson ≤ var_A * var_B)
+    (h_cota_positiva : 0 < cota_robertson) :
+    var_A ≠ 0 ∧ var_B ≠ 0 := by
+  constructor
+  · intro hA
+    rw [hA, zero_mul] at h_robertson
+    linarith
+  · intro hB
+    rw [hB, mul_zero] at h_robertson
+    linarith
+
+end Blindaje
+
+/-! ## Cierre del puente: Cauchy–Schwarz ⇒ Robertson–Schrödinger -/
+
+noncomputable section
+
+namespace ObstruccionGramUnificada
+
+/-- Puente con `Robertson1929`: cualquier par de vectores en un espacio de
+Hilbert complejo produce una `EvaluacionSchrodinger` cuya cota cuadrática no
+se postula como campo libre — se deriva del defecto de Gram no negativo
+(`gramDefectC_nonneg`). La hipótesis `cota_cuadratica` que
+`Robertson1929.EvaluacionSchrodinger` exige como dato queda aquí demostrada. -/
+def evaluacionSchrodingerDeGram {H : Type*} [NormedAddCommGroup H]
+    [InnerProductSpace ℂ H] (x y : H) :
+    Robertson1929.EvaluacionSchrodinger where
+  sigmaA := ‖x‖
+  sigmaB := ‖y‖
+  covarianza := covarianceC x y
+  conmutador := commutatorCoordinateC x y / 2
+  sigmaA_nonneg := norm_nonneg x
+  sigmaB_nonneg := norm_nonneg y
+  cota_cuadratica := by
+    have h := robertsonSchrodinger_from_gram x y
+    simpa [varianceC] using h
+
+/-- El piso Robertson–Schrödinger de la evaluación construida por Gram queda
+dominado por el producto de normas: la misma conclusión de
+`Robertson1929.pisoSchrodinger_le_producto`, instanciada sobre una evaluación
+que ya no es un supuesto sino un teorema. -/
+theorem pisoSchrodinger_evaluacionSchrodingerDeGram_le
+    {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] (x y : H) :
+    Robertson1929.pisoSchrodinger (evaluacionSchrodingerDeGram x y) ≤ ‖x‖ * ‖y‖ := by
+  have h := Robertson1929.pisoSchrodinger_le_producto (evaluacionSchrodingerDeGram x y)
+  simpa [evaluacionSchrodingerDeGram] using h
+
+end ObstruccionGramUnificada
+
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D3_GrafoCamino.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D3_GrafoCamino.lean
new file mode 100644
index 000000000..c7c8d7e35
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D3_GrafoCamino.lean
@@ -0,0 +1,212 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D0_Habitat
+public import Mathlib.Combinatorics.SimpleGraph.Hasse
+
+@[expose] public section
+
+/-!
+# D3 — Operadores de transporte y posición sobre el grafo camino
+
+El soporte discreto no introduce un grafo ad hoc: es literalmente
+`SimpleGraph.pathGraph d` de Mathlib, el camino con `d` vértices
+`0,1,…,d−1` y aristas sólo entre vecinos consecutivos. Sobre esa base se
+definen dos matrices hermitianas: `T_d` (transporte, soportado en las
+aristas del camino) y `P_d` (posición, diagonal, con coordenadas
+centradas en `[-1,1]`).
+
+La segunda mitad del archivo (`CanalPreFuerza`) prueba, sin apelar a
+ninguna elección de diseño, que el grafo camino es la **única** opción
+compatible con dos condiciones puramente combinatorias: localidad
+(ninguna arista salta vecinos) y completitud (no falta ningún paso
+elemental). Cualquier grafo local en `Fin d` que no omita un paso mínimo
+**es** `pathGraph d`; no hay otro candidato.
+-/
+
+namespace TransportePosicion
+
+open SimpleGraph
+
+/-- Grafo de fase del canal transporte–posición: el camino `pathGraph d`
+de Mathlib. -/
+abbrev GrafoTP (d : ℕ) : SimpleGraph (Fin d) :=
+  SimpleGraph.pathGraph d
+
+/-- Adyacencia elemental del camino: sólo hay desplazamiento mínimo de un
+paso. -/
+theorem grafoTP_adj {d : ℕ} {i j : Fin d} :
+    (GrafoTP d).Adj i j ↔ i.val + 1 = j.val ∨ j.val + 1 = i.val := by
+  simpa [GrafoTP] using
+    (SimpleGraph.pathGraph_adj (n := d) (u := i) (v := j))
+
+/-- Predicado decidible del desplazamiento mínimo en la línea discreta. -/
+def PasoMinimo {d : ℕ} (i j : Fin d) : Prop :=
+  i.val + 1 = j.val ∨ j.val + 1 = i.val
+
+instance pasoMinimo_decidable {d : ℕ} (i j : Fin d) :
+    Decidable (PasoMinimo i j) := by
+  unfold PasoMinimo
+  infer_instance
+
+/-- El paso mínimo decidible es exactamente la adyacencia de `pathGraph`. -/
+theorem pasoMinimo_iff_adj {d : ℕ} {i j : Fin d} :
+    PasoMinimo i j ↔ (GrafoTP d).Adj i j := by
+  rw [grafoTP_adj]
+  rfl
+
+/-- El soporte discreto `T_d/P_d` es isomorfo a `pathGraph d` por definición
+canónica. -/
+theorem grafoTP_es_pathGraph (d : ℕ) :
+    Nonempty (GrafoTP d ≃g SimpleGraph.pathGraph d) := by
+  change Nonempty (SimpleGraph.pathGraph d ≃g SimpleGraph.pathGraph d)
+  exact ⟨SimpleGraph.Iso.refl⟩
+
+/-- Matriz de adyacencia compleja del canal de transporte. -/
+noncomputable def Ad (d : ℕ) : Matrix (Fin d) (Fin d) ℂ :=
+  fun i j => if PasoMinimo i j then 1 else 0
+
+/-- Radio espectral usado para normalizar el transporte de la cadena
+finita: `ρ_d = 2 cos(π/(d+1))`. -/
+noncomputable def rho (d : ℕ) : ℝ :=
+  2 * Real.cos (Real.pi / ((d : ℝ) + 1))
+
+/-- Operador de transporte normalizado `T_d = A_d / ρ_d`. -/
+noncomputable def Td (d : ℕ) : Matrix (Fin d) (Fin d) ℂ :=
+  fun i j => Ad d i j / (rho d : ℂ)
+
+/-- Coordenada centrada de posición sobre la base discreta, en `[-1,1]`. -/
+noncomputable def posicionCoord (d : ℕ) (j : Fin d) : ℝ :=
+  (2 * ((j.val : ℝ) + 1) - ((d : ℝ) + 1)) / ((d : ℝ) - 1)
+
+/-- Operador de posición diagonal `P_d`. -/
+noncomputable def Pd (d : ℕ) : Matrix (Fin d) (Fin d) ℂ :=
+  fun i j => if i = j then (posicionCoord d i : ℂ) else 0
+
+theorem Ad_eq_one_iff {d : ℕ} {i j : Fin d} :
+    Ad d i j = 1 ↔ (GrafoTP d).Adj i j := by
+  unfold Ad
+  rw [← pasoMinimo_iff_adj]
+  by_cases h : PasoMinimo i j
+  · simp [h]
+  · simp [h]
+
+theorem Td_eq_zero_of_not_adj {d : ℕ} {i j : Fin d}
+    (h : ¬ (GrafoTP d).Adj i j) :
+    Td d i j = 0 := by
+  have hpaso : ¬ PasoMinimo i j := by
+    intro hp
+    exact h (pasoMinimo_iff_adj.mp hp)
+  simp [Td, Ad, hpaso]
+
+/-- `P_d` es diagonal en la base discreta. -/
+theorem Pd_eq_zero_offdiag {d : ℕ} {i j : Fin d} (hij : i ≠ j) :
+    Pd d i j = 0 := by
+  simp [Pd, hij]
+
+/-- En la diagonal, `P_d` devuelve la coordenada discreta centrada. -/
+theorem Pd_diag {d : ℕ} (i : Fin d) :
+    Pd d i i = (posicionCoord d i : ℂ) := by
+  simp [Pd]
+
+end TransportePosicion
+
+/-!
+## Por qué `pathGraph d` y no otro grafo
+
+Un canal local (toda arista es un paso mínimo entre vecinos) y completo
+(no falta ningún paso mínimo posible) sobre `Fin d` es, por extensionalidad
+de la relación de adyacencia, exactamente `pathGraph d`. No hay una
+"simplificación" que conserve ambas propiedades: quitar una arista rompe
+la completitud.
+-/
+
+namespace CanalPreFuerza
+
+open SimpleGraph
+
+/-- Localidad estricta: toda arista del canal es un paso entre vecinos
+consecutivos. No se permiten saltos ni atajos. -/
+def LocalidadOrdenada {d : ℕ} (G : SimpleGraph (Fin d)) : Prop :=
+  ∀ {i j : Fin d}, G.Adj i j → TransportePosicion.PasoMinimo i j
+
+/-- Completitud: todo paso entre vecinos consecutivos debe estar presente.
+Quitar uno rompe el movimiento local completo entre los extremos de la
+celda. -/
+def PasosElementalesCompletos {d : ℕ} (G : SimpleGraph (Fin d)) : Prop :=
+  ∀ {i j : Fin d}, TransportePosicion.PasoMinimo i j → G.Adj i j
+
+/-- Defecto por intentar simplificar más que `pathGraph d`: se omite al
+menos un paso elemental consecutivo. -/
+def OmitePasoElemental {d : ℕ} (G : SimpleGraph (Fin d)) : Prop :=
+  ∃ i j : Fin d, TransportePosicion.PasoMinimo i j ∧ ¬ G.Adj i j
+
+/-- Canal ordenado, local y completo en la celda discreta. -/
+structure CanalLocalNoRamificadoOrdenado (d : ℕ) where
+  grafo : SimpleGraph (Fin d)
+  localidad_ordenada : LocalidadOrdenada grafo
+  pasos_elementales : PasosElementalesCompletos grafo
+
+/-- En un canal local ordenado completo, la adyacencia es exactamente el
+paso mínimo de la celda. -/
+theorem CanalLocalNoRamificadoOrdenado.adj_iff_paso
+    {d : ℕ} (C : CanalLocalNoRamificadoOrdenado d) {i j : Fin d} :
+    C.grafo.Adj i j ↔ TransportePosicion.PasoMinimo i j := by
+  exact ⟨fun h => C.localidad_ordenada h,
+    fun h => C.pasos_elementales h⟩
+
+/-- Teorema de minimalidad: localidad ordenada y pasos elementales
+completos fuerzan que el soporte sea exactamente `pathGraph d`. -/
+theorem canal_local_no_ramificado_es_pathGraph
+    {d : ℕ} (C : CanalLocalNoRamificadoOrdenado d) :
+    C.grafo = SimpleGraph.pathGraph d := by
+  ext i j
+  rw [C.adj_iff_paso]
+  simpa [TransportePosicion.GrafoTP] using
+    (TransportePosicion.pasoMinimo_iff_adj (d := d) (i := i) (j := j))
+
+/-- Versión isomórfica del mismo cierre. -/
+theorem canal_local_no_ramificado_iso_pathGraph
+    {d : ℕ} (C : CanalLocalNoRamificadoOrdenado d) :
+    Nonempty (C.grafo ≃g SimpleGraph.pathGraph d) := by
+  rw [canal_local_no_ramificado_es_pathGraph C]
+  exact ⟨SimpleGraph.Iso.refl⟩
+
+/-- El canal canónico `T_d/P_d` satisface directamente el certificado local
+ordenado: no tiene saltos y no omite pasos elementales. -/
+def canal_TP_local_no_ramificado (d : ℕ) :
+    CanalLocalNoRamificadoOrdenado d where
+  grafo := TransportePosicion.GrafoTP d
+  localidad_ordenada := by
+    intro i j h
+    exact (TransportePosicion.pasoMinimo_iff_adj
+      (d := d) (i := i) (j := j)).mpr h
+  pasos_elementales := by
+    intro i j h
+    exact (TransportePosicion.pasoMinimo_iff_adj
+      (d := d) (i := i) (j := j)).mp h
+
+/-- Ningún canal que ya satisface el certificado local ordenado puede omitir
+un paso elemental: "no hay una simplificación local más simple que
+`pathGraph d`". -/
+theorem no_hay_canal_local_mas_simple_que_Pd
+    {d : ℕ} (C : CanalLocalNoRamificadoOrdenado d) :
+    ¬ OmitePasoElemental C.grafo := by
+  rintro ⟨i, j, hpaso, hno⟩
+  exact hno (C.pasos_elementales hpaso)
+
+/-- Cierre: el soporte canónico `T_d/P_d` es `pathGraph d`, y cualquier
+intento local de hacerlo "más simple" pierde un paso elemental. -/
+theorem cierre_minimalidad_local_TP (d : ℕ) :
+    (canal_TP_local_no_ramificado d).grafo = SimpleGraph.pathGraph d ∧
+      ¬ OmitePasoElemental (canal_TP_local_no_ramificado d).grafo := by
+  exact ⟨canal_local_no_ramificado_es_pathGraph
+      (canal_TP_local_no_ramificado d),
+    no_hay_canal_local_mas_simple_que_Pd
+      (canal_TP_local_no_ramificado d)⟩
+
+end CanalPreFuerza
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D4_PorQueNoDiagonal.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D4_PorQueNoDiagonal.lean
new file mode 100644
index 000000000..2db08a9d8
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D4_PorQueNoDiagonal.lean
@@ -0,0 +1,253 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D3_GrafoCamino
+
+@[expose] public section
+
+/-!
+# D4 — Por qué no un paso diagonal
+
+`D3_GrafoCamino.lean` fija el soporte de `T_d` en los pasos mínimos
+(vecino a vecino) del camino `pathGraph d`. Este archivo cierra, con dos
+argumentos independientes, la pregunta de por qué nunca se considera un
+paso "diagonal" (cambiar más de una coordenada a la vez) como alternativa:
+
+1. **Si hubiera ≥ 2 ejes genuinos** (el modelo se generalizara a una malla
+   cúbica `Fin dx × Fin dy × Fin dz`), Pitágoras decide: el paso ortogonal
+   (un solo eje) tiene distancia euclidiana exactamente `1`; el paso
+   diagonal doble, exactamente `√2`; el triple, exactamente `√3`. Como
+   `1 < √2` y `1 < √3`, el paso ortogonal es siempre estrictamente más
+   corto. No es una preferencia de diseño: es la adyacencia mínima que
+   Pitágoras obliga.
+2. **En el caso efectivamente usado por `T_d/P_d`** (un solo eje, `dy = dz = 1`),
+   la pregunta ni siquiera se plantea: con un único eje genuino la propia
+   relación "paso diagonal" es la relación **vacía** — no existe ningún par
+   de sitios que la satisfaga, porque un eje trivial (`Fin 1`) no tiene
+   ningún paso mínimo posible. La diagonal presupone, para poder
+   enunciarse de forma no vacía, dos ejes ya distinguidos entre sí.
+-/
+
+noncomputable section
+
+namespace PathGraph3D
+
+/-- Sitio de una malla cúbica: producto de tres mallas 1D. -/
+abbrev Sitio3D (dx dy dz : ℕ) := Fin dx × Fin dy × Fin dz
+
+/-- Adyacencia ortogonal del cubo: cambia una sola coordenada a la vez, por
+un paso mínimo en ese eje. -/
+def Adj3D {dx dy dz : ℕ} (p q : Sitio3D dx dy dz) : Prop :=
+  (TransportePosicion.PasoMinimo p.1 q.1 ∧ p.2 = q.2) ∨
+    (p.1 = q.1 ∧ TransportePosicion.PasoMinimo p.2.1 q.2.1 ∧ p.2.2 = q.2.2) ∨
+    (p.1 = q.1 ∧ p.2.1 = q.2.1 ∧ TransportePosicion.PasoMinimo p.2.2 q.2.2)
+
+end PathGraph3D
+
+/-! ## 1. Pitágoras: el paso ortogonal es siempre estrictamente más corto -/
+
+namespace OrtogonalidadMinimalPitagoras
+
+open PathGraph3D
+open TransportePosicion
+
+/-- Distancia euclidiana entre dos sitios del cubo, viendo cada coordenada
+`Fin` como un real vía el casteo natural. -/
+noncomputable def dist3D {dx dy dz : ℕ} (p q : Sitio3D dx dy dz) : ℝ :=
+  Real.sqrt (
+    ((p.1 : ℝ) - (q.1 : ℝ)) ^ 2 +
+    ((p.2.1 : ℝ) - (q.2.1 : ℝ)) ^ 2 +
+    ((p.2.2 : ℝ) - (q.2.2 : ℝ)) ^ 2)
+
+/-- Dos coordenadas a un `PasoMinimo` difieren, como reales, en exactamente
+`±1`; su diferencia al cuadrado es `1`. -/
+theorem pasoMinimo_sq_diff_eq_one {d : ℕ} {i j : Fin d} (h : PasoMinimo i j) :
+    ((i.val : ℝ) - (j.val : ℝ)) ^ 2 = 1 := by
+  rcases h with h | h
+  · have hij : (j.val : ℝ) = (i.val : ℝ) + 1 := by exact_mod_cast h.symm
+    rw [hij]; ring
+  · have hji : (i.val : ℝ) = (j.val : ℝ) + 1 := by exact_mod_cast h.symm
+    rw [hji]; ring
+
+theorem eq_sq_diff_eq_zero {d : ℕ} {i j : Fin d} (h : i = j) :
+    ((i.val : ℝ) - (j.val : ℝ)) ^ 2 = 0 := by
+  rw [h]; ring
+
+/-! ### Paso ortogonal: distancia exactamente `1` -/
+
+theorem dist3D_eq_one_of_Adj3D
+    {dx dy dz : ℕ} {p q : Sitio3D dx dy dz} (h : Adj3D p q) :
+    dist3D p q = 1 := by
+  unfold dist3D
+  rcases h with ⟨hx, hyz⟩ | ⟨hx, hy, hz⟩ | ⟨hx, hy, hz⟩
+  · have hy0 : ((p.2.1 : ℝ) - (q.2.1 : ℝ)) ^ 2 = 0 :=
+      eq_sq_diff_eq_zero (congrArg Prod.fst hyz)
+    have hz0 : ((p.2.2 : ℝ) - (q.2.2 : ℝ)) ^ 2 = 0 :=
+      eq_sq_diff_eq_zero (congrArg Prod.snd hyz)
+    have hsum :
+        ((p.1 : ℝ) - (q.1 : ℝ)) ^ 2 + ((p.2.1 : ℝ) - (q.2.1 : ℝ)) ^ 2 +
+            ((p.2.2 : ℝ) - (q.2.2 : ℝ)) ^ 2 = 1 := by
+      rw [pasoMinimo_sq_diff_eq_one hx, hy0, hz0]; ring
+    rw [hsum, Real.sqrt_one]
+  · have hx0 : ((p.1 : ℝ) - (q.1 : ℝ)) ^ 2 = 0 := eq_sq_diff_eq_zero hx
+    have hz0 : ((p.2.2 : ℝ) - (q.2.2 : ℝ)) ^ 2 = 0 := eq_sq_diff_eq_zero hz
+    have hsum :
+        ((p.1 : ℝ) - (q.1 : ℝ)) ^ 2 + ((p.2.1 : ℝ) - (q.2.1 : ℝ)) ^ 2 +
+            ((p.2.2 : ℝ) - (q.2.2 : ℝ)) ^ 2 = 1 := by
+      rw [hx0, pasoMinimo_sq_diff_eq_one hy, hz0]; ring
+    rw [hsum, Real.sqrt_one]
+  · have hx0 : ((p.1 : ℝ) - (q.1 : ℝ)) ^ 2 = 0 := eq_sq_diff_eq_zero hx
+    have hy0 : ((p.2.1 : ℝ) - (q.2.1 : ℝ)) ^ 2 = 0 := eq_sq_diff_eq_zero hy
+    have hsum :
+        ((p.1 : ℝ) - (q.1 : ℝ)) ^ 2 + ((p.2.1 : ℝ) - (q.2.1 : ℝ)) ^ 2 +
+            ((p.2.2 : ℝ) - (q.2.2 : ℝ)) ^ 2 = 1 := by
+      rw [hx0, hy0, pasoMinimo_sq_diff_eq_one hz]; ring
+    rw [hsum, Real.sqrt_one]
+
+/-! ### Paso diagonal doble: distancia exactamente `√2` -/
+
+/-- Vecino diagonal en dos ejes: dos coordenadas cambian por `PasoMinimo`
+simultáneamente, la tercera queda fija. `Adj3D` nunca produce este caso. -/
+def PasoDiagonalDoble {dx dy dz : ℕ} (p q : Sitio3D dx dy dz) : Prop :=
+  (PasoMinimo p.1 q.1 ∧ PasoMinimo p.2.1 q.2.1 ∧ p.2.2 = q.2.2) ∨
+  (PasoMinimo p.1 q.1 ∧ p.2.1 = q.2.1 ∧ PasoMinimo p.2.2 q.2.2) ∨
+  (p.1 = q.1 ∧ PasoMinimo p.2.1 q.2.1 ∧ PasoMinimo p.2.2 q.2.2)
+
+theorem dist3D_eq_sqrt_two_of_PasoDiagonalDoble
+    {dx dy dz : ℕ} {p q : Sitio3D dx dy dz} (h : PasoDiagonalDoble p q) :
+    dist3D p q = Real.sqrt 2 := by
+  unfold dist3D
+  rcases h with ⟨hx, hy, hz⟩ | ⟨hx, hy, hz⟩ | ⟨hx, hy, hz⟩
+  · have hz0 : ((p.2.2 : ℝ) - (q.2.2 : ℝ)) ^ 2 = 0 := eq_sq_diff_eq_zero hz
+    have hsum :
+        ((p.1 : ℝ) - (q.1 : ℝ)) ^ 2 + ((p.2.1 : ℝ) - (q.2.1 : ℝ)) ^ 2 +
+            ((p.2.2 : ℝ) - (q.2.2 : ℝ)) ^ 2 = 2 := by
+      rw [pasoMinimo_sq_diff_eq_one hx, pasoMinimo_sq_diff_eq_one hy, hz0]; ring
+    rw [hsum]
+  · have hy0 : ((p.2.1 : ℝ) - (q.2.1 : ℝ)) ^ 2 = 0 := eq_sq_diff_eq_zero hy
+    have hsum :
+        ((p.1 : ℝ) - (q.1 : ℝ)) ^ 2 + ((p.2.1 : ℝ) - (q.2.1 : ℝ)) ^ 2 +
+            ((p.2.2 : ℝ) - (q.2.2 : ℝ)) ^ 2 = 2 := by
+      rw [pasoMinimo_sq_diff_eq_one hx, hy0, pasoMinimo_sq_diff_eq_one hz]; ring
+    rw [hsum]
+  · have hx0 : ((p.1 : ℝ) - (q.1 : ℝ)) ^ 2 = 0 := eq_sq_diff_eq_zero hx
+    have hsum :
+        ((p.1 : ℝ) - (q.1 : ℝ)) ^ 2 + ((p.2.1 : ℝ) - (q.2.1 : ℝ)) ^ 2 +
+            ((p.2.2 : ℝ) - (q.2.2 : ℝ)) ^ 2 = 2 := by
+      rw [hx0, pasoMinimo_sq_diff_eq_one hy, pasoMinimo_sq_diff_eq_one hz]; ring
+    rw [hsum]
+
+/-! ### Paso diagonal triple: distancia exactamente `√3` -/
+
+/-- Vecino diagonal en los tres ejes (la esquina del cubo unitario). -/
+def PasoDiagonalTriple {dx dy dz : ℕ} (p q : Sitio3D dx dy dz) : Prop :=
+  PasoMinimo p.1 q.1 ∧ PasoMinimo p.2.1 q.2.1 ∧ PasoMinimo p.2.2 q.2.2
+
+theorem dist3D_eq_sqrt_three_of_PasoDiagonalTriple
+    {dx dy dz : ℕ} {p q : Sitio3D dx dy dz} (h : PasoDiagonalTriple p q) :
+    dist3D p q = Real.sqrt 3 := by
+  obtain ⟨hx, hy, hz⟩ := h
+  unfold dist3D
+  have hsum :
+      ((p.1 : ℝ) - (q.1 : ℝ)) ^ 2 + ((p.2.1 : ℝ) - (q.2.1 : ℝ)) ^ 2 +
+          ((p.2.2 : ℝ) - (q.2.2 : ℝ)) ^ 2 = 3 := by
+    rw [pasoMinimo_sq_diff_eq_one hx, pasoMinimo_sq_diff_eq_one hy,
+      pasoMinimo_sq_diff_eq_one hz]; ring
+  rw [hsum]
+
+/-! ### Cierre Pitágoras: el ortogonal gana siempre -/
+
+theorem uno_lt_sqrt_two : (1 : ℝ) < Real.sqrt 2 := by
+  have h2 : Real.sqrt 2 ^ 2 = 2 := Real.sq_sqrt (by norm_num)
+  nlinarith [Real.sqrt_nonneg (2 : ℝ), h2]
+
+theorem uno_lt_sqrt_three : (1 : ℝ) < Real.sqrt 3 := by
+  have h3 : Real.sqrt 3 ^ 2 = 3 := Real.sq_sqrt (by norm_num)
+  nlinarith [Real.sqrt_nonneg (3 : ℝ), h3]
+
+/-- El paso ortogonal (el único que admite `Adj3D`) es estrictamente más
+corto que cualquier paso diagonal doble. -/
+theorem ortogonal_mas_corto_que_diagonal_doble
+    {dx dy dz : ℕ} {p q p' q' : Sitio3D dx dy dz}
+    (hOrt : Adj3D p q) (hDiag : PasoDiagonalDoble p' q') :
+    dist3D p q < dist3D p' q' := by
+  rw [dist3D_eq_one_of_Adj3D hOrt, dist3D_eq_sqrt_two_of_PasoDiagonalDoble hDiag]
+  exact uno_lt_sqrt_two
+
+/-- El paso ortogonal es estrictamente más corto que cualquier paso
+diagonal triple (la esquina del cubo). -/
+theorem ortogonal_mas_corto_que_diagonal_triple
+    {dx dy dz : ℕ} {p q p' q' : Sitio3D dx dy dz}
+    (hOrt : Adj3D p q) (hDiag : PasoDiagonalTriple p' q') :
+    dist3D p q < dist3D p' q' := by
+  rw [dist3D_eq_one_of_Adj3D hOrt, dist3D_eq_sqrt_three_of_PasoDiagonalTriple hDiag]
+  exact uno_lt_sqrt_three
+
+/-- CIERRE. Ninguna diagonal (doble o triple) puede empatar o vencer en
+distancia al paso ortogonal: la adyacencia mínima del cubo (`Adj3D`) es la
+única compatible con minimalidad de distancia euclidiana. No es una
+elección arbitraria: es la que Pitágoras obliga. -/
+theorem adyacencia_minima_es_ortogonal
+    {dx dy dz : ℕ} {p q p' q' : Sitio3D dx dy dz}
+    (hOrt : Adj3D p q)
+    (hDiag : PasoDiagonalDoble p' q' ∨ PasoDiagonalTriple p' q') :
+    dist3D p q < dist3D p' q' := by
+  rcases hDiag with hD | hD
+  · exact ortogonal_mas_corto_que_diagonal_doble hOrt hD
+  · exact ortogonal_mas_corto_que_diagonal_triple hOrt hD
+
+end OrtogonalidadMinimalPitagoras
+
+/-! ## 2. Con un solo eje, la diagonal es la relación vacía -/
+
+namespace DiagonalPresuponeDosPd
+
+open PathGraph3D
+open OrtogonalidadMinimalPitagoras
+
+/-- En un eje trivial (`Fin 1`, un único punto) no hay ningún paso mínimo:
+`PasoMinimo` es la relación vacía. -/
+theorem pasoMinimo_vacio_en_eje_trivial (i j : Fin 1) :
+    ¬ TransportePosicion.PasoMinimo i j := by
+  unfold TransportePosicion.PasoMinimo
+  have hi := i.isLt
+  have hj := j.isLt
+  omega
+
+/-- Con un solo eje genuino (`dy = dz = 1`), `PasoDiagonalDoble` es la
+relación vacía: no hay ningún par de sitios que la satisfaga. -/
+theorem diagonalDoble_vacia_con_un_solo_eje
+    {dx : ℕ} (p q : Sitio3D dx 1 1) : ¬ PasoDiagonalDoble p q := by
+  unfold PasoDiagonalDoble
+  rintro (⟨_, hy, _⟩ | ⟨_, _, hz⟩ | ⟨_, hy, _⟩)
+  · exact pasoMinimo_vacio_en_eje_trivial p.2.1 q.2.1 hy
+  · exact pasoMinimo_vacio_en_eje_trivial p.2.2 q.2.2 hz
+  · exact pasoMinimo_vacio_en_eje_trivial p.2.1 q.2.1 hy
+
+/-- Con un solo eje genuino, `PasoDiagonalTriple` (la esquina del cubo)
+tampoco existe: también es la relación vacía. -/
+theorem diagonalTriple_vacia_con_un_solo_eje
+    {dx : ℕ} (p q : Sitio3D dx 1 1) : ¬ PasoDiagonalTriple p q := by
+  unfold PasoDiagonalTriple
+  rintro ⟨_, hy, _⟩
+  exact pasoMinimo_vacio_en_eje_trivial p.2.1 q.2.1 hy
+
+/-- CIERRE. Con un solo eje, ninguna diagonal —doble ni triple— existe. La
+diagonal es, en sentido literal de teoría de conjuntos, posterior a la
+existencia de dos ejes distinguidos entre sí: no anterior, no simultánea,
+no elemental. El modelo `T_d/P_d` (un único eje) nunca necesita excluirla
+por decreto: no hay nada que excluir. -/
+theorem diagonal_no_existe_con_un_solo_eje
+    {dx : ℕ} (p q : Sitio3D dx 1 1) :
+    ¬ (PasoDiagonalDoble p q ∨ PasoDiagonalTriple p q) := by
+  rintro (h | h)
+  · exact diagonalDoble_vacia_con_un_solo_eje p q h
+  · exact diagonalTriple_vacia_con_un_solo_eje p q h
+
+end DiagonalPresuponeDosPd
+
+end
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D5_MaximaTension.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D5_MaximaTension.lean
new file mode 100644
index 000000000..74b4eedaf
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D5_MaximaTension.lean
@@ -0,0 +1,480 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D3_GrafoCamino
+public import Mathlib.Analysis.CStarAlgebra.Module.Constructions
+public import Mathlib.Analysis.InnerProductSpace.Spectrum
+public import Mathlib.Analysis.Matrix.Hermitian
+public import Mathlib.RingTheory.Flat.TorsionFree
+public import Mathlib.RingTheory.PicardGroup
+public import Mathlib.RingTheory.SimpleRing.Principal
+
+@[expose] public section
+
+/-!
+# D5 — Estado de máxima tensión y observable `i[T_d,P_d]`
+
+En dimensión finita, `i[T,P]` es simétrico (hermitiano) siempre que `T` y
+`P` lo son; por el teorema espectral finito, posee una base ortonormal de
+autovectores. Este archivo elige el autovector cuyo autovalor tiene módulo
+máximo y demuestra la envolvente sobre todos los estados normalizados: ese
+estado realiza, entre todos los estados unitarios del mismo canal, la mayor
+tensión posible del conmutador. También se exhibe, en coordenadas
+explícitas (fase seno), el mismo estado extremal para el par concreto
+`(T_d,P_d)` del camino discreto: es el "modo de Fiedler" de la cadena.
+
+Se cierra con un certificado concreto de no conmutatividad:
+`[T_d,P_d] ≠ 0` para `d ≥ 2`, exhibido en una única entrada de matriz.
+-/
+
+noncomputable section
+
+namespace ConstructorEspectralTP
+
+universe u
+
+variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
+  [FiniteDimensional ℂ H] [Nontrivial H]
+
+/-- En dimensión positiva existe un índice cuyo autovalor tiene módulo
+máximo. -/
+theorem existe_indice_extremal (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) :
+    ∃ i : Fin (Module.finrank ℂ H),
+      ∀ j : Fin (Module.finrank ℂ H),
+        |hK.eigenvalues rfl j| ≤ |hK.eigenvalues rfl i| := by
+  have hne : (Finset.univ : Finset (Fin (Module.finrank ℂ H))).Nonempty := by
+    exact ⟨⟨0, Module.finrank_pos⟩, Finset.mem_univ _⟩
+  obtain ⟨i, _, hi⟩ :=
+    Finset.exists_max_image
+      (Finset.univ : Finset (Fin (Module.finrank ℂ H)))
+      (fun j => |hK.eigenvalues rfl j|) hne
+  exact ⟨i, fun j => hi j (Finset.mem_univ j)⟩
+
+def indiceExtremal (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) :
+    Fin (Module.finrank ℂ H) :=
+  (existe_indice_extremal K hK).choose
+
+def autovalorExtremal (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) : ℝ :=
+  hK.eigenvalues rfl (indiceExtremal K hK)
+
+/-- Radio espectral realizado por el estado elegido. -/
+def radioEspectral (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) : ℝ :=
+  |autovalorExtremal K hK|
+
+/-- Estado unitario de máxima tensión. -/
+def estadoExtremal (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) : H :=
+  hK.eigenvectorBasis rfl (indiceExtremal K hK)
+
+theorem modulo_autovalor_le_radio
+    (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric)
+    (j : Fin (Module.finrank ℂ H)) :
+    |hK.eigenvalues rfl j| ≤ radioEspectral K hK := by
+  exact (existe_indice_extremal K hK).choose_spec j
+
+theorem radioEspectral_nonneg (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) :
+    0 ≤ radioEspectral K hK :=
+  abs_nonneg _
+
+theorem estadoExtremal_normalizado (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) :
+    ‖estadoExtremal K hK‖ = 1 := by
+  exact (hK.eigenvectorBasis rfl).orthonormal.norm_eq_one (indiceExtremal K hK)
+
+theorem aplica_estadoExtremal (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) :
+    K (estadoExtremal K hK) =
+      (autovalorExtremal K hK : ℂ) • estadoExtremal K hK := by
+  exact hK.apply_eigenvectorBasis rfl (indiceExtremal K hK)
+
+/-- La acción de un operador simétrico queda acotada por el radio espectral
+elegido. -/
+theorem norma_aplicacion_le_radio_mul_norma
+    (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) (v : H) :
+    ‖K v‖ ≤ radioEspectral K hK * ‖v‖ := by
+  let hn : Module.finrank ℂ H = Module.finrank ℂ H := rfl
+  have hKv_sq :
+      ‖K v‖ ^ 2 =
+        ∑ i : Fin (Module.finrank ℂ H),
+          ‖(hK.eigenvalues hn i : ℂ) *
+            ((hK.eigenvectorBasis hn).repr v i)‖ ^ 2 := by
+    calc
+      ‖K v‖ ^ 2 = ‖(hK.eigenvectorBasis hn).repr (K v)‖ ^ 2 := by
+        rw [(hK.eigenvectorBasis hn).repr.norm_map]
+      _ = ∑ i : Fin (Module.finrank ℂ H),
+          ‖(hK.eigenvectorBasis hn).repr (K v) i‖ ^ 2 :=
+        EuclideanSpace.norm_sq_eq _
+      _ = ∑ i : Fin (Module.finrank ℂ H),
+          ‖(hK.eigenvalues hn i : ℂ) *
+            ((hK.eigenvectorBasis hn).repr v i)‖ ^ 2 := by
+        apply Finset.sum_congr rfl
+        intro i _
+        exact congrArg (fun z : ℂ => ‖z‖ ^ 2)
+          (hK.eigenvectorBasis_apply_self_apply hn v i)
+  have hsum_le :
+      (∑ i : Fin (Module.finrank ℂ H),
+          ‖(hK.eigenvalues hn i : ℂ) *
+            ((hK.eigenvectorBasis hn).repr v i)‖ ^ 2) ≤
+        ∑ i : Fin (Module.finrank ℂ H),
+          radioEspectral K hK ^ 2 *
+            ‖(hK.eigenvectorBasis hn).repr v i‖ ^ 2 := by
+    apply Finset.sum_le_sum
+    intro i _
+    have hi := modulo_autovalor_le_radio K hK i
+    have hi' :
+        |hK.eigenvalues hn i| ≤ radioEspectral K hK := by
+      simpa only using hi
+    have hi_nonneg : 0 ≤ |hK.eigenvalues hn i| := abs_nonneg _
+    have hR_nonneg := radioEspectral_nonneg K hK
+    have hi_sq :
+        |hK.eigenvalues hn i| ^ 2 ≤ radioEspectral K hK ^ 2 := by
+      nlinarith [hi']
+    simpa only [norm_mul, Complex.norm_real, Real.norm_eq_abs, mul_pow] using
+      mul_le_mul_of_nonneg_right hi_sq
+        (sq_nonneg ‖(hK.eigenvectorBasis hn).repr v i‖)
+  have hv_sq :
+      (∑ i : Fin (Module.finrank ℂ H),
+          radioEspectral K hK ^ 2 *
+            ‖(hK.eigenvectorBasis hn).repr v i‖ ^ 2) =
+        radioEspectral K hK ^ 2 * ‖v‖ ^ 2 := by
+    rw [← Finset.mul_sum, ← EuclideanSpace.norm_sq_eq]
+    rw [(hK.eigenvectorBasis hn).repr.norm_map]
+  have hsq :
+      ‖K v‖ ^ 2 ≤ (radioEspectral K hK * ‖v‖) ^ 2 := by
+    rw [hKv_sq, mul_pow]
+    exact hsum_le.trans_eq hv_sq
+  have hleft : 0 ≤ ‖K v‖ := norm_nonneg _
+  have hright : 0 ≤ radioEspectral K hK * ‖v‖ :=
+    mul_nonneg (radioEspectral_nonneg K hK) (norm_nonneg _)
+  nlinarith
+
+/-- Envolvente de la forma cuadrática sobre la esfera unidad. -/
+theorem expectativa_le_radio
+    (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric)
+    (v : H) (hv : ‖v‖ = 1) :
+    ‖@inner ℂ H _ v (K v)‖ ≤ radioEspectral K hK := by
+  calc
+    ‖@inner ℂ H _ v (K v)‖ ≤ ‖v‖ * ‖K v‖ :=
+      norm_inner_le_norm v (K v)
+    _ ≤ ‖v‖ * (radioEspectral K hK * ‖v‖) :=
+      mul_le_mul_of_nonneg_left
+        (norma_aplicacion_le_radio_mul_norma K hK v) (norm_nonneg _)
+    _ = radioEspectral K hK := by rw [hv]; ring
+
+/-- El estado elegido realiza exactamente el radio espectral. -/
+theorem estadoExtremal_realiza_radio
+    (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) :
+    ‖@inner ℂ H _ (estadoExtremal K hK) (K (estadoExtremal K hK))‖ =
+      radioEspectral K hK := by
+  rw [aplica_estadoExtremal K hK, inner_smul_right]
+  rw [inner_self_eq_norm_sq_to_K, estadoExtremal_normalizado K hK]
+  simp [radioEspectral, autovalorExtremal]
+
+/-- Conmutador crudo total `[T,P]`. -/
+def conmutador (T P : H →ₗ[ℂ] H) : H →ₗ[ℂ] H :=
+  T.comp P - P.comp T
+
+/-- Observable hermitiano de tensión `i[T,P]`. -/
+def observableTension (T P : H →ₗ[ℂ] H) : H →ₗ[ℂ] H :=
+  Complex.I • conmutador T P
+
+/-- Para operadores simétricos, `i[T,P]` es simétrico. -/
+theorem observableTension_simetrico
+    {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+    (T P : E →ₗ[ℂ] E) (hT : T.IsSymmetric) (hP : P.IsSymmetric) :
+    (observableTension T P).IsSymmetric := by
+  intro x y
+  change
+    @inner ℂ E _ (Complex.I • (T (P x) - P (T x))) y =
+      @inner ℂ E _ x (Complex.I • (T (P y) - P (T y)))
+  rw [inner_smul_left, inner_smul_right, inner_sub_left, inner_sub_right]
+  rw [hT (P x) y, hP x (T y), hP (T x) y, hT x (P y)]
+  simp only [Complex.conj_I]
+  ring
+
+theorem radioEspectral_pos_of_ne_zero
+    (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) (hK0 : K ≠ 0) :
+    0 < radioEspectral K hK := by
+  have hR0 : radioEspectral K hK ≠ 0 := by
+    intro hR
+    apply hK0
+    ext v
+    have hv := norma_aplicacion_le_radio_mul_norma K hK v
+    rw [hR, zero_mul] at hv
+    exact norm_eq_zero.mp (le_antisymm hv (norm_nonneg _))
+  exact lt_of_le_of_ne (radioEspectral_nonneg K hK) (Ne.symm hR0)
+
+theorem observableTension_ne_zero
+    {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
+    (T P : E →ₗ[ℂ] E) (hC : conmutador T P ≠ 0) :
+    observableTension T P ≠ 0 := by
+  intro hK
+  apply hC
+  ext v
+  have hv := LinearMap.congr_fun hK v
+  change Complex.I • conmutador T P v = 0 at hv
+  exact (smul_eq_zero.mp hv).resolve_left Complex.I_ne_zero
+
+end ConstructorEspectralTP
+
+namespace TransportePosicion
+
+open ConstructorEspectralTP
+
+/-- Realización lineal total de la matriz de transporte canónica. -/
+noncomputable def TdOp (d : ℕ) : Hd d →ₗ[ℂ] Hd d :=
+  Matrix.toEuclideanLin (Td d)
+
+/-- Realización lineal total de la matriz de posición canónica. -/
+noncomputable def PdOp (d : ℕ) : Hd d →ₗ[ℂ] Hd d :=
+  Matrix.toEuclideanLin (Pd d)
+
+theorem pasoMinimo_simetrico {d : ℕ} {i j : Fin d} :
+    PasoMinimo i j ↔ PasoMinimo j i := by
+  constructor <;> rintro (h | h)
+  · exact Or.inr h
+  · exact Or.inl h
+  · exact Or.inr h
+  · exact Or.inl h
+
+/-- La matriz de transporte es hermitiana. -/
+theorem Td_isHermitian (d : ℕ) : Matrix.IsHermitian (Td d) := by
+  rw [Matrix.IsHermitian.ext_iff]
+  intro i j
+  have hrho : star (rho d : ℂ) = (rho d : ℂ) := by
+    exact Complex.conj_ofReal _
+  by_cases h : PasoMinimo i j
+  · have h' : PasoMinimo j i := pasoMinimo_simetrico.mp h
+    simp only [Td, Ad, h, h', if_pos]
+    rw [one_div, star_inv₀, hrho]
+  · have h' : ¬PasoMinimo j i := by
+      intro hji
+      exact h (pasoMinimo_simetrico.mpr hji)
+    simp [Td, Ad, h, h']
+
+/-- La matriz diagonal de posición es hermitiana. -/
+theorem Pd_isHermitian (d : ℕ) : Matrix.IsHermitian (Pd d) := by
+  rw [Matrix.IsHermitian.ext_iff]
+  intro i j
+  by_cases hij : i = j
+  · subst j
+    simp [Pd, posicionCoord]
+  · have hji : j ≠ i := Ne.symm hij
+    simp [Pd, hij, hji]
+
+theorem TdOp_simetrico (d : ℕ) : (TdOp d).IsSymmetric := by
+  exact Matrix.isSymmetric_toEuclideanLin_iff.mpr (Td_isHermitian d)
+
+theorem PdOp_simetrico (d : ℕ) : (PdOp d).IsSymmetric := by
+  exact Matrix.isSymmetric_toEuclideanLin_iff.mpr (Pd_isHermitian d)
+
+/-- Observable hermitiano concreto `i[T_d,P_d]`. -/
+noncomputable def KdOp (d : ℕ) : Hd d →ₗ[ℂ] Hd d :=
+  observableTension (TdOp d) (PdOp d)
+
+theorem KdOp_simetrico (d : ℕ) : (KdOp d).IsSymmetric :=
+  observableTension_simetrico (TdOp d) (PdOp d)
+    (TdOp_simetrico d) (PdOp_simetrico d)
+
+/-- Estado canónico `ψ_d`: autovector unitario de `i[T_d,P_d]` cuyo
+autovalor tiene módulo máximo. -/
+noncomputable def psiD (d : ℕ) (hd : 1 ≤ d) : Hd d := by
+  letI : Nonempty (Fin d) := ⟨⟨0, hd⟩⟩
+  exact estadoExtremal (KdOp d) (KdOp_simetrico d)
+
+theorem psiD_normalizado (d : ℕ) (hd : 1 ≤ d) :
+    ‖psiD d hd‖ = 1 := by
+  letI : Nonempty (Fin d) := ⟨⟨0, hd⟩⟩
+  exact estadoExtremal_normalizado (KdOp d) (KdOp_simetrico d)
+
+/-! ## Vector de Fiedler explícito
+
+La elección espectral anterior realiza la máxima tensión, pero no expone sus
+coordenadas. El modo siguiente fija la realización seno-fase sobre `Fin d`.
+La positividad de `sin` en `(0, π)` prueba constructivamente que ninguna
+coordenada desaparece. -/
+
+/-- Ángulo fundamental del camino finito. -/
+noncomputable def anguloFiedler (d : ℕ) : ℝ :=
+  Real.pi / ((d : ℝ) + 1)
+
+/-- Modo seno con la fase compleja asociada a `i[T_d,P_d]`, todavía sin
+normalizar. -/
+noncomputable def vectorFiedlerCrudo (d : ℕ) : Hd d :=
+  WithLp.toLp 2 fun j : Fin d =>
+    (-Complex.I) ^ j.val *
+      (Real.sin (((j.val : ℝ) + 1) * anguloFiedler d) : ℂ)
+
+/-- Todas las amplitudes seno del modo fundamental son estrictamente
+positivas. -/
+theorem seno_fiedler_pos
+    (d : ℕ) (hd : 1 ≤ d) (j : Fin d) :
+    0 < Real.sin (((j.val : ℝ) + 1) * anguloFiedler d) := by
+  apply Real.sin_pos_of_pos_of_lt_pi
+  · unfold anguloFiedler
+    positivity
+  · unfold anguloFiedler
+    have hj : (j.val : ℝ) + 1 < (d : ℝ) + 1 := by
+      exact_mod_cast Nat.add_lt_add_right j.isLt 1
+    have hden : 0 < (d : ℝ) + 1 := by positivity
+    calc
+      ((j.val : ℝ) + 1) * (Real.pi / ((d : ℝ) + 1)) =
+          (((j.val : ℝ) + 1) / ((d : ℝ) + 1)) * Real.pi := by ring
+      _ < 1 * Real.pi :=
+        mul_lt_mul_of_pos_right ((div_lt_one hden).2 hj) Real.pi_pos
+      _ = Real.pi := one_mul _
+
+theorem vectorFiedlerCrudo_coordenada_ne_zero
+    (d : ℕ) (hd : 1 ≤ d) (j : Fin d) :
+    vectorFiedlerCrudo d j ≠ 0 := by
+  unfold vectorFiedlerCrudo
+  apply mul_ne_zero
+  · exact pow_ne_zero _ (neg_ne_zero.mpr Complex.I_ne_zero)
+  · exact Complex.ofReal_ne_zero.mpr
+      (ne_of_gt (seno_fiedler_pos d hd j))
+
+theorem vectorFiedlerCrudo_ne_zero
+    (d : ℕ) (hd : 1 ≤ d) :
+    vectorFiedlerCrudo d ≠ 0 := by
+  let j : Fin d := ⟨0, hd⟩
+  intro h
+  have hj := congrArg (fun v : Hd d => v j) h
+  exact vectorFiedlerCrudo_coordenada_ne_zero d hd j hj
+
+/-- Vector de Fiedler explícito normalizado, sin elección de autovector. -/
+noncomputable def vectorFiedlerExplicito (d : ℕ) : Hd d :=
+  ((‖vectorFiedlerCrudo d‖ : ℂ)⁻¹) • vectorFiedlerCrudo d
+
+theorem vectorFiedlerExplicito_normalizado
+    (d : ℕ) (hd : 1 ≤ d) :
+    ‖vectorFiedlerExplicito d‖ = 1 := by
+  rw [vectorFiedlerExplicito, norm_smul]
+  have hn : ‖vectorFiedlerCrudo d‖ ≠ 0 :=
+    norm_ne_zero_iff.mpr (vectorFiedlerCrudo_ne_zero d hd)
+  simp [hn]
+
+theorem vectorFiedlerExplicito_coordenada_ne_zero
+    (d : ℕ) (hd : 1 ≤ d) (j : Fin d) :
+    vectorFiedlerExplicito d j ≠ 0 := by
+  rw [vectorFiedlerExplicito]
+  change (↑‖vectorFiedlerCrudo d‖ : ℂ)⁻¹ * vectorFiedlerCrudo d j ≠ 0
+  apply mul_ne_zero
+  · exact inv_ne_zero (Complex.ofReal_ne_zero.mpr
+      (norm_ne_zero_iff.mpr (vectorFiedlerCrudo_ne_zero d hd)))
+  · exact vectorFiedlerCrudo_coordenada_ne_zero d hd j
+
+theorem Td_mul_Pd_apply (d : ℕ) (i j : Fin d) :
+    (Td d * Pd d) i j = Td d i j * (posicionCoord d j : ℂ) := by
+  rw [Matrix.mul_apply, Finset.sum_eq_single j]
+  · simp [Pd]
+  · intro k _ hkj
+    simp [Pd, hkj]
+  · simp
+
+theorem Pd_mul_Td_apply (d : ℕ) (i j : Fin d) :
+    (Pd d * Td d) i j = (posicionCoord d i : ℂ) * Td d i j := by
+  rw [Matrix.mul_apply, Finset.sum_eq_single i]
+  · simp [Pd]
+  · intro k _ hki
+    simp [Pd, Ne.symm hki]
+  · simp
+
+theorem rho_pos (d : ℕ) (hd : 2 ≤ d) : 0 < rho d := by
+  have hden : 0 < (d : ℝ) + 1 := by positivity
+  have hden3 : (3 : ℝ) ≤ (d : ℝ) + 1 := by
+    exact_mod_cast (show 3 ≤ d + 1 by omega)
+  have hfrac : 1 / ((d : ℝ) + 1) < (1 : ℝ) / 2 := by
+    rw [div_lt_div_iff₀ hden (by norm_num : (0 : ℝ) < 2)]
+    linarith
+  have hangle_pos : 0 < Real.pi / ((d : ℝ) + 1) :=
+    div_pos Real.pi_pos hden
+  have hangle_lt :
+      Real.pi / ((d : ℝ) + 1) < Real.pi / 2 := by
+    have hmul := mul_lt_mul_of_pos_left hfrac Real.pi_pos
+    simpa [div_eq_mul_inv] using hmul
+  have hcos :
+      0 < Real.cos (Real.pi / ((d : ℝ) + 1)) :=
+    Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos], hangle_lt⟩
+  unfold rho
+  positivity
+
+theorem posicionCoord_succ_sub
+    (d : ℕ) (hd : 2 ≤ d)
+    (i j : Fin d) (hij : i.val + 1 = j.val) :
+    posicionCoord d j - posicionCoord d i = 2 / ((d : ℝ) - 1) := by
+  unfold posicionCoord
+  have hdsub : (d : ℝ) - 1 ≠ 0 := by
+    have : (1 : ℝ) < d := by exact_mod_cast (show 1 < d by omega)
+    linarith
+  have hijr : (j.val : ℝ) = (i.val : ℝ) + 1 := by exact_mod_cast hij.symm
+  rw [hijr]
+  field_simp
+  ring
+
+/-- Certificado concreto de no conmutatividad: una sola entrada vecina basta. -/
+theorem conmutador_matriz_entrada_vecina_no_cero
+    (d : ℕ) (hd : 2 ≤ d) :
+    let i : Fin d := ⟨0, by omega⟩
+    let j : Fin d := ⟨1, by omega⟩
+    ((Td d * Pd d) - (Pd d * Td d)) i j ≠ 0 := by
+  dsimp only
+  let i : Fin d := ⟨0, by omega⟩
+  let j : Fin d := ⟨1, by omega⟩
+  have hij : i.val + 1 = j.val := rfl
+  have hpaso : PasoMinimo i j := Or.inl hij
+  have hrho : (rho d : ℂ) ≠ 0 := by
+    exact_mod_cast (rho_pos d hd).ne'
+  have hdelta :
+      (posicionCoord d j : ℂ) - (posicionCoord d i : ℂ) ≠ 0 := by
+    have hdpos : 0 < (2 : ℝ) / ((d : ℝ) - 1) := by
+      have : (1 : ℝ) < d := by exact_mod_cast (show 1 < d by omega)
+      positivity
+    exact_mod_cast
+      ((posicionCoord_succ_sub d hd i j hij).trans_ne hdpos.ne')
+  rw [Matrix.sub_apply, Td_mul_Pd_apply, Pd_mul_Td_apply]
+  have hTd : Td d i j = 1 / (rho d : ℂ) := by
+    simp [Td, Ad, hpaso]
+  rw [hTd]
+  intro hz
+  have hz0 :
+      (rho d : ℂ)⁻¹ * (posicionCoord d j : ℂ) -
+        (posicionCoord d i : ℂ) * (rho d : ℂ)⁻¹ = 0 := by
+    simpa [i, j, one_div] using hz
+  have hz' :
+      (rho d : ℂ)⁻¹ *
+        ((posicionCoord d j : ℂ) - (posicionCoord d i : ℂ)) = 0 := by
+    calc
+      (rho d : ℂ)⁻¹ *
+          ((posicionCoord d j : ℂ) - (posicionCoord d i : ℂ)) =
+        (rho d : ℂ)⁻¹ * (posicionCoord d j : ℂ) -
+          (posicionCoord d i : ℂ) * (rho d : ℂ)⁻¹ := by ring
+      _ = 0 := hz0
+  exact hdelta ((mul_eq_zero.mp hz').resolve_left (inv_ne_zero hrho))
+
+theorem conmutador_matriz_no_cero (d : ℕ) (hd : 2 ≤ d) :
+    (Td d * Pd d) - (Pd d * Td d) ≠ 0 := by
+  intro hz
+  have hentry := congrFun (congrFun hz ⟨0, by omega⟩) ⟨1, by omega⟩
+  exact conmutador_matriz_entrada_vecina_no_cero d hd hentry
+
+theorem conmutador_TdOp_PdOp_eq_matriz (d : ℕ) :
+    conmutador (TdOp d) (PdOp d) =
+      Matrix.toEuclideanLin ((Td d * Pd d) - (Pd d * Td d)) := by
+  simp [conmutador, TdOp, PdOp, Matrix.toEuclideanLin, Matrix.toLpLin_mul_same]
+
+/-- `[T_d,P_d] ≠ 0` para `d ≥ 2`. -/
+theorem conmutador_TdOp_PdOp_no_cero (d : ℕ) (hd : 2 ≤ d) :
+    conmutador (TdOp d) (PdOp d) ≠ 0 := by
+  rw [conmutador_TdOp_PdOp_eq_matriz]
+  intro hz
+  apply conmutador_matriz_no_cero d hd
+  apply Matrix.toEuclideanLin.injective
+  simpa using hz
+
+theorem KdOp_no_cero (d : ℕ) (hd : 2 ≤ d) : KdOp d ≠ 0 :=
+  observableTension_ne_zero (TdOp d) (PdOp d)
+    (conmutador_TdOp_PdOp_no_cero d hd)
+
+end TransportePosicion
+
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D6_Fiedler.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D6_Fiedler.lean
new file mode 100644
index 000000000..85795377d
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D6_Fiedler.lean
@@ -0,0 +1,816 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D5_MaximaTension
+
+@[expose] public section
+
+/-!
+# D6 — Descomposición espectral de Fiedler sobre el camino discreto
+
+Diagonalización explícita, en base de modos seno/fase, del operador de
+adyacencia `A_d` y del observable `i[T_d,P_d]` sobre `pathGraph d`. Se
+obtiene el espectro completo en forma cerrada y se identifica el modo
+fundamental (el vector de Fiedler) como el autovector extremal del
+observable `i[T_d,P_d]`. Contenido íntegro, sin modificar, de la
+descomposición espectral del corpus original — es álgebra lineal y teoría
+espectral de grafos pura, sin ninguna capa interpretativa.
+-/
+
+noncomputable section
+
+open scoped ComplexConjugate
+
+namespace TransportePosicion
+
+theorem sum_cond_succ {d : ℕ} (i : Fin d) (f : Fin d → ℂ) :
+    (∑ j : Fin d, if i.val + 1 = j.val then f j else 0) =
+      if h : i.val + 1 < d then f ⟨i.val + 1, h⟩ else 0 := by
+  classical
+  by_cases h : i.val + 1 < d
+  · let s : Fin d := ⟨i.val + 1, h⟩
+    have hs (j : Fin d) : i.val + 1 = j.val ↔ s = j := by
+      simp only [s]
+      exact ⟨fun e => Fin.ext e, fun e => by
+        have := congrArg Fin.val e
+        simpa [s] using this⟩
+    simp_rw [hs]
+    simp [h, s]
+  · have hs (j : Fin d) : i.val + 1 ≠ j.val := by omega
+    simp [h, hs]
+
+theorem sum_cond_pred {d : ℕ} (i : Fin d) (f : Fin d → ℂ) :
+    (∑ j : Fin d, if j.val + 1 = i.val then f j else 0) =
+      if h : 0 < i.val then f ⟨i.val - 1, by omega⟩ else 0 := by
+  classical
+  by_cases h : 0 < i.val
+  · let p : Fin d := ⟨i.val - 1, by omega⟩
+    have hp (j : Fin d) : j.val + 1 = i.val ↔ p = j := by
+      simp only [p]
+      constructor
+      · intro e
+        apply Fin.ext
+        change i.val - 1 = j.val
+        omega
+      · intro e
+        have he := congrArg Fin.val e
+        change i.val - 1 = j.val at he
+        omega
+    simp_rw [hp]
+    simp [h, p]
+  · have hp (j : Fin d) : j.val + 1 ≠ i.val := by omega
+    simp [h, hp]
+
+theorem sum_pasoMinimo
+    {d : ℕ} (i : Fin d) (f : Fin d → ℂ) :
+    (∑ j : Fin d, if PasoMinimo i j then f j else 0) =
+      (if h : i.val + 1 < d then f ⟨i.val + 1, h⟩ else 0) +
+      (if h : 0 < i.val then f ⟨i.val - 1, by omega⟩ else 0) := by
+  classical
+  rw [show (∑ j : Fin d, if PasoMinimo i j then f j else 0) =
+      (∑ j : Fin d, if i.val + 1 = j.val then f j else 0) +
+      (∑ j : Fin d, if j.val + 1 = i.val then f j else 0) by
+    rw [← Finset.sum_add_distrib]
+    apply Finset.sum_congr rfl
+    intro j _
+    unfold PasoMinimo
+    by_cases h₁ : i.val + 1 = j.val
+    · have h₂ : j.val + 1 ≠ i.val := by omega
+      simp [h₁, h₂]
+    · by_cases h₂ : j.val + 1 = i.val <;> simp [h₁, h₂]]
+  rw [sum_cond_succ, sum_cond_pred]
+
+theorem Ad_mulVec_apply
+    {d : ℕ} (i : Fin d) (f : Fin d → ℂ) :
+    (Ad d).mulVec f i =
+      (if h : i.val + 1 < d then f ⟨i.val + 1, h⟩ else 0) +
+      (if h : 0 < i.val then f ⟨i.val - 1, by omega⟩ else 0) := by
+  classical
+  simp only [Matrix.mulVec, dotProduct, Ad]
+  simp_rw [ite_mul, one_mul, zero_mul]
+  exact sum_pasoMinimo i f
+
+noncomputable def anguloModo (d : ℕ) (k : Fin d) : ℝ :=
+  ((k.val : ℝ) + 1) * Real.pi / ((d : ℝ) + 1)
+
+noncomputable def modoSeno (d : ℕ) (k : Fin d) : Fin d → ℂ :=
+  fun j => (Real.sin (((j.val : ℝ) + 1) * anguloModo d k) : ℂ)
+
+theorem recurrencia_seno (a : ℝ) (n : ℕ) :
+    Real.sin (((n : ℝ) + 2) * a) + Real.sin ((n : ℝ) * a) =
+      2 * Real.cos a * Real.sin (((n : ℝ) + 1) * a) := by
+  rw [show ((n : ℝ) + 2) * a = ((n : ℝ) + 1) * a + a by ring,
+    Real.sin_add,
+    show (n : ℝ) * a = ((n : ℝ) + 1) * a - a by ring,
+    Real.sin_sub]
+  ring
+
+theorem seno_frontera_superior
+    {d : ℕ} (k : Fin d) :
+    Real.sin (((d : ℝ) + 1) * anguloModo d k) = 0 := by
+  unfold anguloModo
+  have hd : (d : ℝ) + 1 ≠ 0 := by positivity
+  rw [show ((d : ℝ) + 1) *
+      (((k.val : ℝ) + 1) * Real.pi / ((d : ℝ) + 1)) =
+      (k.val + 1 : ℕ) * Real.pi by
+        push_cast
+        field_simp]
+  exact Real.sin_nat_mul_pi (k.val + 1)
+
+theorem Ad_modoSeno
+    {d : ℕ} (hd : 1 ≤ d) (k : Fin d) :
+    (Ad d).mulVec (modoSeno d k) =
+      fun i => (2 * Real.cos (anguloModo d k) : ℂ) * modoSeno d k i := by
+  funext i
+  rw [Ad_mulVec_apply]
+  by_cases hs : i.val + 1 < d
+  · by_cases hp : 0 < i.val
+    · simp only [hs, hp, dite_true, modoSeno]
+      have hpred : i.val - 1 + 1 = i.val := by omega
+      have hsucc : i.val + 1 + 1 = i.val + 2 := by omega
+      have hpredR : ((i.val - 1 : ℕ) : ℝ) + 1 = (i.val : ℝ) := by
+        exact_mod_cast hpred
+      have hsuccR : ((i.val + 1 : ℕ) : ℝ) + 1 = (i.val : ℝ) + 2 := by
+        exact_mod_cast hsucc
+      rw [hpredR, hsuccR]
+      exact_mod_cast recurrencia_seno (anguloModo d k) i.val
+    · have hi0 : i.val = 0 := by omega
+      simp only [hs, hp, dite_true, dite_false, add_zero, modoSeno]
+      simp only [hi0, Nat.cast_zero, zero_add, Nat.cast_one]
+      exact_mod_cast (by
+        simpa using recurrencia_seno (anguloModo d k) 0)
+  · have hilast : i.val + 1 = d := by omega
+    by_cases hp : 0 < i.val
+    · simp only [hs, hp, dite_false, dite_true, zero_add, modoSeno]
+      have hrec := recurrencia_seno (anguloModo d k) i.val
+      have hzero :
+          Real.sin (((i.val : ℝ) + 2) * anguloModo d k) = 0 := by
+        rw [show ((i.val : ℝ) + 2) = (d : ℝ) + 1 by
+          exact_mod_cast (show i.val + 2 = d + 1 by omega)]
+        exact seno_frontera_superior k
+      rw [hzero, zero_add] at hrec
+      have hpred : i.val - 1 + 1 = i.val := by omega
+      have hpredR : ((i.val - 1 : ℕ) : ℝ) + 1 = (i.val : ℝ) := by
+        exact_mod_cast hpred
+      rw [hpredR]
+      exact_mod_cast hrec
+    · have hd1 : d = 1 := by omega
+      subst d
+      have hk0 : k = 0 := Subsingleton.elim _ _
+      have hi0 : i = 0 := Subsingleton.elim _ _
+      subst k
+      subst i
+      norm_num [modoSeno, anguloModo]
+
+noncomputable def autovalorAd (d : ℕ) (k : Fin d) : ℂ :=
+  (2 * Real.cos (anguloModo d k) : ℝ)
+
+theorem anguloModo_mem_Icc {d : ℕ} (k : Fin d) :
+    anguloModo d k ∈ Set.Icc (0 : ℝ) Real.pi := by
+  constructor
+  · unfold anguloModo
+    positivity
+  · unfold anguloModo
+    have hk : (k.val : ℝ) + 1 ≤ (d : ℝ) + 1 := by
+      exact_mod_cast (show k.val + 1 ≤ d + 1 by omega)
+    have hd : 0 < (d : ℝ) + 1 := by positivity
+    calc
+      ((k.val : ℝ) + 1) * Real.pi / ((d : ℝ) + 1) ≤
+          ((d : ℝ) + 1) * Real.pi / ((d : ℝ) + 1) := by
+            gcongr
+      _ = Real.pi := by field_simp
+
+theorem autovalorAd_injective {d : ℕ} :
+    Function.Injective (autovalorAd d) := by
+  intro k l hkl
+  have hcos :
+      Real.cos (anguloModo d k) = Real.cos (anguloModo d l) := by
+    apply mul_left_cancel₀ (a := (2 : ℝ)) (by norm_num)
+    apply Complex.ofReal_injective
+    simpa [autovalorAd] using hkl
+  have hang : anguloModo d k = anguloModo d l :=
+    Real.strictAntiOn_cos.injOn
+      (anguloModo_mem_Icc k) (anguloModo_mem_Icc l) hcos
+  apply Fin.ext
+  unfold anguloModo at hang
+  have hp : Real.pi ≠ 0 := Real.pi_ne_zero
+  have hd : (d : ℝ) + 1 ≠ 0 := by positivity
+  have : (k.val : ℝ) = (l.val : ℝ) := by
+    field_simp at hang
+    nlinarith
+  exact_mod_cast this
+
+theorem modoSeno_ne_zero {d : ℕ} (hd : 1 ≤ d) (k : Fin d) :
+    modoSeno d k ≠ 0 := by
+  intro h
+  have h0 := congrFun h ⟨0, hd⟩
+  have ha0 : 0 < anguloModo d k := by
+    unfold anguloModo
+    positivity
+  have hapi : anguloModo d k < Real.pi := by
+    unfold anguloModo
+    have hk : (k.val : ℝ) + 1 < (d : ℝ) + 1 := by
+      exact_mod_cast Nat.add_lt_add_right k.isLt 1
+    have hdR : 0 < (d : ℝ) + 1 := by positivity
+    calc
+      ((k.val : ℝ) + 1) * Real.pi / ((d : ℝ) + 1) <
+          ((d : ℝ) + 1) * Real.pi / ((d : ℝ) + 1) := by
+            gcongr
+      _ = Real.pi := by field_simp
+  have hs := (Real.sin_pos_of_pos_of_lt_pi ha0 hapi).ne'
+  apply Complex.ofReal_ne_zero.mpr hs
+  simpa [modoSeno] using h0
+
+theorem modoSeno_hasEigenvector
+    {d : ℕ} (hd : 1 ≤ d) (k : Fin d) :
+    Module.End.HasEigenvector (Matrix.toLin' (Ad d))
+      (autovalorAd d k) (modoSeno d k) := by
+  constructor
+  · rw [Module.End.mem_eigenspace_iff, Matrix.toLin'_apply]
+    ext i
+    simpa [autovalorAd] using congrFun (Ad_modoSeno hd k) i
+  · exact modoSeno_ne_zero hd k
+
+theorem modosSeno_linearIndependent
+    {d : ℕ} (hd : 1 ≤ d) :
+    LinearIndependent ℂ (modoSeno d) :=
+  Module.End.eigenvectors_linearIndependent' (Matrix.toLin' (Ad d))
+    (autovalorAd d) autovalorAd_injective (modoSeno d)
+    (modoSeno_hasEigenvector hd)
+
+noncomputable def baseModosSeno
+    {d : ℕ} (hd : 1 ≤ d) : Module.Basis (Fin d) ℂ (Fin d → ℂ) := by
+  classical
+  exact basisOfPiSpaceOfLinearIndependent (modosSeno_linearIndependent hd)
+
+theorem baseModosSeno_apply
+    {d : ℕ} (hd : 1 ≤ d) (k : Fin d) :
+    baseModosSeno hd k = modoSeno d k := by
+  classical
+  exact congrFun (coe_basisOfPiSpaceOfLinearIndependent
+    (modosSeno_linearIndependent hd)) k
+
+theorem Ad_eq_suma_modos
+    {d : ℕ} (hd : 1 ≤ d) (v : Fin d → ℂ) :
+    Matrix.toLin' (Ad d) v =
+      ∑ k : Fin d,
+        (baseModosSeno hd).repr v k •
+          (autovalorAd d k • modoSeno d k) := by
+  calc
+    Matrix.toLin' (Ad d) v =
+        Matrix.toLin' (Ad d)
+          (∑ k, (baseModosSeno hd).repr v k • baseModosSeno hd k) := by
+            rw [(baseModosSeno hd).sum_repr v]
+    _ = ∑ k, (baseModosSeno hd).repr v k •
+          Matrix.toLin' (Ad d) (baseModosSeno hd k) := by
+            simp only [map_sum, map_smul]
+    _ = _ := by
+      apply Finset.sum_congr rfl
+      intro k _
+      rw [baseModosSeno_apply]
+      congr 1
+      rw [Matrix.toLin'_apply]
+      ext i
+      simpa [autovalorAd] using congrFun (Ad_modoSeno hd k) i
+
+theorem repr_Ad
+    {d : ℕ} (hd : 1 ≤ d) (v : Fin d → ℂ) (k : Fin d) :
+    (baseModosSeno hd).repr (Matrix.toLin' (Ad d) v) k =
+      autovalorAd d k * (baseModosSeno hd).repr v k := by
+  rw [Ad_eq_suma_modos hd v, map_sum]
+  classical
+  simp [← baseModosSeno_apply hd, Finsupp.single_apply, mul_comm]
+
+theorem autovalorAd_agota_espectro
+    {d : ℕ} (hd : 1 ≤ d) {μ : ℂ}
+    (hμ : Module.End.HasEigenvalue (Matrix.toLin' (Ad d)) μ) :
+    ∃ k : Fin d, μ = autovalorAd d k := by
+  obtain ⟨v, hv⟩ := hμ.exists_hasEigenvector
+  have hrepr : (baseModosSeno hd).repr v ≠ 0 := by
+    simpa using (baseModosSeno hd).repr.injective.ne hv.2
+  have hk : ∃ k : Fin d, (baseModosSeno hd).repr v k ≠ 0 := by
+    by_contra h
+    push Not at h
+    apply hrepr
+    apply Finsupp.ext
+    intro k
+    exact h k
+  obtain ⟨k, hk⟩ := hk
+  have heig := Module.End.mem_eigenspace_iff.mp hv.1
+  have hc := congrArg (fun w => (baseModosSeno hd).repr w k) heig
+  rw [repr_Ad] at hc
+  simp only [map_smul] at hc
+  exact ⟨k, (mul_right_cancel₀ hk hc).symm⟩
+
+theorem anguloFiedler_le_anguloModo
+    {d : ℕ} (k : Fin d) :
+    anguloFiedler d ≤ anguloModo d k := by
+  unfold anguloFiedler anguloModo
+  have hd : 0 < (d : ℝ) + 1 := by positivity
+  have hk : (1 : ℝ) ≤ (k.val : ℝ) + 1 := by
+    exact_mod_cast (show 1 ≤ k.val + 1 by omega)
+  calc
+    Real.pi / ((d : ℝ) + 1) =
+        1 * Real.pi / ((d : ℝ) + 1) := by ring
+    _ ≤ ((k.val : ℝ) + 1) * Real.pi / ((d : ℝ) + 1) := by
+      gcongr
+
+theorem anguloModo_le_pi_sub_fiedler
+    {d : ℕ} (k : Fin d) :
+    anguloModo d k ≤ Real.pi - anguloFiedler d := by
+  unfold anguloFiedler anguloModo
+  have hd : 0 < (d : ℝ) + 1 := by positivity
+  have hk : (k.val : ℝ) + 1 ≤ d := by
+    exact_mod_cast (show k.val + 1 ≤ d by omega)
+  calc
+    ((k.val : ℝ) + 1) * Real.pi / ((d : ℝ) + 1) ≤
+        (d : ℝ) * Real.pi / ((d : ℝ) + 1) := by
+          gcongr
+    _ = Real.pi - Real.pi / ((d : ℝ) + 1) := by
+      field_simp
+      ring
+
+theorem abs_cos_anguloModo_le_cos_fiedler
+    {d : ℕ} (hd : 2 ≤ d) (k : Fin d) :
+    |Real.cos (anguloModo d k)| ≤ Real.cos (anguloFiedler d) := by
+  apply abs_le.mpr
+  constructor
+  · rw [← Real.cos_pi_sub]
+    apply Real.cos_le_cos_of_nonneg_of_le_pi
+    · exact (anguloModo_mem_Icc k).1
+    · have hθ : 0 ≤ anguloFiedler d := by
+        unfold anguloFiedler
+        positivity
+      linarith [Real.pi_pos]
+    · exact anguloModo_le_pi_sub_fiedler k
+  · apply Real.cos_le_cos_of_nonneg_of_le_pi
+    · unfold anguloFiedler
+      positivity
+    · exact (anguloModo_mem_Icc k).2
+    · exact anguloFiedler_le_anguloModo k
+
+theorem norma_autovalorAd_le_rho
+    {d : ℕ} (hd : 2 ≤ d) (k : Fin d) :
+    ‖autovalorAd d k‖ ≤ rho d := by
+  rw [autovalorAd, Complex.norm_real, Real.norm_eq_abs,
+    abs_mul, abs_of_nonneg (by norm_num : (0 : ℝ) ≤ 2)]
+  rw [rho]
+  exact mul_le_mul_of_nonneg_left
+    (by simpa [anguloFiedler] using
+      abs_cos_anguloModo_le_cos_fiedler hd k)
+    (by norm_num)
+
+noncomputable def Kmat (d : ℕ) : Matrix (Fin d) (Fin d) ℂ :=
+  Complex.I • ((Td d * Pd d) - (Pd d * Td d))
+
+noncomputable def fase (j : ℕ) : ℂ := (-Complex.I) ^ j
+
+noncomputable def modoFase (d : ℕ) (k : Fin d) : Fin d → ℂ :=
+  fun j => fase j.val * modoSeno d k j
+
+theorem Kmat_apply
+    (d : ℕ) (i j : Fin d) :
+    Kmat d i j =
+      Complex.I * Td d i j *
+        ((posicionCoord d j : ℂ) - posicionCoord d i) := by
+  rw [Kmat]
+  change Complex.I * (((Td d * Pd d) - (Pd d * Td d)) i j) = _
+  rw [Matrix.sub_apply,
+    Td_mul_Pd_apply, Pd_mul_Td_apply]
+  ring
+
+theorem termino_vecino_fase
+    {d : ℕ} (hd : 2 ≤ d) {i j : Fin d}
+    (hpaso : PasoMinimo i j) (z : ℂ) :
+    Kmat d i j * (fase j.val * z) =
+      ((2 / ((d : ℝ) - 1) / rho d : ℝ) : ℂ) *
+        (fase i.val * z) := by
+  have hrhoR : rho d ≠ 0 := (rho_pos d hd).ne'
+  have hrhoC : (rho d : ℂ) ≠ 0 := Complex.ofReal_ne_zero.mpr hrhoR
+  have hdsubR : (d : ℝ) - 1 ≠ 0 := by
+    have : (1 : ℝ) < d := by exact_mod_cast (show 1 < d by omega)
+    linarith
+  rw [Kmat_apply]
+  have hTd : Td d i j = 1 / (rho d : ℂ) := by
+    simp [Td, Ad, hpaso]
+  rw [hTd]
+  rcases hpaso with hij | hji
+  · have hpos := posicionCoord_succ_sub d hd i j hij
+    have hposC :
+        (posicionCoord d j : ℂ) - posicionCoord d i =
+          ((2 / ((d : ℝ) - 1) : ℝ) : ℂ) := by
+      exact_mod_cast hpos
+    have hpow : fase j.val = fase i.val * (-Complex.I) := by
+      unfold fase
+      rw [← hij, pow_succ]
+    rw [hpow, hposC]
+    push_cast
+    field_simp [hrhoR, hdsubR]
+    ring_nf
+    simp [Complex.I_sq]
+  · have hpos := posicionCoord_succ_sub d hd j i hji
+    have hposC :
+        (posicionCoord d j : ℂ) - posicionCoord d i =
+          -((2 / ((d : ℝ) - 1) : ℝ) : ℂ) := by
+      exact_mod_cast (show posicionCoord d j - posicionCoord d i =
+        -(2 / ((d : ℝ) - 1)) by linarith)
+    have hpow : fase i.val = fase j.val * (-Complex.I) := by
+      unfold fase
+      rw [← hji, pow_succ]
+    have hI : fase j.val = fase i.val * Complex.I := by
+      calc
+        fase j.val = fase j.val * ((-Complex.I) * Complex.I) := by
+          rw [show (-Complex.I) * Complex.I = 1 by
+            apply Complex.ext <;> norm_num]
+          ring
+        _ = fase i.val * Complex.I := by rw [hpow]; ring
+    rw [hI, hposC]
+    push_cast
+    field_simp [hrhoR, hdsubR]
+    ring_nf
+    simp [Complex.I_sq]
+
+theorem Kmat_mulVec_modoFase
+    {d : ℕ} (hd : 2 ≤ d) (k : Fin d) :
+    (Kmat d).mulVec (modoFase d k) =
+      fun i =>
+        (((2 / ((d : ℝ) - 1)) *
+          (2 * Real.cos (anguloModo d k) / rho d) : ℝ) : ℂ) *
+          modoFase d k i := by
+  funext i
+  simp only [Matrix.mulVec, dotProduct]
+  rw [show (∑ j : Fin d, Kmat d i j * modoFase d k j) =
+      ∑ j : Fin d,
+        if PasoMinimo i j then
+          (((2 / ((d : ℝ) - 1) / rho d : ℝ) : ℂ) *
+            (fase i.val * modoSeno d k j))
+        else 0 by
+      apply Finset.sum_congr rfl
+      intro j _
+      by_cases hp : PasoMinimo i j
+      · simp only [hp, if_pos, modoFase]
+        exact termino_vecino_fase hd hp (modoSeno d k j)
+      · have hz : Kmat d i j = 0 := by
+          rw [Kmat_apply]
+          simp [Td, Ad, hp]
+        simp [hp, hz]]
+  rw [show (∑ x : Fin d,
+      if PasoMinimo i x then
+        (((2 / ((d : ℝ) - 1) / rho d : ℝ) : ℂ) *
+          (fase i.val * modoSeno d k x))
+      else 0) =
+      (((2 / ((d : ℝ) - 1) / rho d : ℝ) : ℂ) * fase i.val) *
+        ∑ x : Fin d, if PasoMinimo i x then modoSeno d k x else 0 by
+    rw [Finset.mul_sum]
+    apply Finset.sum_congr rfl
+    intro x _
+    by_cases hp : PasoMinimo i x <;> simp [hp]
+    ring]
+  rw [show (∑ j : Fin d, if PasoMinimo i j then modoSeno d k j else 0) =
+      (Ad d).mulVec (modoSeno d k) i by
+        simp only [Matrix.mulVec, dotProduct, Ad]
+        simp_rw [ite_mul, one_mul, zero_mul]]
+  rw [congrFun (Ad_modoSeno (by omega) k) i]
+  simp only [modoFase]
+  push_cast
+  ring
+
+noncomputable def autovalorK (d : ℕ) (k : Fin d) : ℂ :=
+  ((((2 / ((d : ℝ) - 1)) / rho d : ℝ) : ℂ) * autovalorAd d k)
+
+theorem Kmat_modoFase
+    {d : ℕ} (hd : 2 ≤ d) (k : Fin d) :
+    Matrix.toLin' (Kmat d) (modoFase d k) =
+      autovalorK d k • modoFase d k := by
+  rw [Matrix.toLin'_apply]
+  ext i
+  change (Kmat d).mulVec (modoFase d k) i =
+    autovalorK d k * modoFase d k i
+  rw [congrFun (Kmat_mulVec_modoFase hd k) i]
+  simp only [autovalorK, autovalorAd]
+  push_cast
+  ring
+
+theorem modoFase_ne_zero
+    {d : ℕ} (hd : 2 ≤ d) (k : Fin d) :
+    modoFase d k ≠ 0 := by
+  intro h
+  have h0 := congrFun h ⟨0, by omega⟩
+  have hs := modoSeno_ne_zero (by omega : 1 ≤ d) k
+  apply hs
+  funext j
+  have hf : fase j.val ≠ 0 := by
+    exact pow_ne_zero _ (neg_ne_zero.mpr Complex.I_ne_zero)
+  have hj := congrFun h j
+  simp only [modoFase] at hj
+  exact (mul_eq_zero.mp hj).resolve_left hf
+
+theorem autovalorK_injective
+    {d : ℕ} (hd : 2 ≤ d) :
+    Function.Injective (autovalorK d) := by
+  intro k l hkl
+  apply autovalorAd_injective
+  unfold autovalorK at hkl
+  have hcR : (2 / ((d : ℝ) - 1)) / rho d ≠ 0 := by
+    have hdR : (d : ℝ) - 1 ≠ 0 := by
+      have : (1 : ℝ) < d := by exact_mod_cast (show 1 < d by omega)
+      linarith
+    exact div_ne_zero (div_ne_zero (by norm_num) hdR) (rho_pos d hd).ne'
+  exact mul_left_cancel₀ (Complex.ofReal_ne_zero.mpr hcR) hkl
+
+theorem modoFase_hasEigenvector
+    {d : ℕ} (hd : 2 ≤ d) (k : Fin d) :
+    Module.End.HasEigenvector (Matrix.toLin' (Kmat d))
+      (autovalorK d k) (modoFase d k) := by
+  constructor
+  · rw [Module.End.mem_eigenspace_iff]
+    exact Kmat_modoFase hd k
+  · exact modoFase_ne_zero hd k
+
+theorem modosFase_linearIndependent
+    {d : ℕ} (hd : 2 ≤ d) :
+    LinearIndependent ℂ (modoFase d) :=
+  Module.End.eigenvectors_linearIndependent' (Matrix.toLin' (Kmat d))
+    (autovalorK d) (autovalorK_injective hd) (modoFase d)
+    (modoFase_hasEigenvector hd)
+
+noncomputable def baseModosFase
+    {d : ℕ} (hd : 2 ≤ d) : Module.Basis (Fin d) ℂ (Fin d → ℂ) := by
+  classical
+  exact basisOfPiSpaceOfLinearIndependent (modosFase_linearIndependent hd)
+
+theorem baseModosFase_apply
+    {d : ℕ} (hd : 2 ≤ d) (k : Fin d) :
+    baseModosFase hd k = modoFase d k := by
+  classical
+  exact congrFun (coe_basisOfPiSpaceOfLinearIndependent
+    (modosFase_linearIndependent hd)) k
+
+theorem Kmat_eq_suma_modos
+    {d : ℕ} (hd : 2 ≤ d) (v : Fin d → ℂ) :
+    Matrix.toLin' (Kmat d) v =
+      ∑ k : Fin d,
+        (baseModosFase hd).repr v k •
+          (autovalorK d k • modoFase d k) := by
+  calc
+    Matrix.toLin' (Kmat d) v =
+        Matrix.toLin' (Kmat d)
+          (∑ k, (baseModosFase hd).repr v k • baseModosFase hd k) := by
+            rw [(baseModosFase hd).sum_repr v]
+    _ = ∑ k, (baseModosFase hd).repr v k •
+          Matrix.toLin' (Kmat d) (baseModosFase hd k) := by
+            simp only [map_sum, map_smul]
+    _ = _ := by
+      apply Finset.sum_congr rfl
+      intro k _
+      rw [baseModosFase_apply]
+      congr 1
+      exact Kmat_modoFase hd k
+
+theorem repr_Kmat
+    {d : ℕ} (hd : 2 ≤ d) (v : Fin d → ℂ) (k : Fin d) :
+    (baseModosFase hd).repr (Matrix.toLin' (Kmat d) v) k =
+      autovalorK d k * (baseModosFase hd).repr v k := by
+  rw [Kmat_eq_suma_modos hd v, map_sum]
+  classical
+  simp [← baseModosFase_apply hd, Finsupp.single_apply, mul_comm]
+
+theorem Kmat_autovalor_agota_espectro
+    {d : ℕ} (hd : 2 ≤ d) {μ : ℂ}
+    (hμ : Module.End.HasEigenvalue (Matrix.toLin' (Kmat d)) μ) :
+    ∃ k : Fin d, μ = autovalorK d k := by
+  obtain ⟨v, hv⟩ := hμ.exists_hasEigenvector
+  have hrepr : (baseModosFase hd).repr v ≠ 0 := by
+    simpa using (baseModosFase hd).repr.injective.ne hv.2
+  obtain ⟨k, hk⟩ :
+      ∃ k : Fin d, (baseModosFase hd).repr v k ≠ 0 := by
+    by_contra h
+    push Not at h
+    apply hrepr
+    apply Finsupp.ext
+    intro k
+    exact h k
+  have heig := Module.End.mem_eigenspace_iff.mp hv.1
+  have hcoord := congrArg (fun w => (baseModosFase hd).repr w k) heig
+  rw [repr_Kmat] at hcoord
+  simp only [map_smul] at hcoord
+  exact ⟨k, (mul_right_cancel₀ hk hcoord).symm⟩
+
+theorem norma_autovalorK_le_paso
+    {d : ℕ} (hd : 2 ≤ d) (k : Fin d) :
+    ‖autovalorK d k‖ ≤ 2 / ((d : ℝ) - 1) := by
+  have hδ : 0 ≤ 2 / ((d : ℝ) - 1) := by
+    have hdsub : 0 < (d : ℝ) - 1 := by
+      have : (1 : ℝ) < d := by exact_mod_cast (show 1 < d by omega)
+      linarith
+    exact div_nonneg (by norm_num) hdsub.le
+  have hρ := rho_pos d hd
+  rw [autovalorK, norm_mul, Complex.norm_real, Real.norm_eq_abs,
+    abs_of_nonneg (div_nonneg hδ hρ.le)]
+  calc
+    (2 / ((d : ℝ) - 1) / rho d) * ‖autovalorAd d k‖ ≤
+        (2 / ((d : ℝ) - 1) / rho d) * rho d := by
+          gcongr
+          exact norma_autovalorAd_le_rho hd k
+    _ = 2 / ((d : ℝ) - 1) := by
+      field_simp [(rho_pos d hd).ne']
+
+theorem KdOp_eq_Kmat (d : ℕ) :
+    KdOp d = Matrix.toEuclideanLin (Kmat d) := by
+  unfold KdOp ConstructorEspectralTP.observableTension Kmat
+  rw [conmutador_TdOp_PdOp_eq_matriz]
+  exact (Matrix.toEuclideanLin :
+    Matrix (Fin d) (Fin d) ℂ ≃ₗ[ℂ] (Hd d →ₗ[ℂ] Hd d)).map_smul
+      Complex.I ((Td d * Pd d) - (Pd d * Td d)) |>.symm
+
+noncomputable def modoFaseHd (d : ℕ) (k : Fin d) : Hd d :=
+  WithLp.toLp 2 (modoFase d k)
+
+theorem KdOp_modoFaseHd
+    {d : ℕ} (hd : 2 ≤ d) (k : Fin d) :
+    KdOp d (modoFaseHd d k) =
+      autovalorK d k • modoFaseHd d k := by
+  rw [KdOp_eq_Kmat]
+  change WithLp.toLp 2 ((Kmat d).mulVec (modoFase d k)) =
+    WithLp.toLp 2 (fun i => autovalorK d k * modoFase d k i)
+  congr 1
+  funext i
+  simpa [smul_eq_mul] using congrFun (Kmat_modoFase hd k) i
+
+theorem modoFaseHd_ne_zero
+    {d : ℕ} (hd : 2 ≤ d) (k : Fin d) :
+    modoFaseHd d k ≠ 0 := by
+  exact (WithLp.linearEquiv 2 ℂ (Fin d → ℂ)).symm.injective.ne
+    (modoFase_ne_zero hd k)
+
+theorem modoFaseHd_hasEigenvector
+    {d : ℕ} (hd : 2 ≤ d) (k : Fin d) :
+    Module.End.HasEigenvector (KdOp d)
+      (autovalorK d k) (modoFaseHd d k) := by
+  constructor
+  · rw [Module.End.mem_eigenspace_iff]
+    exact KdOp_modoFaseHd hd k
+  · exact modoFaseHd_ne_zero hd k
+
+theorem modosFaseHd_linearIndependent
+    {d : ℕ} (hd : 2 ≤ d) :
+    LinearIndependent ℂ (modoFaseHd d) :=
+  Module.End.eigenvectors_linearIndependent' (KdOp d)
+    (autovalorK d) (autovalorK_injective hd) (modoFaseHd d)
+    (modoFaseHd_hasEigenvector hd)
+
+noncomputable def baseModosFaseHd
+    {d : ℕ} (hd : 2 ≤ d) : Module.Basis (Fin d) ℂ (Hd d) :=
+  (baseModosFase hd).map (WithLp.linearEquiv 2 ℂ (Fin d → ℂ)).symm
+
+theorem baseModosFaseHd_apply
+    {d : ℕ} (hd : 2 ≤ d) (k : Fin d) :
+    baseModosFaseHd hd k = modoFaseHd d k := by
+  simp [baseModosFaseHd, modoFaseHd, baseModosFase_apply]
+
+theorem KdOp_eq_suma_modos
+    {d : ℕ} (hd : 2 ≤ d) (v : Hd d) :
+    KdOp d v =
+      ∑ k : Fin d,
+        (baseModosFaseHd hd).repr v k •
+          (autovalorK d k • modoFaseHd d k) := by
+  calc
+    KdOp d v =
+        KdOp d
+          (∑ k, (baseModosFaseHd hd).repr v k •
+            baseModosFaseHd hd k) := by
+              rw [(baseModosFaseHd hd).sum_repr v]
+    _ = ∑ k, (baseModosFaseHd hd).repr v k •
+          KdOp d (baseModosFaseHd hd k) := by
+            simp only [map_sum, map_smul]
+    _ = _ := by
+      apply Finset.sum_congr rfl
+      intro k _
+      rw [baseModosFaseHd_apply]
+      congr 1
+      exact KdOp_modoFaseHd hd k
+
+theorem repr_KdOp
+    {d : ℕ} (hd : 2 ≤ d) (v : Hd d) (k : Fin d) :
+    (baseModosFaseHd hd).repr (KdOp d v) k =
+      autovalorK d k * (baseModosFaseHd hd).repr v k := by
+  rw [KdOp_eq_suma_modos hd v, map_sum]
+  classical
+  simp [← baseModosFaseHd_apply hd, Finsupp.single_apply, mul_comm]
+
+theorem KdOp_autovalor_agota_espectro
+    {d : ℕ} (hd : 2 ≤ d) {μ : ℂ}
+    (hμ : Module.End.HasEigenvalue (KdOp d) μ) :
+    ∃ k : Fin d, μ = autovalorK d k := by
+  obtain ⟨v, hv⟩ := hμ.exists_hasEigenvector
+  have hrepr : (baseModosFaseHd hd).repr v ≠ 0 := by
+    simpa using (baseModosFaseHd hd).repr.injective.ne hv.2
+  obtain ⟨k, hk⟩ :
+      ∃ k : Fin d, (baseModosFaseHd hd).repr v k ≠ 0 := by
+    by_contra h
+    push Not at h
+    apply hrepr
+    apply Finsupp.ext
+    intro k
+    exact h k
+  have heig := Module.End.mem_eigenspace_iff.mp hv.1
+  have hcoord := congrArg (fun w => (baseModosFaseHd hd).repr w k) heig
+  rw [repr_KdOp] at hcoord
+  simp only [map_smul] at hcoord
+  exact ⟨k, (mul_right_cancel₀ hk hcoord).symm⟩
+
+theorem todo_autovalor_KdOp_acotado
+    {d : ℕ} (hd : 2 ≤ d) {μ : ℂ}
+    (hμ : Module.End.HasEigenvalue (KdOp d) μ) :
+    ‖μ‖ ≤ 2 / ((d : ℝ) - 1) := by
+  obtain ⟨k, rfl⟩ := KdOp_autovalor_agota_espectro hd hμ
+  exact norma_autovalorK_le_paso hd k
+
+theorem autovalorK_fundamental
+    (d : ℕ) (hd : 2 ≤ d) :
+    autovalorK d ⟨0, by omega⟩ =
+      ((2 / ((d : ℝ) - 1) : ℝ) : ℂ) := by
+  have hρ : rho d ≠ 0 := (rho_pos d hd).ne'
+  have hdsub : (d : ℝ) - 1 ≠ 0 := by
+    have : (1 : ℝ) < d := by exact_mod_cast (show 1 < d by omega)
+    linarith
+  simp only [autovalorK, autovalorAd, anguloModo,
+    Nat.cast_zero, zero_add]
+  rw [show (1 : ℝ) * Real.pi / ((d : ℝ) + 1) =
+    Real.pi / ((d : ℝ) + 1) by ring]
+  rw [show (2 * Real.cos (Real.pi / ((d : ℝ) + 1)) : ℝ) = rho d by
+    rfl]
+  push_cast
+  field_simp [hρ, hdsub]
+
+theorem modoFaseHd_fundamental_eq_vectorFiedlerCrudo
+    (d : ℕ) (hd : 2 ≤ d) :
+    modoFaseHd d ⟨0, by omega⟩ = vectorFiedlerCrudo d := by
+  change WithLp.toLp 2 (fun j : Fin d =>
+      (-Complex.I) ^ j.val *
+        (Real.sin (((j.val : ℝ) + 1) *
+          ((((⟨0, by omega⟩ : Fin d).val : ℝ) + 1) * Real.pi /
+            ((d : ℝ) + 1))) : ℂ)) =
+    WithLp.toLp 2 (fun j : Fin d =>
+      (-Complex.I) ^ j.val *
+        (Real.sin (((j.val : ℝ) + 1) *
+          (Real.pi / ((d : ℝ) + 1))) : ℂ))
+  congr 1
+  funext j
+  congr 3
+  norm_num
+
+theorem KdOp_vectorFiedlerCrudo
+    (d : ℕ) (hd : 2 ≤ d) :
+    KdOp d (vectorFiedlerCrudo d) =
+      ((2 / ((d : ℝ) - 1) : ℝ) : ℂ) • vectorFiedlerCrudo d := by
+  rw [← modoFaseHd_fundamental_eq_vectorFiedlerCrudo d hd,
+    KdOp_modoFaseHd hd, autovalorK_fundamental d hd]
+
+theorem KdOp_vectorFiedlerExplicito
+    (d : ℕ) (hd : 2 ≤ d) :
+    KdOp d (vectorFiedlerExplicito d) =
+      ((2 / ((d : ℝ) - 1) : ℝ) : ℂ) • vectorFiedlerExplicito d := by
+  rw [vectorFiedlerExplicito, map_smul, KdOp_vectorFiedlerCrudo d hd]
+  module
+
+theorem radioEspectral_KdOp_eq_paso
+    (d : ℕ) (hd : 2 ≤ d) :
+    letI : Nonempty (Fin d) := ⟨⟨0, by omega⟩⟩
+    letI : Nontrivial (Hd d) := inferInstance
+    ConstructorEspectralTP.radioEspectral (KdOp d) (KdOp_simetrico d) =
+      2 / ((d : ℝ) - 1) := by
+  letI : Nonempty (Fin d) := ⟨⟨0, by omega⟩⟩
+  letI : Nontrivial (Hd d) := inferInstance
+  have hdsub : 0 < (d : ℝ) - 1 := by
+    have : (1 : ℝ) < d := by exact_mod_cast (show 1 < d by omega)
+    linarith
+  have hδ : 0 ≤ 2 / ((d : ℝ) - 1) := by
+    exact div_nonneg (by norm_num) hdsub.le
+  apply le_antisymm
+  · have heig :=
+      (KdOp_simetrico d).hasEigenvalue_eigenvalues rfl
+        (ConstructorEspectralTP.indiceExtremal (KdOp d) (KdOp_simetrico d))
+    have hb := todo_autovalor_KdOp_acotado hd heig
+    simpa [ConstructorEspectralTP.radioEspectral,
+      ConstructorEspectralTP.autovalorExtremal,
+      Complex.norm_real, Real.norm_eq_abs] using hb
+  · have hb :=
+      ConstructorEspectralTP.norma_aplicacion_le_radio_mul_norma
+        (KdOp d) (KdOp_simetrico d) (vectorFiedlerExplicito d)
+    rw [KdOp_vectorFiedlerExplicito d hd, norm_smul,
+      vectorFiedlerExplicito_normalizado d (by omega)] at hb
+    have hb' :
+        2 / ‖(((d : ℝ) : ℂ) - 1)‖ ≤
+          ConstructorEspectralTP.radioEspectral (KdOp d) (KdOp_simetrico d) := by
+      simpa [Complex.norm_real, abs_of_nonneg hδ] using hb
+    rw [show (((d : ℝ) : ℂ) - 1) = (((d : ℝ) - 1 : ℝ) : ℂ) by
+      push_cast
+      ring, Complex.norm_real, Real.norm_of_nonneg hdsub.le] at hb'
+    exact hb'
+
+end TransportePosicion
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D7_Niven.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D7_Niven.lean
new file mode 100644
index 000000000..4dbd6e7d8
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D7_Niven.lean
@@ -0,0 +1,92 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D2_Robertson
+
+@[expose] public section
+
+/-!
+# D7 — Teorema de Niven: la saturación sólo ocurre en `d ∈ {2,3}`
+
+Tricotomía de saturación: la ecuación trigonométrica
+
+`cos²(π/(d+1)) = (d−1)/4`
+
+—que es exactamente la condición para que la cota de Robertson se sature
+sobre el modo fundamental del camino discreto— se cumple **si y sólo si**
+`d = 2` o `d = 3`. No hay más soluciones naturales: la prueba distingue
+`d = 4` (reductio algebraico explícito) de `d ≥ 5` (cota de coseno,
+`Blindaje.R3_techo_coseno`). En consecuencia, para todo `d ≥ 4` la brecha
+`C_Nava(d) − 1` es estrictamente positiva (segunda mitad de este archivo,
+`Constructor_DeltaGeom_Pos`).
+-/
+
+open Real
+
+namespace Gnomon
+
+/-- Semilla `d=2`: saturación unitaria exacta `cos²(π/3) = 1/4`. -/
+theorem semilla_d2 : Real.cos (π / 3) ^ 2 = 1 / 4 := by
+  rw [Real.cos_pi_div_three]; norm_num
+
+/-- Semilla `d=3`: saturación unitaria exacta `cos²(π/4) = 1/2`. -/
+theorem semilla_d3 : Real.cos (π / 4) ^ 2 = 1 / 2 := by
+  rw [Real.cos_pi_div_four]
+  rw [div_pow, sq_sqrt (by norm_num : (2:ℝ) ≥ 0)]
+  norm_num
+
+/-- `d=4` no admite saturación unitaria: `cos²(π/5) ≠ 3/4`. -/
+theorem no_saturacion_d4 : Real.cos (π / 5) ^ 2 ≠ 3 / 4 := by
+  rw [Real.cos_pi_div_five]
+  intro h
+  have hs : Real.sqrt 5 ^ 2 = 5 := Real.sq_sqrt (by norm_num)
+  have hnn : 0 ≤ Real.sqrt 5 := Real.sqrt_nonneg 5
+  nlinarith [hs, hnn, h]
+
+/-- TEOREMA DE NIVEN (tricotomía de saturación): para `d ≥ 2`,
+`cos²(π/(d+1)) = (d−1)/4 ↔ d ∈ {2,3}`. -/
+theorem saturacion_iff (d : ℕ) (hd : 2 ≤ d) :
+    Real.cos (π / (d + 1)) ^ 2 = ((d : ℝ) - 1) / 4 ↔ d = 2 ∨ d = 3 := by
+  constructor
+  · intro h
+    by_contra hne
+    push Not at hne
+    obtain ⟨h2, h3⟩ := hne
+    rcases Nat.lt_or_ge d 5 with h5 | h5
+    · have hd4 : d = 4 := by omega
+      subst hd4
+      have hc : ((4 : ℕ) : ℝ) + 1 = 5 := by norm_num
+      rw [hc] at h
+      have h34 : (((4 : ℕ) : ℝ) - 1) / 4 = 3 / 4 := by norm_num
+      rw [h34] at h
+      exact no_saturacion_d4 h
+    · exact absurd h (ne_of_lt (Blindaje.R3_techo_coseno d h5))
+  · rintro (rfl | rfl)
+    · have hc : ((2 : ℕ) : ℝ) + 1 = 3 := by norm_num
+      rw [hc, semilla_d2]; norm_num
+    · have hc : ((3 : ℕ) : ℝ) + 1 = 4 := by norm_num
+      rw [hc, semilla_d3]; norm_num
+
+/-- La cota unitaria no se repone: para `d ≥ 4` la saturación es imposible. -/
+theorem no_reposición_saturacion_camino (d : ℕ) (hd : 4 ≤ d) :
+    Real.cos (π / (d + 1)) ^ 2 ≠ ((d : ℝ) - 1) / 4 := by
+  intro h
+  have hsem : d = 2 ∨ d = 3 := (saturacion_iff d (by omega)).mp h
+  omega
+
+/-- Alias citable: las únicas semillas de saturación son `d = 2` y `d = 3`. -/
+theorem semillas_niven_unicas (d : ℕ) (hd : 2 ≤ d) :
+    Real.cos (π / (d + 1)) ^ 2 = ((d : ℝ) - 1) / 4 → d = 2 ∨ d = 3 :=
+  (saturacion_iff d hd).mp
+
+theorem apertura_no_es_semilla_niven (d : ℕ) (hd : 4 ≤ d) :
+    ¬ (d = 2 ∨ d = 3) := by
+  omega
+
+end Gnomon
+
+

From 086c1fcf3dc37d3aa03dd6b5435ce08c32e28cfa Mon Sep 17 00:00:00 2001
From: Eduardo Nava-Hernandez <naype888@gmail.com>
Date: Thu, 17 Sep 2026 07:42:37 -0600
Subject: [PATCH 05/10] feat(PhyslibAlpha): prove dimensional uncertainty above
 dimension three

Co-authored-by: Claude Opus 4.8 <no-reply+claude-opus-4-8@anthropic.com>
---
 .../CStarAlgebra/DimensionalUncertainty.lean  |  16 +
 .../D10_Certificado.lean                      | 174 +++++
 .../D11_CuantoMinimoArea.lean                 | 153 ++++
 .../D12_ConmutadorEscalarFinito.lean          |  96 +++
 .../D13_PrimeraRupturaCombinatoria.lean       | 116 +++
 .../D14_ExcesoBrechaSzego.lean                |  61 ++
 .../D15_IdentidadCosecanteChebyshev.lean      | 267 +++++++
 .../DimensionalUncertainty/D8_Szego.lean      | 661 ++++++++++++++++++
 .../DimensionalUncertainty/D9_Monotonia.lean  | 557 +++++++++++++++
 9 files changed, 2101 insertions(+)
 create mode 100644 PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D10_Certificado.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D11_CuantoMinimoArea.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D12_ConmutadorEscalarFinito.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D13_PrimeraRupturaCombinatoria.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D14_ExcesoBrechaSzego.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D15_IdentidadCosecanteChebyshev.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D8_Szego.lean
 create mode 100644 PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D9_Monotonia.lean

diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty.lean
new file mode 100644
index 000000000..159cfb6d8
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty.lean
@@ -0,0 +1,16 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D8_Szego
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D9_Monotonia
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D10_Certificado
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D11_CuantoMinimoArea
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D12_ConmutadorEscalarFinito
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D13_PrimeraRupturaCombinatoria
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D14_ExcesoBrechaSzego
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D15_IdentidadCosecanteChebyshev
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D10_Certificado.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D10_Certificado.lean
new file mode 100644
index 000000000..4b0bd8876
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D10_Certificado.lean
@@ -0,0 +1,174 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D6_Fiedler
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D7_Niven
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D8_Szego
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D9_Monotonia
+
+@[expose] public section
+
+/-!
+# D10 — Certificado conjunto: Fiedler + Niven + Szegő en `H_d`
+
+Reúne, en un único certificado citable, los tres pilares que se apoyan sobre
+el hábitat común `H_d = ℂ^d` (`D0_Habitat.lean`): la descomposición
+espectral de Fiedler (`D6_Fiedler.lean`), el teorema de Niven
+(`D7_Niven.lean`) y el límite de Szegő con la positividad de la brecha
+(`D8_Szego.lean`). Este es el teorema terminal del paquete: aquí se acaba
+la matemática que se demuestra en este repositorio.
+
+# Blindaje de `δ_geom(d)` en el Hilbert finito `H_d`
+
+**Hábitat:** \(H_d=\mathtt{EuclideanSpace}\,\mathbb{C}\,(\mathtt{Fin}\,d)\).
+No se abandona ese espacio: es el que acoge la derivación del marco.
+
+**Terna de escudos** (todo sobre el discreto):
+
+| Escudo | Contenido Lean |
+|--------|----------------|
+| **Fiedler** | modo fundamental / `KdOp` / radio espectral en \(H_d\) |
+| **Niven** | `saturacion_iff` + `no_reposición_saturacion_camino` + `deltaGeom_pos_of_four_le` |
+| **Szegő** | `limite_szego_CNava` + `deltaInf_pos` + ∞ no es dimensión |
+| **Monotonía** | `deltaGeom_four_le`: `δ_geom(4)` es el piso global para todo `d ≥ 4` |
+
+Lectura: los productos trigonométricos simultáneamente racionales de la
+saturación del camino **solo** existen en \(d\in\{2,3\}\). No hay más
+semillas; por eso **nada repone la cota unitaria** después de \(d=4\).
+Además, la monotonía certificada fija a \(d=4\) como el menor defecto
+realizado: cualquier medición en un \(H_d\) físico con \(d\ge4\) queda
+separada del cero por al menos \(\delta_{\rm geom}(4)\). Al crecer la
+familia finita, el defecto no se apaga: converge a
+\(\delta_\infty>0\).
+
+**Cierre del hábitat:** \(H_d = \mathbb{C}^d \cong \mathbb{R}^{2d}\), finito.
+Punto. Si quieren continuo infinito, aquí no es hotel — \(d=\infty\) no se
+hospeda en este paquete; a lo más se le ve llegar por la ventana como límite
+(`D8_Szego.lean`), pero nunca cruza la puerta.
+-/
+
+noncomputable section
+
+open Real
+open Filter
+open scoped Topology
+
+namespace BlindajeHd
+
+open TransportePosicion
+open Gnomon
+
+/-! ## Habitat: no se sale de \(H_d\) -/
+
+def HabitatHilbertFinito (d : ℕ) : Prop :=
+  Hd d = EuclideanSpace ℂ (Fin d)
+
+theorem habitatHilbertFinito (d : ℕ) : HabitatHilbertFinito d :=
+  Hd_eq_euclidean d
+
+theorem infinito_no_es_habitat :
+    Tendsto deltaGeom atTop (𝓝 deltaInf) ∧
+      deltaInf = Cinf - 1 ∧
+      0 < deltaInf :=
+  infinito_no_es_dimension_sino_limite
+
+/-! ## Niven: cota unitaria no se repone -/
+
+theorem niven_saturacion_solo_semillas (d : ℕ) (hd : 2 ≤ d) :
+    cos (π / (d + 1)) ^ 2 = ((d : ℝ) - 1) / 4 ↔ d = 2 ∨ d = 3 :=
+  saturacion_iff d hd
+
+/-- **Nada repone la cota** tras \(d=4\). -/
+theorem niven_cota_unitaria_no_se_repone (d : ℕ) (hd : 4 ≤ d) :
+    cos (π / (d + 1)) ^ 2 ≠ ((d : ℝ) - 1) / 4 :=
+  no_reposición_saturacion_camino d hd
+
+theorem niven_deltaGeom_pos_en_Hd (d : ℕ) (hd : 4 ≤ d) :
+    0 < deltaGeom d :=
+  deltaGeom_pos_of_four_le d hd
+
+theorem piso_precision_deltaGeom_d4_en_Hd (d : ℕ) (hd : 4 ≤ d) :
+    deltaGeom 4 ≤ deltaGeom d :=
+  deltaGeom_four_le d hd
+
+/-- En el régimen físico finito `H_d`, `d ≥ 4`, no existe lectura con defecto
+por debajo del piso elemental `δ_geom(4)`. -/
+theorem no_medicion_absoluta_bajo_piso_d4_en_Hd
+    (d : ℕ) (hd : 4 ≤ d) (ε : ℝ) (hε : ε < deltaGeom 4) :
+    ε < deltaGeom d :=
+  lt_of_lt_of_le hε (piso_precision_deltaGeom_d4_en_Hd d hd)
+
+/-! ## Fiedler: espectro y banda en \(H_d\) -/
+
+theorem fiedler_autovector_en_Hd (d : ℕ) (hd : 2 ≤ d) :
+    KdOp d (vectorFiedlerExplicito d) =
+      ((2 / ((d : ℝ) - 1) : ℝ) : ℂ) • vectorFiedlerExplicito d :=
+  KdOp_vectorFiedlerExplicito d hd
+
+theorem fiedler_radio_banda (d : ℕ) (hd : 2 ≤ d) :
+    letI : Nonempty (Fin d) := ⟨⟨0, by omega⟩⟩
+    letI : Nontrivial (Hd d) := inferInstance
+    ConstructorEspectralTP.radioEspectral (KdOp d) (KdOp_simetrico d) =
+      2 / ((d : ℝ) - 1) := by
+  letI : Nonempty (Fin d) := ⟨⟨0, by omega⟩⟩
+  letI : Nontrivial (Hd d) := inferInstance
+  exact radioEspectral_KdOp_eq_paso d hd
+
+/-! ## Szegő: asintótica de la familia finita -/
+
+theorem szego_limite_familia_finita :
+    Tendsto CNava atTop (𝓝 Cinf) :=
+  limite_szego_CNava
+
+theorem szego_deltaInf_pos : 0 < deltaInf :=
+  deltaInf_pos
+
+theorem defecto_real_positivo_desde_Hd4_hasta_limite :
+    (∀ d : ℕ, 4 ≤ d → 0 < deltaGeom d) ∧
+      Tendsto deltaGeom atTop (𝓝 deltaInf) ∧
+      0 < deltaInf :=
+  ⟨niven_deltaGeom_pos_en_Hd, limite_defecto_geometrico, szego_deltaInf_pos⟩
+
+/-! ## Certificado conjunto citable -/
+
+structure CertificadoBlindajeHd where
+  habitat : ∀ d : ℕ, HabitatHilbertFinito d
+  niven_iff :
+    ∀ d : ℕ, 2 ≤ d →
+      (cos (π / ((d : ℝ) + 1)) ^ 2 = ((d : ℝ) - 1) / 4 ↔ d = 2 ∨ d = 3)
+  niven_no_reposición :
+    ∀ d : ℕ, 4 ≤ d →
+      cos (π / ((d : ℝ) + 1)) ^ 2 ≠ ((d : ℝ) - 1) / 4
+  deltaGeom_pos : ∀ d : ℕ, 4 ≤ d → 0 < deltaGeom d
+  deltaGeom_piso_d4 : ∀ d : ℕ, 4 ≤ d → deltaGeom 4 ≤ deltaGeom d
+  fiedler_autovector :
+    ∀ d : ℕ, 2 ≤ d →
+      KdOp d (vectorFiedlerExplicito d) =
+        ((2 / ((d : ℝ) - 1) : ℝ) : ℂ) • vectorFiedlerExplicito d
+  szego_limite : Tendsto CNava atTop (𝓝 Cinf)
+  szego_deltaInf : 0 < deltaInf
+  defecto_real_positivo :
+    (∀ d : ℕ, 4 ≤ d → 0 < deltaGeom d) ∧
+      Tendsto deltaGeom atTop (𝓝 deltaInf) ∧
+      0 < deltaInf
+  infinito_limite :
+    Tendsto deltaGeom atTop (𝓝 deltaInf) ∧
+      deltaInf = Cinf - 1 ∧ 0 < deltaInf
+
+theorem certificadoBlindajeHd_OK : Nonempty CertificadoBlindajeHd :=
+  ⟨{ habitat := habitatHilbertFinito
+     niven_iff := fun d hd => niven_saturacion_solo_semillas d hd
+     niven_no_reposición := fun d hd => niven_cota_unitaria_no_se_repone d hd
+     deltaGeom_pos := fun d hd => niven_deltaGeom_pos_en_Hd d hd
+     deltaGeom_piso_d4 := fun d hd => piso_precision_deltaGeom_d4_en_Hd d hd
+     fiedler_autovector := fun d hd => fiedler_autovector_en_Hd d hd
+     szego_limite := szego_limite_familia_finita
+     szego_deltaInf := szego_deltaInf_pos
+     defecto_real_positivo := defecto_real_positivo_desde_Hd4_hasta_limite
+     infinito_limite := infinito_no_es_habitat }⟩
+
+end BlindajeHd
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D11_CuantoMinimoArea.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D11_CuantoMinimoArea.lean
new file mode 100644
index 000000000..b5b3a13e1
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D11_CuantoMinimoArea.lean
@@ -0,0 +1,153 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D10_Certificado
+
+@[expose] public section
+
+/-!
+# D11 — Cuanto cuántico elemental de área
+
+Este módulo baja a Lean la lectura estrictamente matemática del cuanto
+cuántico elemental:
+
+* `deltaGeom 4` es el primer defecto lineal positivo de la cola `d ≥ 4`.
+* `deltaGeom 4 ^ 2` es el primer cuanto cuántico elemental de área.
+* Por monotonía, ninguna resolución de área realizada en `H_d`, `d ≥ 4`,
+  queda por debajo de ese cuanto.
+
+No introduce unidades físicas, bariones ni escala de Planck. Si luego se
+quiere poner una unidad externa, basta multiplicar por una escala no negativa:
+la cota sobrevive por orden.
+
+La dependencia matemática es la cadena del paquete:
+Cauchy--Gram → Robertson--Schrödinger → instancia `T_d/P_d` →
+Niven/Szegő/monotonía. No modifica Robertson 1929; lo usa como ancla y
+deriva el piso de área de su realización discreta.
+-/
+
+noncomputable section
+
+namespace CuantoMinimoArea
+
+open Gnomon
+
+/-- Cuanto cuántico elemental de área de la cola `H_d`, `d ≥ 4`. -/
+def cuantoCuanticoElemental : ℝ :=
+  deltaGeom 4 ^ 2
+
+/-- Alias operativo: el "cuadrito" mínimo es el cuanto cuántico elemental. -/
+def cuadritoMinimo : ℝ :=
+  cuantoCuanticoElemental
+
+/-- Área de resolución inducida por el defecto geométrico en `H_d`. -/
+def areaResolucionHd (d : ℕ) : ℝ :=
+  deltaGeom d ^ 2
+
+/-- El nombre citable y el alias operativo son la misma cantidad. -/
+theorem cuadritoMinimo_eq_cuantoCuanticoElemental :
+    cuadritoMinimo = cuantoCuanticoElemental := by
+  rfl
+
+/-- El "cuadrito" es exactamente la resolución de área en `H_4`. -/
+theorem cuadritoMinimo_eq_areaResolucionH4 :
+    cuadritoMinimo = areaResolucionHd 4 := by
+  rfl
+
+/-- El cuanto cuántico elemental es exactamente la resolución de área en `H_4`. -/
+theorem cuantoCuanticoElemental_eq_areaResolucionH4 :
+    cuantoCuanticoElemental = areaResolucionHd 4 := by
+  rfl
+
+/-- El cuanto cuántico elemental de área es estrictamente positivo. -/
+theorem cuantoCuanticoElemental_pos : 0 < cuantoCuanticoElemental := by
+  unfold cuantoCuanticoElemental
+  have hδ : 0 < deltaGeom 4 :=
+    deltaGeom_pos_of_four_le 4 (by omega)
+  positivity
+
+/-- Alias de positividad para el nombre operativo. -/
+theorem cuadritoMinimo_pos : 0 < cuadritoMinimo := by
+  simpa [cuadritoMinimo] using cuantoCuanticoElemental_pos
+
+/-- Toda área de resolución en `H_d`, `d ≥ 4`, está por encima del cuanto. -/
+theorem cuantoCuanticoElemental_le_areaResolucionHd (d : ℕ) (hd : 4 ≤ d) :
+    cuantoCuanticoElemental ≤ areaResolucionHd d := by
+  unfold cuantoCuanticoElemental areaResolucionHd
+  exact deltaGeom_sq_four_le d hd
+
+/-- Toda área de resolución en `H_d`, `d ≥ 4`, está por encima del cuadrito. -/
+theorem cuadritoMinimo_le_areaResolucionHd (d : ℕ) (hd : 4 ≤ d) :
+    cuadritoMinimo ≤ areaResolucionHd d := by
+  simpa [cuadritoMinimo] using cuantoCuanticoElemental_le_areaResolucionHd d hd
+
+/-- No existe una resolución realizada en `H_d`, `d ≥ 4`, estrictamente menor
+que el cuanto cuántico elemental. -/
+theorem no_hay_resolucion_menor_que_cuanto_cuantico
+    (d : ℕ) (hd : 4 ≤ d) :
+    ¬ areaResolucionHd d < cuantoCuanticoElemental := by
+  exact not_lt.mpr (cuantoCuanticoElemental_le_areaResolucionHd d hd)
+
+/-- Alias operativo: no hay resolución menor que el cuadrito mínimo. -/
+theorem no_hay_resolucion_menor_que_cuadrito
+    (d : ℕ) (hd : 4 ≤ d) :
+    ¬ areaResolucionHd d < cuadritoMinimo := by
+  simpa [cuadritoMinimo] using no_hay_resolucion_menor_que_cuanto_cuantico d hd
+
+/-- Cualquier umbral por debajo del cuadrito queda por debajo de toda
+resolución realizada en la cola `d ≥ 4`. -/
+theorem umbral_bajo_cuadrito_no_alcanza_Hd
+    (d : ℕ) (hd : 4 ≤ d) (ε : ℝ) (hε : ε < cuadritoMinimo) :
+    ε < areaResolucionHd d :=
+  lt_of_lt_of_le hε (cuadritoMinimo_le_areaResolucionHd d hd)
+
+/-- Poner una escala externa no negativa conserva la cota mínima. -/
+theorem escala_no_negativa_conserva_cuanto_cuantico
+    (escala : ℝ) (hesc : 0 ≤ escala) (d : ℕ) (hd : 4 ≤ d) :
+    escala * cuantoCuanticoElemental ≤ escala * areaResolucionHd d :=
+  mul_le_mul_of_nonneg_left (cuantoCuanticoElemental_le_areaResolucionHd d hd) hesc
+
+/-- Alias operativo para la escala externa no negativa. -/
+theorem escala_no_negativa_conserva_cuadrito
+    (escala : ℝ) (hesc : 0 ≤ escala) (d : ℕ) (hd : 4 ≤ d) :
+    escala * cuadritoMinimo ≤ escala * areaResolucionHd d :=
+by
+  simpa [cuadritoMinimo] using escala_no_negativa_conserva_cuanto_cuantico escala hesc d hd
+
+/-- Con una escala externa positiva, el cuanto escalado sigue siendo
+estrictamente positivo. -/
+theorem cuanto_cuantico_escalado_pos
+    (escala : ℝ) (hesc : 0 < escala) :
+    0 < escala * cuantoCuanticoElemental :=
+  mul_pos hesc cuantoCuanticoElemental_pos
+
+/-- Alias operativo: con escala positiva, el cuadrito escalado sigue siendo
+estrictamente positivo. -/
+theorem cuadrito_escalado_pos
+    (escala : ℝ) (hesc : 0 < escala) :
+    0 < escala * cuadritoMinimo :=
+by
+  simpa [cuadritoMinimo] using cuanto_cuantico_escalado_pos escala hesc
+
+/-- Certificado citable del cuanto cuántico elemental de área. -/
+structure CertificadoCuantoMinimoArea where
+  cuanto_pos : 0 < cuantoCuanticoElemental
+  area_minima : ∀ d : ℕ, 4 ≤ d → cuantoCuanticoElemental ≤ areaResolucionHd d
+  no_menor : ∀ d : ℕ, 4 ≤ d → ¬ areaResolucionHd d < cuantoCuanticoElemental
+  escala_conserva :
+    ∀ escala : ℝ, 0 ≤ escala →
+      ∀ d : ℕ, 4 ≤ d →
+        escala * cuantoCuanticoElemental ≤ escala * areaResolucionHd d
+
+theorem certificadoCuantoMinimoArea_OK :
+    Nonempty CertificadoCuantoMinimoArea :=
+  ⟨{ cuanto_pos := cuantoCuanticoElemental_pos
+     area_minima := cuantoCuanticoElemental_le_areaResolucionHd
+     no_menor := no_hay_resolucion_menor_que_cuanto_cuantico
+     escala_conserva := escala_no_negativa_conserva_cuanto_cuantico }⟩
+
+end CuantoMinimoArea
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D12_ConmutadorEscalarFinito.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D12_ConmutadorEscalarFinito.lean
new file mode 100644
index 000000000..830bccc2b
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D12_ConmutadorEscalarFinito.lean
@@ -0,0 +1,96 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import Mathlib.Analysis.InnerProductSpace.PiL2
+public import Mathlib.LinearAlgebra.Matrix.Notation
+public import Mathlib.LinearAlgebra.Matrix.Trace
+
+@[expose] public section
+
+/-!
+# Obstrucción de traza al conmutador escalar en dimensión finita
+
+Complemento algebraico a la construcción de `T_d`, `P_d` en
+`D3_GrafoCamino.lean`: en dimensión finita ningún conmutador matricial
+puede ser un múltiplo escalar no nulo de la identidad, mientras que sí
+existen pares de matrices unitarias que anticonmutan exactamente.
+
+Dos afirmaciones, independientes entre sí:
+
+1. Todo conmutador matricial `[Q,P] = QP - PQ` tiene traza cero
+   (`Matrix.trace_mul_comm`), así que nunca puede igualar `c • 1` para
+   `c ≠ 0` (`no_nonzero_scalar_exact_commutator`).
+2. Esa obstrucción es sobre el conmutador aditivo; no impide la
+   no-conmutatividad multiplicativa: el par de matrices `2×2`
+   `W₁ = !![0,1;1,0]`, `W₂ = !![1,0;0,-1]` satisface exactamente
+   `W₂ W₁ = -(W₁ W₂)` (`parWeyl_anticonmuta`).
+-/
+
+noncomputable section
+
+namespace ConmutadorEscalarFinito
+
+/-- Conmutador matricial. -/
+def commutator {d : ℕ}
+    (Q P : Matrix (Fin d) (Fin d) ℂ) : Matrix (Fin d) (Fin d) ℂ :=
+  Q * P - P * Q
+
+/-- La traza de todo conmutador matricial finito es cero. -/
+theorem trace_commutator_zero {d : ℕ}
+    (Q P : Matrix (Fin d) (Fin d) ℂ) :
+    Matrix.trace (commutator Q P) = 0 := by
+  rw [commutator, Matrix.trace_sub, Matrix.trace_mul_comm Q P, sub_self]
+
+/-- En dimensión finita positiva, un conmutador no puede ser un múltiplo
+escalar no nulo de la identidad. -/
+theorem no_nonzero_scalar_exact_commutator {d : ℕ} (hd : 0 < d)
+    (Q P : Matrix (Fin d) (Fin d) ℂ) (c : ℂ) (hc : c ≠ 0) :
+    commutator Q P ≠ c • (1 : Matrix (Fin d) (Fin d) ℂ) := by
+  intro h
+  have ht := congrArg Matrix.trace h
+  have hleft : Matrix.trace (commutator Q P) = 0 :=
+    trace_commutator_zero Q P
+  have hright : Matrix.trace (c • (1 : Matrix (Fin d) (Fin d) ℂ)) = c * d := by
+    simp
+  rw [hleft, hright] at ht
+  have hd0 : (d : ℂ) ≠ 0 := by
+    exact_mod_cast (Nat.ne_of_gt hd)
+  exact (mul_ne_zero hc hd0) ht.symm
+
+/-- Corolario: en particular, el conmutador tampoco puede igualar un
+múltiplo imaginario `i·c` de la identidad para ningún real `c ≠ 0`. -/
+theorem commutador_ne_escalar_imaginario {d : ℕ} (hd : 0 < d)
+    (Q P : Matrix (Fin d) (Fin d) ℂ) (c : ℝ) (hc : c ≠ 0) :
+    commutator Q P ≠ (Complex.I * (c : ℂ)) • (1 : Matrix (Fin d) (Fin d) ℂ) := by
+  apply no_nonzero_scalar_exact_commutator hd Q P
+  exact mul_ne_zero Complex.I_ne_zero (Complex.ofReal_ne_zero.mpr hc)
+
+/-! ## Un par de Weyl exacto en dimensión dos -/
+
+/-- Primera matriz del par de Weyl `2×2`. -/
+def W1 : Matrix (Fin 2) (Fin 2) ℂ :=
+  !![0, 1; 1, 0]
+
+/-- Segunda matriz del par de Weyl `2×2`. -/
+def W2 : Matrix (Fin 2) (Fin 2) ℂ :=
+  !![1, 0; 0, -1]
+
+/-- Relación de Weyl exacta: `W₂ W₁ = -(W₁ W₂)`. La obstrucción de traza de
+arriba es sobre el conmutador *aditivo*; no impide esta anticonmutación
+*multiplicativa* exacta en dimensión finita. -/
+theorem parWeyl_anticonmuta : W2 * W1 = -(W1 * W2) := by
+  ext i j
+  fin_cases i <;> fin_cases j <;>
+    norm_num [W1, W2, Matrix.mul_apply, Fin.sum_univ_two]
+
+/-- En particular, `W₁` y `W₂` no conmutan. -/
+theorem parWeyl_no_conmuta : W2 * W1 ≠ W1 * W2 := by
+  intro h
+  have hij := congrFun (congrFun h (0 : Fin 2)) (1 : Fin 2)
+  norm_num [W1, W2, Matrix.mul_apply, Fin.sum_univ_two] at hij
+
+end ConmutadorEscalarFinito
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D13_PrimeraRupturaCombinatoria.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D13_PrimeraRupturaCombinatoria.lean
new file mode 100644
index 000000000..17b6ffd07
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D13_PrimeraRupturaCombinatoria.lean
@@ -0,0 +1,116 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D3_GrafoCamino
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D7_Niven
+
+@[expose] public section
+
+/-!
+# La primera ruptura combinatoria es `d = 4`
+
+Complemento combinatorio al teorema de Niven (`D7_Niven.lean`): la
+saturación espectral de Robertson–Schrödinger sobre `P_d` deja de
+cumplirse exactamente cuando `P_d` adquiere su primera arista *interior*
+(una arista entre dos vértices que no son extremos del camino), y esa
+coincidencia ocurre exactamente en `d = 4`.
+
+Dos rutas independientes hacia la misma dimensión:
+
+* **espectral** (`D7_Niven.lean`): `cos²(π/(d+1)) = (d-1)/4 ↔ d ∈ {2,3}`;
+* **combinatoria** (aquí): `P_d` tiene una arista entre dos vértices
+  interiores si y sólo si `4 ≤ d`.
+
+`primera_ruptura_iff_dimension_cuatro` certifica que ambas rutas señalan
+la misma dimensión `d = 4`, sin usar ninguna ecuación espectral en la
+mitad combinatoria.
+-/
+
+namespace PrimeraRuptura
+
+open SimpleGraph
+
+/-- Ecuación aritmético-espectral que representa la saturación del camino
+(la misma de `Gnomon.saturacion_iff`, escrita como predicado). -/
+def SaturacionCamino (d : ℕ) : Prop :=
+  Real.cos (Real.pi / (d + 1)) ^ 2 = ((d : ℝ) - 1) / 4
+
+/-- Ruptura: negación de la igualdad de saturación del camino. -/
+def RupturaCamino (d : ℕ) : Prop := ¬ SaturacionCamino d
+
+/-- Desde la dimensión mínima `2`, la ruptura ocurre exactamente desde `4`. -/
+theorem ruptura_camino_iff_cuatro_le (d : ℕ) (hd : 2 ≤ d) :
+    RupturaCamino d ↔ 4 ≤ d := by
+  unfold RupturaCamino SaturacionCamino
+  rw [Gnomon.saturacion_iff d hd]
+  omega
+
+/-- La dimensión cuatro ya está en ruptura. -/
+theorem ruptura_camino_cuatro : RupturaCamino 4 := by
+  exact (ruptura_camino_iff_cuatro_le 4 (by norm_num)).2 (by norm_num)
+
+/-- Predicado puramente combinatorio de vértice no terminal del camino. -/
+def VerticeInterior {d : ℕ} (i : Fin d) : Prop :=
+  0 < i.val ∧ i.val + 1 < d
+
+/-- Existe una arista genuinamente interior cuando dos vértices no
+terminales del camino son adyacentes. -/
+def TieneAristaInterior (d : ℕ) : Prop :=
+  ∃ i j : Fin d,
+    VerticeInterior i ∧ VerticeInterior j ∧
+      (SimpleGraph.pathGraph d).Adj i j
+
+/-- El camino tiene una arista interior si y sólo si posee al menos cuatro
+vértices. Esta equivalencia no usa Robertson ni la ecuación de saturación:
+es pura combinatoria del camino. -/
+theorem tiene_arista_interior_iff_cuatro_le (d : ℕ) :
+    TieneAristaInterior d ↔ 4 ≤ d := by
+  constructor
+  · rintro ⟨i, j, hi, hj, hadj⟩
+    rcases hi with ⟨hi0, hiend⟩
+    rcases hj with ⟨hj0, hjend⟩
+    rw [SimpleGraph.pathGraph_adj] at hadj
+    rcases hadj with hij | hji <;> omega
+  · intro hd
+    let i : Fin d := ⟨1, by omega⟩
+    let j : Fin d := ⟨2, by omega⟩
+    refine ⟨i, j, ?_, ?_, ?_⟩
+    · simp [VerticeInterior, i]
+      omega
+    · simp [VerticeInterior, j]
+      omega
+    · rw [SimpleGraph.pathGraph_adj]
+      exact Or.inl rfl
+
+/-- Coincidencia central: dentro del régimen `d ≥ 2`, tener una arista entre
+dos vértices interiores equivale exactamente a romper la saturación. Las dos
+caras se demuestran por rutas independientes: combinatoria y espectral. -/
+theorem transporte_interior_iff_ruptura (d : ℕ) (hd : 2 ≤ d) :
+    TieneAristaInterior d ↔ RupturaCamino d := by
+  exact (tiene_arista_interior_iff_cuatro_le d).trans
+    (ruptura_camino_iff_cuatro_le d hd).symm
+
+/-- La primera ruptura es una propiedad de orden: hay ruptura en `d`, y `d`
+es menor o igual que cualquier otra dimensión admisible que también rompa. -/
+def EsPrimeraRuptura (d : ℕ) : Prop :=
+  2 ≤ d ∧ RupturaCamino d ∧
+    ∀ n : ℕ, 2 ≤ n → RupturaCamino n → d ≤ n
+
+/-- Caracterización dimensional: la primera ruptura es exactamente `d = 4`. -/
+theorem primera_ruptura_iff_dimension_cuatro (d : ℕ) :
+    EsPrimeraRuptura d ↔ d = 4 := by
+  constructor
+  · rintro ⟨hd2, hdR, hmin⟩
+    have h4d : 4 ≤ d := (ruptura_camino_iff_cuatro_le d hd2).1 hdR
+    have hd4 : d ≤ 4 := hmin 4 (by norm_num) ruptura_camino_cuatro
+    omega
+  · rintro rfl
+    refine ⟨by norm_num, ruptura_camino_cuatro, ?_⟩
+    intro n hn2 hnR
+    exact (ruptura_camino_iff_cuatro_le n hn2).1 hnR
+
+end PrimeraRuptura
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D14_ExcesoBrechaSzego.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D14_ExcesoBrechaSzego.lean
new file mode 100644
index 000000000..8fc66ab05
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D14_ExcesoBrechaSzego.lean
@@ -0,0 +1,61 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D9_Monotonia
+
+@[expose] public section
+
+/-!
+# El exceso sobre el límite de Szegő
+
+Corolario aritmético directo de la monotonía (`D9_Monotonia.lean`) y el
+límite de Szegő (`D8_Szego.lean`): el "exceso" `Cinf - CNava(d)` —cuánto
+le falta a `CNava(d)` para alcanzar el límite `C∞`— es positivo, máximo
+exactamente en `d = 4`, estrictamente decreciente en `d`, y se disuelve a
+`0`. No es un pilar nuevo: es la misma cadena de `D9_Monotonia.lean` leída
+desde el lado del remanente en vez del valor mismo.
+-/
+
+open Filter
+open scoped Topology
+
+namespace Gnomon
+
+/-- El exceso de coherencia: cuánto le falta a `CNava(d)` para alcanzar el
+límite de Szegő `C∞`. -/
+noncomputable def excesoBrecha (d : ℕ) : ℝ := Cinf - CNava d
+
+/-- El exceso es siempre positivo: `CNava(d)` nunca alcanza `C∞` a `d`
+finito. -/
+theorem excesoBrecha_pos (d : ℕ) (hd : 4 ≤ d) : 0 < excesoBrecha d := by
+  unfold excesoBrecha
+  linarith [CNava_lt_Cinf d hd]
+
+/-- El exceso es estrictamente decreciente en `d`, heredado de la
+monotonía de `CNava`. -/
+theorem excesoBrecha_strictAnti {a b : ℕ} (ha : 4 ≤ a) (hb : 4 ≤ b) (hab : a < b) :
+    excesoBrecha b < excesoBrecha a := by
+  unfold excesoBrecha
+  linarith [CNava_strictMonoOn_ge_four ha hb hab]
+
+/-- El exceso máximo de toda la cola `d ≥ 4` se alcanza exactamente en
+`d = 4`: el mínimo global de `CNava` es el techo del exceso. -/
+theorem excesoBrecha_le_four (d : ℕ) (hd : 4 ≤ d) :
+    excesoBrecha d ≤ excesoBrecha 4 := by
+  unfold excesoBrecha
+  linarith [CNava_four_le d hd]
+
+/-- El exceso se apaga por completo: `Cinf − CNava(d) → 0`, acotado arriba
+por `excesoBrecha 4` y llevado a `0` por el límite de Szegő. -/
+theorem excesoBrecha_tendsto_zero :
+    Tendsto (fun d : ℕ => excesoBrecha d) atTop (𝓝 0) := by
+  unfold excesoBrecha
+  have h : Tendsto (fun d : ℕ => Cinf - CNava d) atTop (𝓝 (Cinf - Cinf)) :=
+    limite_szego_CNava.const_sub Cinf
+  simpa using h
+
+end Gnomon
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D15_IdentidadCosecanteChebyshev.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D15_IdentidadCosecanteChebyshev.lean
new file mode 100644
index 000000000..1a67035d5
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D15_IdentidadCosecanteChebyshev.lean
@@ -0,0 +1,267 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import Mathlib.RingTheory.Polynomial.Chebyshev
+public import Mathlib.Analysis.Calculus.Deriv.Polynomial
+public import Mathlib.Analysis.Calculus.Deriv.Comp
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Basic
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema
+public import Mathlib.Algebra.Polynomial.Splits
+
+@[expose] public section
+
+/-!
+# Identidad cosecante clásica vía Chebyshev
+
+Identidad clásica de análisis, autocontenida:
+\[
+\sum_{k=1}^{N-1}\csc^2(k\pi/N)=(N^2-1)/3.
+\]
+Prueba vía el polinomio de Chebyshev de segunda especie `U_{N-1}`: sus
+raíces son `cos(kπ/N)`, y la derivada logarítmica evaluada en `±1`
+(descomponiendo `1/(1-x²) = ½(1/(1-x) + 1/(1+x))`) da la suma cerrada.
+
+No depende de ningún objeto definido en otro archivo de este paquete: es
+un resultado de análisis clásico, completo en sí mismo sobre `Mathlib`.
+-/
+
+noncomputable section
+
+set_option maxHeartbeats 1000000
+
+open Polynomial Polynomial.Chebyshev Real
+open scoped BigOperators
+
+namespace IdentidadCosecanteChebyshev
+
+/-! ## Chebyshev: `U_n` se parte en lineales reales -/
+
+theorem U_splits_real (n : ℕ) : (U ℝ n).Splits := by
+  rw [splits_iff_card_roots]
+  cases n with
+  | zero =>
+    simp [U_zero]
+  | succ m =>
+    set N := m + 1 with hN
+    have hdeg : (U ℝ N).natDegree = N := natDegree_U_natCast (R := ℝ) N
+    rw [roots_U_real N]
+    have hinj :
+        Set.InjOn (fun k : ℕ ↦ cos ((k + 1) * π / (N + 1))) (Finset.range N) :=
+      (Finset.range N).nodup_map_iff_injOn.mp (roots_U_real_nodup N)
+    have hcard :
+        ((Finset.range N).image fun k : ℕ ↦
+            cos ((k + 1) * π / (N + 1))).card = N := by
+      rw [Finset.card_image_of_injOn hinj, Finset.card_range]
+    show (Multiset.card _) = _
+    rw [hdeg]
+    exact hcard
+
+/-! ## Derivadas de `U_n` en `±1` -/
+
+theorem U_deriv_eval_one (n : ℕ) :
+    (derivative (U ℝ (n : ℤ))).eval (1 : ℝ) =
+      ((n : ℝ) + 2) * ((n : ℝ) + 1) * (n : ℝ) / 3 := by
+  have h := derivative_U_eval_one (R := ℝ) (n : ℤ)
+  -- `3 * U'(1) = (n+2)(n+1)n` con coerciones enteras
+  push_cast at h
+  linarith
+
+theorem U_deriv_eval_neg_one (n : ℕ) :
+    (derivative (U ℝ (n : ℤ))).eval (-1 : ℝ) =
+      -((-1 : ℝ) ^ n) *
+        (((n : ℝ) + 2) * ((n : ℝ) + 1) * (n : ℝ) / 3) := by
+  -- Paridad: U_n(-x) = (-1)^n U_n(x)
+  have hfun :
+      (fun x : ℝ ↦ (U ℝ (n : ℤ)).eval (-x)) =
+        fun x ↦ (-1 : ℝ) ^ n * (U ℝ (n : ℤ)).eval x := by
+    funext x
+    rw [U_eval_neg (R := ℝ) n x, Int.cast_negOnePow_natCast]
+  have hL : HasDerivAt (fun x : ℝ ↦ (U ℝ (n : ℤ)).eval (-x))
+      (-(derivative (U ℝ (n : ℤ))).eval (-1)) 1 := by
+    have h := (U ℝ (n : ℤ)).hasDerivAt (-1 : ℝ)
+    have hc := h.comp (1 : ℝ) (hasDerivAt_id' (𝕜 := ℝ) 1).neg
+    rw [mul_neg_one] at hc
+    exact hc
+  have hR : HasDerivAt
+      (fun x : ℝ ↦ (-1 : ℝ) ^ n * (U ℝ (n : ℤ)).eval x)
+      ((-1 : ℝ) ^ n * (derivative (U ℝ (n : ℤ))).eval 1) 1 :=
+    ((U ℝ (n : ℤ)).hasDerivAt (1 : ℝ)).const_mul _
+  have heq : HasDerivAt (fun x : ℝ ↦ (U ℝ (n : ℤ)).eval (-x))
+      ((-1 : ℝ) ^ n * (derivative (U ℝ (n : ℤ))).eval 1) 1 :=
+    hfun ▸ hR
+  have hder := HasDerivAt.unique hL heq
+  -- -U'(-1) = (-1)^n U'(1)
+  have hUd1 := U_deriv_eval_one n
+  rw [hUd1] at hder
+  linarith
+
+/-! ## Sumas sobre raíces -/
+
+theorem sum_one_div_one_sub_roots (n : ℕ) (hn : 1 ≤ n) :
+    ((U ℝ (n : ℤ)).roots.map fun z : ℝ ↦ (1 : ℝ) / (1 - z)).sum =
+      (((n : ℝ) + 1) ^ 2 - 1) / 3 := by
+  have hsplit : (U ℝ (n : ℤ)).Splits := U_splits_real n
+  have hne : (U ℝ (n : ℤ)).eval (1 : ℝ) ≠ 0 := by
+    rw [U_eval_one]; positivity
+  have hlog := hsplit.eval_derivative_div_eval_of_ne_zero hne
+  have hU1 : (U ℝ (n : ℤ)).eval (1 : ℝ) = (n : ℝ) + 1 := by
+    simp
+  have hUd := U_deriv_eval_one n
+  have hratio :
+      (derivative (U ℝ (n : ℤ))).eval (1 : ℝ) / (U ℝ (n : ℤ)).eval (1 : ℝ) =
+        (((n : ℝ) + 1) ^ 2 - 1) / 3 := by
+    rw [hUd, hU1]
+    have : (n : ℝ) + 1 ≠ 0 := by positivity
+    field_simp [this]; ring
+  rw [← hratio, hlog]
+
+theorem sum_one_div_one_add_roots (n : ℕ) (hn : 1 ≤ n) :
+    ((U ℝ (n : ℤ)).roots.map fun z : ℝ ↦ (1 : ℝ) / (1 + z)).sum =
+      (((n : ℝ) + 1) ^ 2 - 1) / 3 := by
+  have hsplit : (U ℝ (n : ℤ)).Splits := U_splits_real n
+  have hU : (U ℝ (n : ℤ)).eval (-1 : ℝ) = (-1 : ℝ) ^ n * ((n : ℝ) + 1) := by
+    rw [U_eval_neg_one (R := ℝ) (n : ℤ), Int.cast_negOnePow_natCast]
+    push_cast; ring
+  have hn1 : (n : ℝ) + 1 ≠ 0 := by positivity
+  have hpow : (-1 : ℝ) ^ n ≠ 0 := pow_ne_zero n (by norm_num)
+  have hne : (U ℝ (n : ℤ)).eval (-1 : ℝ) ≠ 0 := by
+    rw [hU]; exact mul_ne_zero hpow hn1
+  have hlog := hsplit.eval_derivative_div_eval_of_ne_zero hne
+  -- U'(-1)/U(-1) = ∑ 1/(-1-z) = -∑ 1/(1+z)
+  have hUd := U_deriv_eval_neg_one n
+  have hratio :
+      (derivative (U ℝ (n : ℤ))).eval (-1 : ℝ) / (U ℝ (n : ℤ)).eval (-1 : ℝ) =
+        -((((n : ℝ) + 1) ^ 2 - 1) / 3) := by
+    rw [hUd, hU]
+    field_simp [hpow, hn1]; ring
+  have hmap :
+      ((U ℝ (n : ℤ)).roots.map fun z : ℝ ↦ (1 : ℝ) / (-1 - z)).sum =
+        -((U ℝ (n : ℤ)).roots.map fun z : ℝ ↦ (1 : ℝ) / (1 + z)).sum := by
+    have hpt :
+        ((U ℝ (n : ℤ)).roots.map fun z : ℝ ↦ (1 : ℝ) / (-1 - z)) =
+          (U ℝ (n : ℤ)).roots.map fun z : ℝ ↦ -((1 : ℝ) / (1 + z)) := by
+      refine Multiset.map_congr rfl fun z _ => ?_
+      have : (-1 - z : ℝ) = -(1 + z) := by ring
+      rw [this, div_neg]
+    rw [hpt, Multiset.sum_map_neg]
+  have : -((U ℝ (n : ℤ)).roots.map fun z : ℝ ↦ (1 : ℝ) / (1 + z)).sum =
+      -((((n : ℝ) + 1) ^ 2 - 1) / 3) := by
+    rwa [← hmap, ← hlog]
+  linarith
+
+theorem root_abs_lt_one {n : ℕ} {z : ℝ}
+    (hz : z ∈ (U ℝ (n : ℤ)).roots) (hn : 1 ≤ n) : |z| < 1 := by
+  have hroots := roots_U_real n
+  rw [hroots, Finset.mem_val, Finset.mem_image] at hz
+  obtain ⟨k, hk, rfl⟩ := hz
+  have hklt : k < n := Finset.mem_range.mp hk
+  have hθpos : 0 < (k + 1 : ℝ) * π / (n + 1) := by positivity
+  have hθlt : (k + 1 : ℝ) * π / (n + 1) < π := by
+    have : (k + 1 : ℝ) ≤ n := by exact_mod_cast Nat.succ_le_of_lt hklt
+    calc
+      (k + 1 : ℝ) * π / (n + 1) ≤ n * π / (n + 1) := by gcongr
+      _ < π := by
+        rw [div_lt_iff₀ (by positivity)]
+        nlinarith [pi_pos]
+  have hsin : 0 < sin ((k + 1 : ℝ) * π / (n + 1)) :=
+    sin_pos_of_pos_of_lt_pi hθpos hθlt
+  have hpyth := sin_sq_add_cos_sq ((k + 1 : ℝ) * π / (n + 1))
+  have hsq : cos ((k + 1 : ℝ) * π / (n + 1)) ^ 2 < 1 := by
+    nlinarith [mul_pos hsin hsin]
+  exact (sq_lt_one_iff_abs_lt_one _).mp hsq
+
+theorem sum_one_div_one_sub_sq_roots (n : ℕ) (hn : 1 ≤ n) :
+    ((U ℝ (n : ℤ)).roots.map fun z : ℝ ↦ (1 : ℝ) / (1 - z ^ 2)).sum =
+      (((n : ℝ) + 1) ^ 2 - 1) / 3 := by
+  have h1 := sum_one_div_one_sub_roots n hn
+  have h2 := sum_one_div_one_add_roots n hn
+  have hpoint (z : ℝ) (hz : z ∈ (U ℝ (n : ℤ)).roots) :
+      (1 : ℝ) / (1 - z ^ 2) =
+        (1 / 2 : ℝ) * ((1 : ℝ) / (1 - z) + (1 : ℝ) / (1 + z)) := by
+    have habs := root_abs_lt_one hz hn
+    have hz1 : z ≠ 1 := by
+      intro h; rw [h, abs_one] at habs; linarith
+    have hzm1 : z ≠ -1 := by
+      intro h; rw [h, abs_neg, abs_one] at habs; linarith
+    have hden : (1 - z ^ 2 : ℝ) ≠ 0 := by
+      intro h
+      have : z ^ 2 = 1 := by linarith
+      have : |z| = 1 := (sq_eq_one_iff.mp this).elim (by intro; simp [*]) (by intro; simp [*])
+      -- |z|=1 contradice |z|<1
+      linarith [habs]
+    have hz1' : (1 - z : ℝ) ≠ 0 := sub_ne_zero.mpr (Ne.symm hz1)
+    have hzm : (1 + z : ℝ) ≠ 0 := by
+      intro h; exact hzm1 (by linarith)
+    field_simp [hz1', hzm, hden]
+    ring
+  -- levantar la identidad puntual a la suma sobre el multiconjunto
+  have hdecomp :
+      ((U ℝ (n : ℤ)).roots.map fun z : ℝ ↦ (1 : ℝ) / (1 - z ^ 2)).sum =
+        (1 / 2 : ℝ) *
+          (((U ℝ (n : ℤ)).roots.map fun z ↦ (1 : ℝ) / (1 - z)).sum +
+            ((U ℝ (n : ℤ)).roots.map fun z ↦ (1 : ℝ) / (1 + z)).sum) := by
+    classical
+    calc
+      ((U ℝ (n : ℤ)).roots.map fun z : ℝ ↦ (1 : ℝ) / (1 - z ^ 2)).sum
+          = ((U ℝ (n : ℤ)).roots.map fun z : ℝ ↦
+              (1 / 2 : ℝ) * ((1 : ℝ) / (1 - z) + (1 : ℝ) / (1 + z))).sum := by
+            refine congr_arg Multiset.sum (Multiset.map_congr rfl fun z hz => hpoint z hz)
+      _ = (1 / 2 : ℝ) *
+            ((U ℝ (n : ℤ)).roots.map fun z : ℝ ↦
+              ((1 : ℝ) / (1 - z) + (1 : ℝ) / (1 + z))).sum := by
+            simp [Multiset.sum_map_mul_left]
+      _ = (1 / 2 : ℝ) *
+            (((U ℝ (n : ℤ)).roots.map fun z ↦ (1 : ℝ) / (1 - z)).sum +
+              ((U ℝ (n : ℤ)).roots.map fun z ↦ (1 : ℝ) / (1 + z)).sum) := by
+            simp [Multiset.sum_map_add]
+  rw [hdecomp, h1, h2]
+  ring
+
+/-- **Identidad cosecante clásica.** `∑_{k=1}^{N-1} csc²(kπ/N) = (N²-1)/3`. -/
+theorem sum_csc_sq (N : ℕ) (hN : 2 ≤ N) :
+    ∑ k ∈ Finset.Ico 1 N, (sin ((k : ℝ) * π / N))⁻¹ ^ 2 =
+      ((N : ℝ) ^ 2 - 1) / 3 := by
+  obtain ⟨n, rfl⟩ : ∃ n, N = n + 1 := ⟨N - 1, by omega⟩
+  have hn : 1 ≤ n := by omega
+  have hsum := sum_one_div_one_sub_sq_roots n hn
+  classical
+  have hinj :
+      Set.InjOn (fun k : ℕ ↦ cos ((k + 1) * π / (n + 1))) (Finset.range n) :=
+    (Finset.range n).nodup_map_iff_injOn.mp (roots_U_real_nodup n)
+  -- suma sobre raíces vista como multiconjunto → suma sobre `range n`
+  have hfin :
+      ((U ℝ (n : ℤ)).roots.map fun z : ℝ ↦ (1 : ℝ) / (1 - z ^ 2)).sum =
+        ∑ k ∈ Finset.range n,
+          (1 : ℝ) / (1 - cos ((k + 1 : ℝ) * π / (n + 1)) ^ 2) := by
+    rw [roots_U_real n, Finset.image_val_of_injOn hinj, Multiset.map_map]
+    rfl
+  -- 1 - cos² = sin²
+  have hsin :
+      ∑ k ∈ Finset.range n,
+          (1 : ℝ) / (1 - cos ((k + 1 : ℝ) * π / (n + 1)) ^ 2) =
+        ∑ k ∈ Finset.range n, (sin ((k + 1 : ℝ) * π / (n + 1)))⁻¹ ^ 2 := by
+    refine Finset.sum_congr rfl fun k hk => ?_
+    have h1 : (1 : ℝ) - cos ((k + 1 : ℝ) * π / (n + 1)) ^ 2 =
+        sin ((k + 1 : ℝ) * π / (n + 1)) ^ 2 := by
+      linarith [sin_sq_add_cos_sq ((k + 1 : ℝ) * π / (n + 1))]
+    rw [h1, one_div, inv_pow]
+  -- reindexar `Ico 1 (n+1)` como imagen de `range n` bajo `·+1`
+  have himg :
+      Finset.Ico 1 (n + 1) = (Finset.range n).image (fun k ↦ k + 1) := by
+    ext k
+    simp only [Finset.mem_Ico, Finset.mem_image, Finset.mem_range]
+    constructor
+    · rintro ⟨h1, h2⟩
+      exact ⟨k - 1, by omega, by omega⟩
+    · rintro ⟨j, hj, rfl⟩
+      omega
+  rw [himg, Finset.sum_image (fun x _ y _ h => by simpa using h)]
+  push_cast
+  rw [← hsin, ← hfin]
+  exact hsum
+
+end IdentidadCosecanteChebyshev
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D8_Szego.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D8_Szego.lean
new file mode 100644
index 000000000..92122fc92
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D8_Szego.lean
@@ -0,0 +1,661 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import Mathlib.Algebra.Order.Ring.Star
+public import Mathlib.Algebra.Order.Star.Real
+public import Mathlib.Algebra.Ring.IsFormallyReal
+public import Mathlib.Analysis.Real.Pi.Bounds
+public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc
+public import Mathlib.Tactic.IntervalCases
+
+@[expose] public section
+
+/-!
+# D8 — El límite de Szegő y la positividad de la brecha
+
+Define la constante de coherencia finita `C_Nava(d)` (forma cerrada exacta
+en `cos`/`sin` de `π/(d+1)`) y su brecha `deltaGeom(d) = C_Nava(d) − 1`.
+Dos resultados centrales:
+
+1. **Positividad** (`deltaGeom_pos_of_four_le`): `deltaGeom(d) > 0` para
+   todo `d ≥ 4`. Se verifica `d = 4, 5` en forma cerrada exacta y `d ≥ 6`
+   mediante cotas de Taylor certificadas para seno y coseno — sin apelar a
+   ningún cálculo numérico externo, sólo álgebra racional y las cotas
+   `Real.pi_gt_d2`/`Real.pi_lt_d2` de Mathlib.
+2. **Límite de Szegő** (`limite_szego_CNava`): `C_Nava(d) → C_∞ = √(π²/3−2)`
+   cuando `d → ∞` (como filtro `atTop` sobre la sucesión de espacios
+   finitos, no como un nuevo espacio de Hilbert en `d = ∞`; ver
+   `infinito_no_es_dimension_sino_limite`). En particular
+   `deltaInf = C_∞ − 1 > 0`, consecuencia exacta de `π > 3`.
+-/
+
+noncomputable section
+
+open Real
+open Filter
+open scoped Topology
+
+namespace Gnomon
+
+/-! ## Forma cerrada y límite asintótico -/
+
+/-- `N = d + 1`, notación para el grafo camino `pathGraph d`. -/
+noncomputable def Nreal (d : ℕ) : ℝ := (d : ℝ) + 1
+
+/-- Ángulo espectral fundamental `θ_d = π/(d+1)`. -/
+noncomputable def theta (d : ℕ) : ℝ := π / Nreal d
+
+/-- Forma cerrada exacta de `C_Nava(d)²`. -/
+noncomputable def CNavaSq (d : ℕ) : ℝ :=
+  2 * ((d : ℝ) - 1) / (Nreal d * Real.cos (theta d) ^ 2) *
+    (((Nreal d ^ 2 + 2) / 6) * Real.sin (theta d) ^ 2 - 1)
+
+/-- Constante de coherencia finita. -/
+noncomputable def CNava (d : ℕ) : ℝ := Real.sqrt (CNavaSq d)
+
+/-- Límite universal de Szegő. -/
+noncomputable def Cinf : ℝ := Real.sqrt (π ^ 2 / 3 - 2)
+
+/-- Defecto geométrico finito `δ_geom(d) = C_Nava(d) - 1`. -/
+noncomputable def deltaGeom (d : ℕ) : ℝ := CNava d - 1
+
+/-- Defecto asintótico `δ_∞ = C_∞ - 1`. -/
+noncomputable def deltaInf : ℝ := Cinf - 1
+
+/-- Término principal de la expansión de Szegő. -/
+noncomputable def deltaSzegoPrincipal (d : ℕ) : ℝ :=
+  deltaInf - Cinf / Nreal d
+
+theorem CNavaSq_forma_cerrada (d : ℕ) :
+    CNavaSq d =
+      2 * ((d : ℝ) - 1) / (Nreal d * Real.cos (theta d) ^ 2) *
+        (((Nreal d ^ 2 + 2) / 6) * Real.sin (theta d) ^ 2 - 1) := rfl
+
+/-- La forma cerrada reescrita mediante `sinc`. Elimina la singularidad
+aparente y permite tomar el límite en Lean. -/
+theorem CNavaSq_forma_regularizada (d : ℕ) :
+    CNavaSq d =
+      2 * (1 - 2 / Nreal d) / Real.cos (theta d) ^ 2 *
+        ((π ^ 2 * Real.sinc (theta d) ^ 2 +
+          2 * Real.sin (theta d) ^ 2) / 6 - 1) := by
+  have hN : Nreal d ≠ 0 := by
+    unfold Nreal
+    positivity
+  have ht : theta d ≠ 0 := by
+    unfold theta
+    exact div_ne_zero Real.pi_ne_zero hN
+  rw [CNavaSq, Real.sinc_of_ne_zero ht]
+  unfold theta Nreal
+  field_simp
+  ring
+
+theorem Nreal_tendsto_atTop : Tendsto Nreal atTop atTop := by
+  unfold Nreal
+  exact tendsto_atTop_add_const_right atTop 1 tendsto_natCast_atTop_atTop
+
+theorem theta_tendsto_zero : Tendsto theta atTop (𝓝 0) := by
+  unfold theta
+  exact Nreal_tendsto_atTop.const_div_atTop π
+
+private theorem CNavaSq_regularizada_tendsto :
+    Tendsto
+      (fun d : ℕ =>
+        2 * (1 - 2 / Nreal d) / Real.cos (theta d) ^ 2 *
+          ((π ^ 2 * Real.sinc (theta d) ^ 2 +
+            2 * Real.sin (theta d) ^ 2) / 6 - 1))
+      atTop (𝓝 (π ^ 2 / 3 - 2)) := by
+  have hratio : Tendsto (fun d : ℕ => 1 - 2 / Nreal d) atTop (𝓝 1) := by
+    convert tendsto_const_nhds.sub
+      (Nreal_tendsto_atTop.const_div_atTop 2) using 1
+    norm_num
+  have hcos : Tendsto
+      (fun d : ℕ => Real.cos (theta d) ^ 2) atTop (𝓝 1) := by
+    simpa [Real.cos_zero] using
+      (Real.continuous_cos.continuousAt.tendsto.comp theta_tendsto_zero).pow 2
+  have hsinc : Tendsto
+      (fun d : ℕ => Real.sinc (theta d) ^ 2) atTop (𝓝 1) := by
+    simpa [Real.sinc_zero] using
+      (Real.continuous_sinc.continuousAt.tendsto.comp theta_tendsto_zero).pow 2
+  have hsin : Tendsto
+      (fun d : ℕ => Real.sin (theta d) ^ 2) atTop (𝓝 0) := by
+    simpa [Real.sin_zero] using
+      (Real.continuous_sin.continuousAt.tendsto.comp theta_tendsto_zero).pow 2
+  have hbracket : Tendsto
+      (fun d : ℕ =>
+        (π ^ 2 * Real.sinc (theta d) ^ 2 +
+          2 * Real.sin (theta d) ^ 2) / 6 - 1)
+      atTop (𝓝 (π ^ 2 / 6 - 1)) := by
+    convert ((tendsto_const_nhds.mul hsinc).add
+      (tendsto_const_nhds.mul hsin)).div_const 6 |>.sub_const 1 using 1
+    ring_nf
+  have hfactor : Tendsto
+      (fun d : ℕ => 2 * (1 - 2 / Nreal d) / Real.cos (theta d) ^ 2)
+      atTop (𝓝 2) := by
+    have htworatio : Tendsto
+        (fun d : ℕ => (2 : ℝ) * (1 - 2 / Nreal d))
+        atTop (𝓝 ((2 : ℝ) * 1)) :=
+      tendsto_const_nhds.mul hratio
+    convert htworatio.div hcos (by norm_num : (1 : ℝ) ≠ 0) using 1
+    · ext d
+      simp [Pi.div_apply]
+    · norm_num
+  convert hfactor.mul hbracket using 1
+  · ext d
+    ring_nf
+
+/-- TEOREMA DE SZEGŐ, forma cuadrática: `C_Nava(d)² → (π²-6)/3`. -/
+theorem limite_szego_CNavaSq :
+    Tendsto CNavaSq atTop (𝓝 (π ^ 2 / 3 - 2)) := by
+  apply CNavaSq_regularizada_tendsto.congr'
+  filter_upwards with d
+  exact (CNavaSq_forma_regularizada d).symm
+
+/-- LÍMITE DE SZEGŐ: `C_Nava(d) → C_∞ = √((π²-6)/3)`, construido sobre la
+teoría clásica de distribución espectral de Szegő. -/
+theorem limite_szego_CNava : Tendsto CNava atTop (𝓝 Cinf) := by
+  unfold CNava Cinf
+  exact Real.continuous_sqrt.continuousAt.tendsto.comp limite_szego_CNavaSq
+
+/-- Nombre citable de la especialización. Es un alias del resultado ya
+demostrado, no una rederivación de la teoría clásica de Toeplitz/Szegő. -/
+theorem limite_nava_szego_CNava : Tendsto CNava atTop (𝓝 Cinf) :=
+  limite_szego_CNava
+
+/-- El defecto geométrico converge al defecto universal asintótico. -/
+theorem limite_defecto_geometrico :
+    Tendsto deltaGeom atTop (𝓝 deltaInf) := by
+  unfold deltaGeom deltaInf
+  exact limite_szego_CNava.sub_const 1
+
+/-- El defecto universal jamás se anula: `δ_∞ > 0`, consecuencia exacta de
+`π > 3`. -/
+theorem deltaInf_pos : 0 < deltaInf := by
+  unfold deltaInf Cinf
+  have hpi : (3 : ℝ) < π := Real.pi_gt_three
+  have hx : (1 : ℝ) < π ^ 2 / 3 - 2 := by
+    nlinarith [Real.pi_pos]
+  have hs := Real.sqrt_lt_sqrt (by norm_num : (0 : ℝ) ≤ 1) hx
+  rw [Real.sqrt_one] at hs
+  linarith
+
+/-- El infinito no se añade como una dimensión realizada: la sucesión de
+defectos finitos sólo converge al valor límite estricto `δ∞ = C∞ - 1`. -/
+theorem infinito_no_es_dimension_sino_limite :
+    Tendsto deltaGeom atTop (𝓝 deltaInf) ∧ deltaInf = Cinf - 1 ∧ 0 < deltaInf :=
+  ⟨limite_defecto_geometrico, rfl, deltaInf_pos⟩
+
+theorem Cinf_pos : 0 < Cinf := by
+  have h := deltaInf_pos
+  unfold deltaInf at h
+  linarith
+
+/-- Monotonía asintótica: el término principal `δ_∞ - C_∞/(d+1)` de la
+expansión de Szegő es estrictamente creciente. -/
+theorem deltaSzegoPrincipal_strictMono : StrictMono deltaSzegoPrincipal := by
+  intro a b hab
+  have hNa : 0 < Nreal a := by
+    unfold Nreal
+    positivity
+  have hNlt : Nreal a < Nreal b := by
+    have habr : (a : ℝ) < b := by exact_mod_cast hab
+    unfold Nreal
+    linarith
+  have hinv : 1 / Nreal b < 1 / Nreal a :=
+    one_div_lt_one_div_of_lt hNa hNlt
+  unfold deltaSzegoPrincipal
+  have hmul := mul_lt_mul_of_pos_left hinv Cinf_pos
+  simpa [div_eq_mul_inv] using sub_lt_sub_left hmul deltaInf
+
+/-! ## Positividad de la brecha para `d ≥ 4`
+
+Niven cierra la ecuación de saturación en `d ∈ {2,3}` (`D7_Niven.lean`).
+Lo que sigue traduce ese hecho trigonométrico a la desigualdad algebraica
+`1 < CNava(d)` (equivalentemente `0 < deltaGeom d`) para todo `d ≥ 4`,
+verificando los casos `d = 4, 5` en forma cerrada exacta y `d ≥ 6` mediante
+las mismas cotas de Taylor certificadas usadas arriba para el límite. -/
+
+/-! ## Consecuencia: `δ_geom(d) > 0` para todo `d ≥ 4`
+
+Niven cierra la ecuación de saturación en `d ∈ {2,3}`. El resto de esta
+sección traduce ese hecho trigonométrico a la desigualdad algebraica
+`1 < CNava(d)` (equivalentemente `0 < deltaGeom d`) para todo `d ≥ 4`,
+verificando los casos `d = 4, 5` en forma cerrada exacta y `d ≥ 6` mediante
+cotas de Taylor certificadas para seno y coseno (sin apelar a ningún
+resultado numérico externo: sólo `Real.pi_gt_d2`/`pi_lt_d2` de Mathlib y
+álgebra racional). -/
+
+theorem CNavaSq_two : CNavaSq 2 = 1 := by
+  simp only [CNavaSq, Nreal, theta, Nat.cast_ofNat]
+  rw [show (2:ℝ)+1 = 3 by norm_num, show (2:ℝ)-1 = 1 by norm_num]
+  rw [cos_pi_div_three, sin_pi_div_three]
+  have h3 : (√3)^2 = (3:ℝ) := sq_sqrt (by norm_num)
+  have hs : (√3 / 2)^2 = (3:ℝ)/4 := by rw [div_pow, h3]; norm_num
+  simp [hs]; norm_num
+
+theorem CNavaSq_three : CNavaSq 3 = 1 := by
+  simp only [CNavaSq, Nreal, theta, Nat.cast_ofNat]
+  rw [show (3:ℝ)+1 = 4 by norm_num, show (3:ℝ)-1 = 2 by norm_num]
+  rw [cos_pi_div_four, sin_pi_div_four]
+  have h2 : (√2)^2 = (2:ℝ) := sq_sqrt (by norm_num)
+  have hs : (√2 / 2)^2 = (2:ℝ)/4 := by rw [div_pow, h2]; norm_num
+  simp [hs]; norm_num
+
+theorem CNavaSq_four_eq : CNavaSq 4 = (99 - 42 * √5) / 5 := by
+  simp only [CNavaSq, Nreal, theta, Nat.cast_ofNat]
+  rw [show (4:ℝ)+1 = 5 by norm_num, show (4:ℝ)-1 = 3 by norm_num]
+  have hc0 : cos (π / 5) = (1 + √5) / 4 := cos_pi_div_five
+  have hs5 : (√5)^2 = (5:ℝ) := sq_sqrt (by norm_num)
+  have hcos2 : cos (π / 5)^2 = (3 + √5) / 8 := by
+    rw [hc0]; ring_nf; simp [hs5]; ring
+  have hsin2 : sin (π / 5)^2 = (5 - √5) / 8 := by
+    have : sin (π / 5)^2 = 1 - cos (π / 5)^2 := by
+      rw [← sin_sq_add_cos_sq (π / 5)]; ring
+    rw [this, hcos2]; ring
+  rw [hcos2, hsin2]
+  ring_nf; field_simp; ring_nf; simp [hs5]; ring
+
+theorem one_lt_CNavaSq_four : 1 < CNavaSq 4 := by
+  rw [CNavaSq_four_eq]
+  have h5 : (0:ℝ) < 5 := by norm_num
+  rw [one_lt_div h5]
+  have h : √5 < (47:ℝ) / 21 := by
+    rw [sqrt_lt (by norm_num : (0:ℝ) ≤ 5) (by positivity)]; norm_num
+  have hs : (√5)^2 = (5:ℝ) := sq_sqrt (by norm_num)
+  nlinarith [hs, h, sqrt_nonneg 5]
+
+theorem CNavaSq_five : CNavaSq 5 = (28:ℝ)/27 := by
+  simp only [CNavaSq, Nreal, theta, Nat.cast_ofNat]
+  rw [show (5:ℝ)+1 = 6 by norm_num, show (5:ℝ)-1 = 4 by norm_num]
+  rw [cos_pi_div_six, sin_pi_div_six]
+  have h3 : (√3)^2 = (3:ℝ) := sq_sqrt (by norm_num)
+  have hc : (√3 / 2)^2 = (3:ℝ)/4 := by rw [div_pow, h3]; norm_num
+  simp [hc]; norm_num
+
+theorem one_lt_CNavaSq_five : 1 < CNavaSq 5 := by
+  rw [CNavaSq_five]; norm_num
+
+theorem one_lt_CNavaSq_six : 1 < CNavaSq 6 := by
+  simp only [CNavaSq, Nreal, theta, Nat.cast_ofNat]
+  rw [show (6:ℝ)+1 = 7 by norm_num, show (6:ℝ)-1 = 5 by norm_num]
+  set θ := π / 7
+  have h7 : (0:ℝ) < 7 := by norm_num
+  have hθpos : 0 < θ := div_pos pi_pos h7
+  have hθlt : θ < π / 2 := by
+    rw [div_lt_div_iff₀ h7 (by norm_num : (0:ℝ)<2)]; nlinarith [pi_pos]
+  have hπlo : (314:ℝ)/100 < π := by have := pi_gt_d2; norm_num at this ⊢; linarith
+  have hπhi : π < (315:ℝ)/100 := by have := pi_lt_d2; norm_num at this ⊢; linarith
+  have hθ_lt_one : θ < 1 := by
+    have : θ < (315:ℝ)/100 / 7 := by
+      simp only [θ]; exact div_lt_div_of_pos_right hπhi h7
+    exact lt_trans this (by norm_num)
+  have hs0pos : 0 < θ - θ^3/6 := by
+    have : θ^2 < 6 := by nlinarith [hθpos, hθ_lt_one]
+    nlinarith [hθpos, this]
+  have hsin : θ - θ^3/6 < sin θ := sin_gt_sub_cube hθpos
+  set s0 := θ - θ^3/6
+  have hcos_pos : 0 < cos θ := cos_pos_of_mem_Ioo ⟨by linarith [pi_pos, hθpos], hθlt⟩
+  have hcos2 : cos θ ^ 2 < 1 - s0 ^ 2 := by
+    have hsq : s0^2 < sin θ ^ 2 :=
+      pow_lt_pow_left₀ hsin (le_of_lt hs0pos) (by norm_num)
+    have : cos θ ^ 2 = 1 - sin θ ^ 2 := by rw [← sin_sq_add_cos_sq θ]; ring
+    linarith
+  have hs0_bound :
+      s0 ≥ ((314:ℝ)/100 / 7) * (1 - ((315:ℝ)/100 / 7)^2 / 6) := by
+    have hs0θ : s0 = θ * (1 - θ^2/6) := by ring
+    have hθlo : (314:ℝ)/100 / 7 ≤ θ := by
+      simp only [θ]; exact div_le_div_of_nonneg_right hπlo.le (le_of_lt h7)
+    have hθhi2 : θ ≤ (315:ℝ)/100 / 7 := by
+      simp only [θ]; exact div_le_div_of_nonneg_right hπhi.le (le_of_lt h7)
+    have hfac : 1 - θ^2/6 ≥ 1 - ((315:ℝ)/100 / 7)^2 / 6 := by
+      nlinarith [hθpos, hθhi2]
+    rw [hs0θ]; nlinarith [hθlo, hfac, hθpos]
+  have hpos_lo :
+      (0:ℝ) ≤ ((314:ℝ)/100 / 7) * (1 - ((315:ℝ)/100 / 7)^2 / 6) := by
+    norm_num
+  have hs0sq :
+      s0^2 ≥ (((314:ℝ)/100 / 7) * (1 - ((315:ℝ)/100 / 7)^2 / 6))^2 :=
+    pow_le_pow_left₀ hpos_lo hs0_bound 2
+  have h1s0 :
+      1 - s0^2 ≤
+        1 - (((314:ℝ)/100 / 7) * (1 - ((315:ℝ)/100 / 7)^2 / 6))^2 := by
+    nlinarith [hs0sq]
+  have hlt_thr :
+      1 - (((314:ℝ)/100 / 7) * (1 - ((315:ℝ)/100 / 7)^2 / 6))^2 <
+        (75:ℝ)/92 := by
+    norm_num
+  have hcos_thr : cos θ ^ 2 < (75:ℝ)/92 :=
+    lt_of_lt_of_le hcos2 (le_trans h1s0 hlt_thr.le)
+  have hs : sin θ ^ 2 = 1 - cos θ ^ 2 := by
+    rw [← sin_sq_add_cos_sq θ]; ring
+  rw [hs]
+  set c := cos θ ^ 2
+  have hcpos : 0 < c := sq_pos_of_pos hcos_pos
+  have hc_thr : c < (75:ℝ)/92 := hcos_thr
+  have hsimp :
+      2 * 5 / (7 * c) * (((7:ℝ)^2 + 2) / 6 * (1 - c) - 1) =
+        5 * (15 - 17 * c) / (7 * c) := by
+    field_simp; ring
+  rw [hsimp]
+  have hden : 0 < 7 * c := by positivity
+  rw [one_lt_div hden]
+  nlinarith [hc_thr, hcpos]
+
+noncomputable def thrCNava (d : ℕ) : ℝ :=
+  ((d : ℝ) - 1) * (((d : ℝ) + 1) ^ 2 - 4) /
+    (((d : ℝ) - 1) * (((d : ℝ) + 1) ^ 2 + 2) + 3 * ((d : ℝ) + 1))
+
+theorem CNavaSq_eq_cos_form (d : ℕ) (hd : 2 ≤ d)
+    (hcos_ne : cos (π / ((d : ℝ) + 1)) ≠ 0) :
+    CNavaSq d =
+      ((d : ℝ) - 1) / (3 * ((d : ℝ) + 1) * cos (π / ((d : ℝ) + 1)) ^ 2) *
+        ((((d : ℝ) + 1) ^ 2 - 4) -
+          (((d : ℝ) + 1) ^ 2 + 2) * cos (π / ((d : ℝ) + 1)) ^ 2) := by
+  set N := (d : ℝ) + 1
+  set c := cos (π / N) ^ 2
+  have hs : sin (π / N) ^ 2 = 1 - c := by
+    have := sin_sq_add_cos_sq (π / N)
+    simp only [c] at *; linarith
+  have hcos_ne' : cos (π / N) ≠ 0 := by simpa [N] using hcos_ne
+  simp only [CNavaSq, Nreal, theta]
+  change
+      2 * ((d : ℝ) - 1) / (N * cos (π / N) ^ 2) *
+          (((N ^ 2 + 2) / 6) * sin (π / N) ^ 2 - 1) =
+        ((d : ℝ) - 1) / (3 * N * cos (π / N) ^ 2) *
+          ((N ^ 2 - 4) - (N ^ 2 + 2) * cos (π / N) ^ 2)
+  rw [hs]
+  have hc0 : c ≠ 0 := by
+    have : cos (π / N) ^ 2 ≠ 0 := pow_ne_zero 2 hcos_ne'
+    simpa [c] using this
+  have hN0 : N ≠ 0 := by positivity
+  simp only [c] at hc0 ⊢
+  field_simp [hcos_ne', hN0]
+  ring
+
+theorem one_lt_CNavaSq_of_cos_lt_thr
+    (d : ℕ) (hd : 2 ≤ d)
+    (hcos_pos : 0 < cos (π / ((d : ℝ) + 1)))
+    (hthr : cos (π / ((d : ℝ) + 1)) ^ 2 < thrCNava d) :
+    1 < CNavaSq d := by
+  set N := (d : ℝ) + 1
+  set c := cos (π / N) ^ 2
+  have hcpos : 0 < c := by
+    change 0 < cos (π / N) ^ 2
+    exact sq_pos_of_pos (by simpa [N] using hcos_pos)
+  have hform : CNavaSq d =
+      ((d : ℝ) - 1) / (3 * N * c) * ((N ^ 2 - 4) - (N ^ 2 + 2) * c) := by
+    have hne : cos (π / ((d : ℝ) + 1)) ≠ 0 := hcos_pos.ne'
+    simpa [N, c] using CNavaSq_eq_cos_form d hd hne
+  have hden : 0 < 3 * N * c := by
+    have : 0 < N := by positivity
+    positivity
+  have ha : 0 < (d : ℝ) - 1 := by
+    have : (2:ℝ) ≤ d := by exact_mod_cast hd
+    linarith
+  set e := (N ^ 2 - 4) - (N ^ 2 + 2) * c with hedef
+  have hthr' : c <
+      ((d : ℝ) - 1) * (N ^ 2 - 4) /
+        (((d : ℝ) - 1) * (N ^ 2 + 2) + 3 * N) := by
+    simpa [c, N, thrCNava] using hthr
+  have hden' : 0 < ((d : ℝ) - 1) * (N ^ 2 + 2) + 3 * N := by positivity
+  have hN2 : 0 < N ^ 2 - 4 := by
+    have : (3:ℝ) ≤ N := by
+      have : (2:ℝ) ≤ d := by exact_mod_cast hd
+      linarith
+    nlinarith
+  have hthr_mul :
+      c * (((d : ℝ) - 1) * (N ^ 2 + 2) + 3 * N) <
+        ((d : ℝ) - 1) * (N ^ 2 - 4) :=
+    (lt_div_iff₀ hden').mp hthr'
+  have he : 0 < e := by
+    have hcmp :
+        ((d : ℝ) - 1) * (N ^ 2 - 4) /
+            (((d : ℝ) - 1) * (N ^ 2 + 2) + 3 * N) <
+          (N ^ 2 - 4) / (N ^ 2 + 2) := by
+      rw [div_lt_div_iff₀ hden' (by positivity)]
+      nlinarith [hN2, ha, show 0 < N by positivity]
+    have hc_mid : c < (N ^ 2 - 4) / (N ^ 2 + 2) := lt_trans hthr' hcmp
+    have : c * (N ^ 2 + 2) < N ^ 2 - 4 := (lt_div_iff₀ (by positivity)).mp hc_mid
+    simp only [e]; linarith
+  have hmul : ((d : ℝ) - 1) * e > 3 * N * c := by
+    simp only [e]
+    nlinarith [hthr_mul]
+  have hgt : ((d : ℝ) - 1) / (3 * N * c) * e > 1 := by
+    have : ((d : ℝ) - 1) * e / (3 * N * c) > 1 :=
+      (one_lt_div hden).mpr hmul
+    convert this using 1; ring
+  rwa [hform]
+
+theorem key_poly_nat (n : ℕ) (hn : 7 ≤ n) :
+    ((314:ℝ)/100)^2 * (1 - ((315:ℝ)/100)^2 / (3 * (n:ℝ)^2)) *
+        ((n:ℝ)^3 - 2*(n:ℝ)^2 + 5*(n:ℝ) - 4) >
+      (9*(n:ℝ) - 12) * (n:ℝ)^2 := by
+  by_cases hle : n ≤ 40
+  · interval_cases n <;> norm_num
+  · have hnR : (41:ℝ) ≤ n := by exact_mod_cast (show 41 ≤ n by omega)
+    have hnpos : (0:ℝ) < n := lt_of_lt_of_le (by norm_num : (0:ℝ) < 41) hnR
+    have h314 : ((314:ℝ)/100)^2 ≥ (985:ℝ)/100 := by norm_num
+    have hfac : 1 - ((315:ℝ)/100)^2 / (3 * (n:ℝ)^2) ≥ (99:ℝ)/100 := by
+      have hle' : ((315:ℝ)/100)^2 / (3 * (n:ℝ)^2) ≤
+          ((315:ℝ)/100)^2 / (3 * 41 ^ 2) := by
+        apply div_le_div_of_nonneg_left (by positivity) (by positivity)
+        nlinarith [hnR]
+      have : ((315:ℝ)/100)^2 / (3 * 41 ^ 2) ≤ (1:ℝ)/100 := by norm_num
+      linarith
+    have hden : (n:ℝ)^3 - 2*(n:ℝ)^2 + 5*(n:ℝ) - 4 ≥ (n:ℝ)^3 - 2*(n:ℝ)^2 := by
+      nlinarith [hnpos]
+    have hden' : (n:ℝ)^3 - 2*(n:ℝ)^2 = (n:ℝ)^2 * ((n:ℝ) - 2) := by ring
+    have hmain : ((985:ℝ)/100) * ((99:ℝ)/100) * ((n:ℝ) - 2) >
+        9 * (n:ℝ) - 12 := by
+      nlinarith [hnR]
+    have hfacpos : (0:ℝ) < 1 - ((315:ℝ)/100)^2 / (3 * (n:ℝ)^2) := by
+      linarith [hfac]
+    have hdenpos : (0:ℝ) < (n:ℝ)^3 - 2*(n:ℝ)^2 + 5*(n:ℝ) - 4 := by
+      nlinarith [hnR]
+    nlinarith [h314, hfac, hden, hden', hmain, hfacpos, hdenpos,
+      show (0:ℝ) ≤ (n:ℝ)^2 by positivity]
+
+theorem cos_sq_bound_of_seven_le (N : ℝ) (hN : (7:ℝ) ≤ N) :
+    cos (π / N) ^ 2 <
+      1 - (((314:ℝ)/100 / N) * (1 - ((315:ℝ)/100 / N)^2 / 6)) ^ 2 := by
+  have hNpos : 0 < N := lt_of_lt_of_le (by norm_num : (0:ℝ) < 7) hN
+  set θ := π / N
+  have hθpos : 0 < θ := div_pos pi_pos hNpos
+  have hθlt : θ < π / 2 := by
+    rw [div_lt_div_iff₀ hNpos (by norm_num : (0:ℝ)<2)]
+    nlinarith [pi_pos, hN]
+  have hπlo : (314:ℝ)/100 < π := by
+    have := pi_gt_d2; norm_num at this ⊢; linarith
+  have hπhi : π < (315:ℝ)/100 := by
+    have := pi_lt_d2; norm_num at this ⊢; linarith
+  have hθle : θ ≤ π / 7 := by
+    rw [div_le_div_iff₀ hNpos (by norm_num : (0:ℝ)<7)]
+    nlinarith [pi_pos, hN]
+  have hθlt1 : θ < 1 := by
+    have h1 : π / 7 < (315:ℝ)/100 / 7 :=
+      div_lt_div_of_pos_right hπhi (by norm_num)
+    have h2 : ((315:ℝ)/100 / 7) < 1 := by norm_num
+    exact lt_of_le_of_lt hθle (lt_trans h1 h2)
+  have hs0pos : 0 < θ - θ^3/6 := by
+    have : θ^2 < 6 := by nlinarith [hθpos, hθlt1]
+    nlinarith [hθpos, this]
+  have hsin : θ - θ^3/6 < sin θ := sin_gt_sub_cube hθpos
+  set s0 := θ - θ^3/6
+  set s0lo := ((314:ℝ)/100 / N) * (1 - ((315:ℝ)/100 / N)^2 / 6)
+  have hcos2 : cos θ ^ 2 < 1 - s0 ^ 2 := by
+    have hsq : s0^2 < sin θ ^ 2 :=
+      pow_lt_pow_left₀ hsin hs0pos.le (by norm_num)
+    have : cos θ ^ 2 = 1 - sin θ ^ 2 := by
+      rw [← sin_sq_add_cos_sq θ]; ring
+    linarith
+  have hfac_s0lo_pos : 0 ≤ 1 - ((315:ℝ)/100 / N)^2 / 6 := by
+    have hle : ((315:ℝ)/100 / N)^2 / 6 ≤ ((315:ℝ)/100 / 7)^2 / 6 := by
+      have : (315:ℝ)/100 / N ≤ (315:ℝ)/100 / 7 :=
+        div_le_div_of_nonneg_left (by positivity) (by norm_num) hN
+      nlinarith [this, show (0:ℝ) ≤ 315/100/N by positivity]
+    have : ((315:ℝ)/100 / 7)^2 / 6 < 1 := by norm_num
+    linarith
+  have hs0_ge : s0 ≥ s0lo := by
+    have hs0θ : s0 = θ * (1 - θ^2/6) := by ring
+    have hθlo : (314:ℝ)/100 / N ≤ θ := by
+      change (314:ℝ)/100 / N ≤ π / N
+      exact div_le_div_of_nonneg_right hπlo.le hNpos.le
+    have hθhi2 : θ ≤ (315:ℝ)/100 / N := by
+      change π / N ≤ (315:ℝ)/100 / N
+      exact div_le_div_of_nonneg_right hπhi.le hNpos.le
+    have hfac : 1 - θ^2/6 ≥ 1 - ((315:ℝ)/100 / N)^2 / 6 := by
+      nlinarith [hθpos, hθhi2]
+    have hfacθ : 0 ≤ 1 - θ^2/6 := by
+      have : θ^2 ≤ 1 := by nlinarith [hθpos, hθlt1]
+      nlinarith
+    have ha : (0:ℝ) ≤ (314:ℝ)/100 / N := by positivity
+    rw [hs0θ]
+    calc
+      θ * (1 - θ^2/6)
+          ≥ ((314:ℝ)/100 / N) * (1 - θ^2/6) :=
+            mul_le_mul_of_nonneg_right hθlo hfacθ
+      _ ≥ ((314:ℝ)/100 / N) * (1 - ((315:ℝ)/100 / N)^2 / 6) :=
+            mul_le_mul_of_nonneg_left hfac ha
+  have hpos_lo : 0 ≤ s0lo := by positivity
+  have hs0sq : s0 ^ 2 ≥ s0lo ^ 2 :=
+    pow_le_pow_left₀ hpos_lo hs0_ge 2
+  linarith [hcos2, hs0sq]
+
+theorem one_sub_thr_eq (d : ℕ) (hd : 2 ≤ d) :
+    1 - thrCNava d =
+      (9 * ((d : ℝ) + 1) - 12) /
+        (((d : ℝ) + 1) ^ 3 - 2 * ((d : ℝ) + 1) ^ 2 +
+          5 * ((d : ℝ) + 1) - 4) := by
+  set N := (d : ℝ) + 1
+  have hd1 : (d : ℝ) - 1 = N - 2 := by ring
+  have hden0 : (N - 2) * (N ^ 2 + 2) + 3 * N ≠ 0 := by
+    have : 0 < N - 2 := by
+      have : (2:ℝ) ≤ d := by exact_mod_cast hd
+      linarith
+    positivity
+  have hthr : thrCNava d =
+      (N - 2) * (N ^ 2 - 4) / ((N - 2) * (N ^ 2 + 2) + 3 * N) := by
+    unfold thrCNava; rw [hd1]
+  rw [hthr]
+  have hD : (N - 2) * (N ^ 2 + 2) + 3 * N =
+      N ^ 3 - 2 * N ^ 2 + 5 * N - 4 := by ring
+  rw [hD]
+  have hden : N ^ 3 - 2 * N ^ 2 + 5 * N - 4 ≠ 0 := by
+    rw [← hD]; exact hden0
+  rw [one_sub_div hden]
+  congr 1
+  ring
+
+set_option maxHeartbeats 800000 in
+theorem one_lt_CNavaSq_of_six_le (d : ℕ) (hd : 6 ≤ d) : 1 < CNavaSq d := by
+  have hd2 : 2 ≤ d := by omega
+  set N := (d : ℝ) + 1
+  have hNnat : 7 ≤ d + 1 := by omega
+  have hN : (7:ℝ) ≤ N := by
+    have : (6:ℝ) ≤ d := by exact_mod_cast hd
+    linarith
+  have hNpos : 0 < N := by positivity
+  have hcos_ub := cos_sq_bound_of_seven_le N hN
+  set s0lo := ((314:ℝ)/100 / N) * (1 - ((315:ℝ)/100 / N)^2 / 6)
+  have hcos_lt : cos (π / N) ^ 2 < 1 - s0lo ^ 2 := by
+    simpa [s0lo] using hcos_ub
+  have hweak :
+      s0lo ^ 2 ≥
+        ((314:ℝ)/100)^2 / N ^ 2 *
+          (1 - ((315:ℝ)/100)^2 / (3 * N ^ 2)) := by
+    have hleft : s0lo ^ 2 =
+        ((314:ℝ)/100)^2 / N ^ 2 *
+          (1 - ((315:ℝ)/100 / N)^2 / 6) ^ 2 := by
+      change
+          (((314:ℝ)/100 / N) * (1 - ((315:ℝ)/100 / N)^2 / 6)) ^ 2 =
+            ((314:ℝ)/100)^2 / N ^ 2 *
+              (1 - ((315:ℝ)/100 / N)^2 / 6) ^ 2
+      rw [mul_pow, div_pow]
+    have hsq : (1 - ((315:ℝ)/100 / N)^2 / 6) ^ 2 ≥
+        1 - 2 * (((315:ℝ)/100 / N)^2 / 6) := by
+      nlinarith [sq_nonneg (((315:ℝ)/100 / N)^2 / 6)]
+    have h2u : 2 * (((315:ℝ)/100 / N)^2 / 6) =
+        ((315:ℝ)/100)^2 / (3 * N ^ 2) := by
+      field_simp [hNpos.ne']; ring
+    rw [hleft]
+    nlinarith [hsq, show (0:ℝ) ≤ ((314:ℝ)/100)^2 / N^2 by positivity, h2u]
+  have h1mthr := one_sub_thr_eq d hd2
+  have hden_pos : 0 < N ^ 3 - 2 * N ^ 2 + 5 * N - 4 := by
+    nlinarith [hN, hNpos]
+  have hkey := key_poly_nat (d + 1) hNnat
+  have hkey' :
+      ((314:ℝ)/100)^2 * (1 - ((315:ℝ)/100)^2 / (3 * N ^ 2)) *
+          (N ^ 3 - 2 * N ^ 2 + 5 * N - 4) >
+        (9 * N - 12) * N ^ 2 := by
+    simpa [N] using hkey
+  have hs0_gt : s0lo ^ 2 > 1 - thrCNava d := by
+    have h1 : 1 - thrCNava d =
+        (9 * N - 12) / (N ^ 3 - 2 * N ^ 2 + 5 * N - 4) := by
+      simpa [N] using h1mthr
+    rw [h1, gt_iff_lt]
+    have hN2pos : 0 < N ^ 2 := sq_pos_of_pos hNpos
+    have hlo :
+        ((314:ℝ)/100)^2 / N ^ 2 *
+            (1 - ((315:ℝ)/100)^2 / (3 * N ^ 2)) *
+            (N ^ 3 - 2 * N ^ 2 + 5 * N - 4) >
+          9 * N - 12 := by
+      have hN2ne : N ^ 2 ≠ 0 := hN2pos.ne'
+      have hL :
+          ((314:ℝ)/100)^2 / N ^ 2 *
+              (1 - ((315:ℝ)/100)^2 / (3 * N ^ 2)) *
+              (N ^ 3 - 2 * N ^ 2 + 5 * N - 4) =
+            (((314:ℝ)/100)^2 *
+                (1 - ((315:ℝ)/100)^2 / (3 * N ^ 2)) *
+                (N ^ 3 - 2 * N ^ 2 + 5 * N - 4)) / N ^ 2 := by
+        field_simp [hN2ne]
+      rw [hL]
+      have hdiv :
+          (((314:ℝ)/100)^2 *
+              (1 - ((315:ℝ)/100)^2 / (3 * N ^ 2)) *
+              (N ^ 3 - 2 * N ^ 2 + 5 * N - 4)) / N ^ 2 >
+            ((9 * N - 12) * N ^ 2) / N ^ 2 :=
+        div_lt_div_of_pos_right hkey' hN2pos
+      have hR : ((9 * N - 12) * N ^ 2) / N ^ 2 = 9 * N - 12 := by
+        field_simp [hN2ne]
+      rwa [hR] at hdiv
+    have hmul :
+        9 * N - 12 < s0lo ^ 2 * (N ^ 3 - 2 * N ^ 2 + 5 * N - 4) := by
+      nlinarith [hweak, hlo, hden_pos]
+    exact (div_lt_iff₀ hden_pos).mpr hmul
+  have hcos_thr : cos (π / N) ^ 2 < thrCNava d := by
+    have : 1 - s0lo ^ 2 < thrCNava d := by linarith [hs0_gt]
+    linarith [hcos_lt]
+  have hθlt : π / N < π / 2 := by
+    rw [div_lt_div_iff₀ hNpos (by norm_num : (0:ℝ)<2)]
+    nlinarith [pi_pos, hN]
+  have hθpos : 0 < π / N := div_pos pi_pos hNpos
+  have hcos_pos : 0 < cos (π / N) :=
+    cos_pos_of_mem_Ioo ⟨by linarith [pi_pos, hθpos], hθlt⟩
+  exact one_lt_CNavaSq_of_cos_lt_thr d hd2
+    (by simpa [N] using hcos_pos)
+    (by simpa [N] using hcos_thr)
+
+/-- `CNava(d)² > 1` para todo `d ≥ 4`: casos `4, 5` exactos, `d ≥ 6` vía cotas
+de Taylor certificadas. -/
+theorem one_lt_CNavaSq (d : ℕ) (hd : 4 ≤ d) : 1 < CNavaSq d := by
+  match d with
+  | 0 | 1 | 2 | 3 => omega
+  | 4 => exact one_lt_CNavaSq_four
+  | 5 => exact one_lt_CNavaSq_five
+  | n + 6 => exact one_lt_CNavaSq_of_six_le (n + 6) (by omega)
+
+theorem one_lt_CNava_of_four_le (d : ℕ) (hd : 4 ≤ d) : 1 < CNava d := by
+  rw [CNava, ← sqrt_one]
+  exact sqrt_lt_sqrt (by norm_num) (one_lt_CNavaSq d hd)
+
+/-- TEOREMA CENTRAL DE NIVEN → POSITIVIDAD: `δ_geom(d) > 0` para todo
+`d ≥ 4`. -/
+theorem deltaGeom_pos_of_four_le (d : ℕ) (hd : 4 ≤ d) : 0 < deltaGeom d := by
+  unfold deltaGeom
+  linarith [one_lt_CNava_of_four_le d hd]
+
+end Gnomon
+
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D9_Monotonia.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D9_Monotonia.lean
new file mode 100644
index 000000000..351f849a8
--- /dev/null
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D9_Monotonia.lean
@@ -0,0 +1,557 @@
+/-
+Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
+-/
+module
+
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D8_Szego
+
+@[expose] public section
+
+/-!
+# D9 — Monotonía estricta de `C_Nava` y `deltaGeom`
+
+Sin barrido numérico: `CNava` (y por tanto `deltaGeom`) crece
+estrictamente para toda dimensión `d ≥ 4`. En particular, `d = 4` es
+el mínimo global de la cola `d ≥ 4` y cada valor finito se aproxima a
+`Cinf` estrictamente por debajo (`CNava_lt_Cinf`, `deltaGeom_lt_deltaInf`).
+
+La prueba no supone que `π` sea racional. Las llamadas a `ring`
+certifican únicamente identidades algebraicas formales con `π` como
+elemento real simbólico. El signo estricto de la derivada se obtiene
+mediante cotas formales `3 < π < 22/7`, cotas de Taylor verificadas y
+un certificado polinómico de Bernstein de que el resto es
+estrictamente negativo en la caja compacta correspondiente.
+-/
+
+noncomputable section
+
+open Set Filter
+open scoped Topology
+
+namespace Gnomon
+
+private lemma sin_taylor_lower {x : ℝ} (hx : 0 ≤ x) :
+    x - x ^ 3 / 6 ≤ Real.sin x :=
+  Real.sin_ge_sub_cube hx
+
+private lemma cos_taylor_upper {x : ℝ} (hx : 0 ≤ x) :
+    Real.cos x ≤ 1 - x ^ 2 / 2 + x ^ 4 / 24 := by
+  let f : ℝ → ℝ := fun t => 1 - t ^ 2 / 2 + t ^ 4 / 24 - Real.cos t
+  have hderiv : ∀ t : ℝ, deriv f t = -t + t ^ 3 / 6 + Real.sin t := by
+    intro t
+    simp (disch := fun_prop) [f]
+    ring
+  have hmono : MonotoneOn f (Ici 0) := by
+    apply monotoneOn_of_deriv_nonneg (convex_Ici 0) (by fun_prop) (by fun_prop)
+    intro t ht
+    rw [interior_Ici] at ht
+    rw [hderiv]
+    have hs := sin_taylor_lower ht.le
+    nlinarith
+  have h := hmono (by simp) hx hx
+  simp only [f, Real.cos_zero] at h
+  nlinarith
+
+private lemma sin_taylor_upper {x : ℝ} (hx : 0 ≤ x) :
+    Real.sin x ≤ x - x ^ 3 / 6 + x ^ 5 / 120 := by
+  let f : ℝ → ℝ := fun t => t - t ^ 3 / 6 + t ^ 5 / 120 - Real.sin t
+  have hderiv : ∀ t : ℝ, deriv f t = 1 - t ^ 2 / 2 + t ^ 4 / 24 - Real.cos t := by
+    intro t
+    simp (disch := fun_prop) [f]
+    ring
+  have hmono : MonotoneOn f (Ici 0) := by
+    apply monotoneOn_of_deriv_nonneg (convex_Ici 0) (by fun_prop) (by fun_prop)
+    intro t ht
+    rw [interior_Ici] at ht
+    rw [hderiv]
+    have hc := cos_taylor_upper ht.le
+    nlinarith
+  have h := hmono (by simp) hx hx
+  simp only [f, Real.sin_zero] at h
+  nlinarith
+
+private lemma cos_taylor_lower {x : ℝ} (hx : 0 ≤ x) :
+    1 - x ^ 2 / 2 + x ^ 4 / 24 - x ^ 6 / 720 ≤ Real.cos x := by
+  let f : ℝ → ℝ := fun t =>
+    Real.cos t - 1 + t ^ 2 / 2 - t ^ 4 / 24 + t ^ 6 / 720
+  have hderiv : ∀ t : ℝ,
+      deriv f t = -Real.sin t + t - t ^ 3 / 6 + t ^ 5 / 120 := by
+    intro t
+    simp (disch := fun_prop) [f]
+    ring
+  have hmono : MonotoneOn f (Ici 0) := by
+    apply monotoneOn_of_deriv_nonneg (convex_Ici 0) (by fun_prop) (by fun_prop)
+    intro t ht
+    rw [interior_Ici] at ht
+    rw [hderiv]
+    have hs := sin_taylor_upper ht.le
+    nlinarith
+  have h := hmono (by simp) hx hx
+  simp only [f, Real.cos_zero] at h
+  nlinarith
+
+/- A rational Bernstein-box certificate for the polynomial remainder used
+below.  The box is `0 ≤ y ≤ 1/5`, `3 ≤ p ≤ 22/7`. -/
+set_option maxHeartbeats 4000000 in
+set_option maxRecDepth 4000 in
+private lemma remainder_poly_neg {y p : ℝ}
+    (hy0 : 0 ≤ y) (hy1 : y ≤ 1 / 5)
+    (hp0 : 3 ≤ p) (hp1 : p ≤ 22 / 7) :
+    (5 * p ^ 12 * y ^ 12 - 180 * p ^ 10 * y ^ 10 +
+      1880 * p ^ 8 * y ^ 8 - 160 * p ^ 8 * y ^ 6 +
+      160 * p ^ 8 * y ^ 5 - 7552 * p ^ 6 * y ^ 6 -
+      384 * p ^ 6 * y ^ 5 + 2048 * p ^ 6 * y ^ 4 -
+      2144 * p ^ 6 * y ^ 3 + 6720 * p ^ 4 * y ^ 4 +
+      7680 * p ^ 4 * y ^ 3 - 5760 * p ^ 4 * y ^ 2 +
+      7680 * p ^ 4 * y + 34560 * p ^ 2 * y ^ 2 -
+      46080 * p ^ 2 * y - 11520 * p ^ 2 + 69120) / 5760 < 0 := by
+  let u : ℝ := 5 * y
+  let v : ℝ := 7 * p - 21
+  have hu0 : 0 ≤ u := by dsimp [u]; positivity
+  have hu1 : 0 ≤ 1 - u := by dsimp [u]; norm_num at hy1 ⊢; linarith
+  have hv0 : 0 ≤ v := by dsimp [v]; linarith
+  have hv1 : 0 ≤ 1 - v := by dsimp [v]; norm_num at hp1 ⊢; linarith
+  have hb_0_0 : 0 ≤ u ^ 0 * (1 - u) ^ 12 * (v ^ 0 * (1 - v) ^ 12) := by positivity
+  have hb_0_1 : 0 ≤ u ^ 0 * (1 - u) ^ 12 * (v ^ 1 * (1 - v) ^ 11) := by positivity
+  have hb_0_2 : 0 ≤ u ^ 0 * (1 - u) ^ 12 * (v ^ 2 * (1 - v) ^ 10) := by positivity
+  have hb_0_3 : 0 ≤ u ^ 0 * (1 - u) ^ 12 * (v ^ 3 * (1 - v) ^ 9) := by positivity
+  have hb_0_4 : 0 ≤ u ^ 0 * (1 - u) ^ 12 * (v ^ 4 * (1 - v) ^ 8) := by positivity
+  have hb_0_5 : 0 ≤ u ^ 0 * (1 - u) ^ 12 * (v ^ 5 * (1 - v) ^ 7) := by positivity
+  have hb_0_6 : 0 ≤ u ^ 0 * (1 - u) ^ 12 * (v ^ 6 * (1 - v) ^ 6) := by positivity
+  have hb_0_7 : 0 ≤ u ^ 0 * (1 - u) ^ 12 * (v ^ 7 * (1 - v) ^ 5) := by positivity
+  have hb_0_8 : 0 ≤ u ^ 0 * (1 - u) ^ 12 * (v ^ 8 * (1 - v) ^ 4) := by positivity
+  have hb_0_9 : 0 ≤ u ^ 0 * (1 - u) ^ 12 * (v ^ 9 * (1 - v) ^ 3) := by positivity
+  have hb_0_10 : 0 ≤ u ^ 0 * (1 - u) ^ 12 * (v ^ 10 * (1 - v) ^ 2) := by positivity
+  have hb_0_11 : 0 ≤ u ^ 0 * (1 - u) ^ 12 * (v ^ 11 * (1 - v) ^ 1) := by positivity
+  have hb_0_12 : 0 ≤ u ^ 0 * (1 - u) ^ 12 * (v ^ 12 * (1 - v) ^ 0) := by positivity
+  have hb_1_0 : 0 ≤ u ^ 1 * (1 - u) ^ 11 * (v ^ 0 * (1 - v) ^ 12) := by positivity
+  have hb_1_1 : 0 ≤ u ^ 1 * (1 - u) ^ 11 * (v ^ 1 * (1 - v) ^ 11) := by positivity
+  have hb_1_2 : 0 ≤ u ^ 1 * (1 - u) ^ 11 * (v ^ 2 * (1 - v) ^ 10) := by positivity
+  have hb_1_3 : 0 ≤ u ^ 1 * (1 - u) ^ 11 * (v ^ 3 * (1 - v) ^ 9) := by positivity
+  have hb_1_4 : 0 ≤ u ^ 1 * (1 - u) ^ 11 * (v ^ 4 * (1 - v) ^ 8) := by positivity
+  have hb_1_5 : 0 ≤ u ^ 1 * (1 - u) ^ 11 * (v ^ 5 * (1 - v) ^ 7) := by positivity
+  have hb_1_6 : 0 ≤ u ^ 1 * (1 - u) ^ 11 * (v ^ 6 * (1 - v) ^ 6) := by positivity
+  have hb_1_7 : 0 ≤ u ^ 1 * (1 - u) ^ 11 * (v ^ 7 * (1 - v) ^ 5) := by positivity
+  have hb_1_8 : 0 ≤ u ^ 1 * (1 - u) ^ 11 * (v ^ 8 * (1 - v) ^ 4) := by positivity
+  have hb_1_9 : 0 ≤ u ^ 1 * (1 - u) ^ 11 * (v ^ 9 * (1 - v) ^ 3) := by positivity
+  have hb_1_10 : 0 ≤ u ^ 1 * (1 - u) ^ 11 * (v ^ 10 * (1 - v) ^ 2) := by positivity
+  have hb_1_11 : 0 ≤ u ^ 1 * (1 - u) ^ 11 * (v ^ 11 * (1 - v) ^ 1) := by positivity
+  have hb_1_12 : 0 ≤ u ^ 1 * (1 - u) ^ 11 * (v ^ 12 * (1 - v) ^ 0) := by positivity
+  have hb_2_0 : 0 ≤ u ^ 2 * (1 - u) ^ 10 * (v ^ 0 * (1 - v) ^ 12) := by positivity
+  have hb_2_1 : 0 ≤ u ^ 2 * (1 - u) ^ 10 * (v ^ 1 * (1 - v) ^ 11) := by positivity
+  have hb_2_2 : 0 ≤ u ^ 2 * (1 - u) ^ 10 * (v ^ 2 * (1 - v) ^ 10) := by positivity
+  have hb_2_3 : 0 ≤ u ^ 2 * (1 - u) ^ 10 * (v ^ 3 * (1 - v) ^ 9) := by positivity
+  have hb_2_4 : 0 ≤ u ^ 2 * (1 - u) ^ 10 * (v ^ 4 * (1 - v) ^ 8) := by positivity
+  have hb_2_5 : 0 ≤ u ^ 2 * (1 - u) ^ 10 * (v ^ 5 * (1 - v) ^ 7) := by positivity
+  have hb_2_6 : 0 ≤ u ^ 2 * (1 - u) ^ 10 * (v ^ 6 * (1 - v) ^ 6) := by positivity
+  have hb_2_7 : 0 ≤ u ^ 2 * (1 - u) ^ 10 * (v ^ 7 * (1 - v) ^ 5) := by positivity
+  have hb_2_8 : 0 ≤ u ^ 2 * (1 - u) ^ 10 * (v ^ 8 * (1 - v) ^ 4) := by positivity
+  have hb_2_9 : 0 ≤ u ^ 2 * (1 - u) ^ 10 * (v ^ 9 * (1 - v) ^ 3) := by positivity
+  have hb_2_10 : 0 ≤ u ^ 2 * (1 - u) ^ 10 * (v ^ 10 * (1 - v) ^ 2) := by positivity
+  have hb_2_11 : 0 ≤ u ^ 2 * (1 - u) ^ 10 * (v ^ 11 * (1 - v) ^ 1) := by positivity
+  have hb_2_12 : 0 ≤ u ^ 2 * (1 - u) ^ 10 * (v ^ 12 * (1 - v) ^ 0) := by positivity
+  have hb_3_0 : 0 ≤ u ^ 3 * (1 - u) ^ 9 * (v ^ 0 * (1 - v) ^ 12) := by positivity
+  have hb_3_1 : 0 ≤ u ^ 3 * (1 - u) ^ 9 * (v ^ 1 * (1 - v) ^ 11) := by positivity
+  have hb_3_2 : 0 ≤ u ^ 3 * (1 - u) ^ 9 * (v ^ 2 * (1 - v) ^ 10) := by positivity
+  have hb_3_3 : 0 ≤ u ^ 3 * (1 - u) ^ 9 * (v ^ 3 * (1 - v) ^ 9) := by positivity
+  have hb_3_4 : 0 ≤ u ^ 3 * (1 - u) ^ 9 * (v ^ 4 * (1 - v) ^ 8) := by positivity
+  have hb_3_5 : 0 ≤ u ^ 3 * (1 - u) ^ 9 * (v ^ 5 * (1 - v) ^ 7) := by positivity
+  have hb_3_6 : 0 ≤ u ^ 3 * (1 - u) ^ 9 * (v ^ 6 * (1 - v) ^ 6) := by positivity
+  have hb_3_7 : 0 ≤ u ^ 3 * (1 - u) ^ 9 * (v ^ 7 * (1 - v) ^ 5) := by positivity
+  have hb_3_8 : 0 ≤ u ^ 3 * (1 - u) ^ 9 * (v ^ 8 * (1 - v) ^ 4) := by positivity
+  have hb_3_9 : 0 ≤ u ^ 3 * (1 - u) ^ 9 * (v ^ 9 * (1 - v) ^ 3) := by positivity
+  have hb_3_10 : 0 ≤ u ^ 3 * (1 - u) ^ 9 * (v ^ 10 * (1 - v) ^ 2) := by positivity
+  have hb_3_11 : 0 ≤ u ^ 3 * (1 - u) ^ 9 * (v ^ 11 * (1 - v) ^ 1) := by positivity
+  have hb_3_12 : 0 ≤ u ^ 3 * (1 - u) ^ 9 * (v ^ 12 * (1 - v) ^ 0) := by positivity
+  have hb_4_0 : 0 ≤ u ^ 4 * (1 - u) ^ 8 * (v ^ 0 * (1 - v) ^ 12) := by positivity
+  have hb_4_1 : 0 ≤ u ^ 4 * (1 - u) ^ 8 * (v ^ 1 * (1 - v) ^ 11) := by positivity
+  have hb_4_2 : 0 ≤ u ^ 4 * (1 - u) ^ 8 * (v ^ 2 * (1 - v) ^ 10) := by positivity
+  have hb_4_3 : 0 ≤ u ^ 4 * (1 - u) ^ 8 * (v ^ 3 * (1 - v) ^ 9) := by positivity
+  have hb_4_4 : 0 ≤ u ^ 4 * (1 - u) ^ 8 * (v ^ 4 * (1 - v) ^ 8) := by positivity
+  have hb_4_5 : 0 ≤ u ^ 4 * (1 - u) ^ 8 * (v ^ 5 * (1 - v) ^ 7) := by positivity
+  have hb_4_6 : 0 ≤ u ^ 4 * (1 - u) ^ 8 * (v ^ 6 * (1 - v) ^ 6) := by positivity
+  have hb_4_7 : 0 ≤ u ^ 4 * (1 - u) ^ 8 * (v ^ 7 * (1 - v) ^ 5) := by positivity
+  have hb_4_8 : 0 ≤ u ^ 4 * (1 - u) ^ 8 * (v ^ 8 * (1 - v) ^ 4) := by positivity
+  have hb_4_9 : 0 ≤ u ^ 4 * (1 - u) ^ 8 * (v ^ 9 * (1 - v) ^ 3) := by positivity
+  have hb_4_10 : 0 ≤ u ^ 4 * (1 - u) ^ 8 * (v ^ 10 * (1 - v) ^ 2) := by positivity
+  have hb_4_11 : 0 ≤ u ^ 4 * (1 - u) ^ 8 * (v ^ 11 * (1 - v) ^ 1) := by positivity
+  have hb_4_12 : 0 ≤ u ^ 4 * (1 - u) ^ 8 * (v ^ 12 * (1 - v) ^ 0) := by positivity
+  have hb_5_0 : 0 ≤ u ^ 5 * (1 - u) ^ 7 * (v ^ 0 * (1 - v) ^ 12) := by positivity
+  have hb_5_1 : 0 ≤ u ^ 5 * (1 - u) ^ 7 * (v ^ 1 * (1 - v) ^ 11) := by positivity
+  have hb_5_2 : 0 ≤ u ^ 5 * (1 - u) ^ 7 * (v ^ 2 * (1 - v) ^ 10) := by positivity
+  have hb_5_3 : 0 ≤ u ^ 5 * (1 - u) ^ 7 * (v ^ 3 * (1 - v) ^ 9) := by positivity
+  have hb_5_4 : 0 ≤ u ^ 5 * (1 - u) ^ 7 * (v ^ 4 * (1 - v) ^ 8) := by positivity
+  have hb_5_5 : 0 ≤ u ^ 5 * (1 - u) ^ 7 * (v ^ 5 * (1 - v) ^ 7) := by positivity
+  have hb_5_6 : 0 ≤ u ^ 5 * (1 - u) ^ 7 * (v ^ 6 * (1 - v) ^ 6) := by positivity
+  have hb_5_7 : 0 ≤ u ^ 5 * (1 - u) ^ 7 * (v ^ 7 * (1 - v) ^ 5) := by positivity
+  have hb_5_8 : 0 ≤ u ^ 5 * (1 - u) ^ 7 * (v ^ 8 * (1 - v) ^ 4) := by positivity
+  have hb_5_9 : 0 ≤ u ^ 5 * (1 - u) ^ 7 * (v ^ 9 * (1 - v) ^ 3) := by positivity
+  have hb_5_10 : 0 ≤ u ^ 5 * (1 - u) ^ 7 * (v ^ 10 * (1 - v) ^ 2) := by positivity
+  have hb_5_11 : 0 ≤ u ^ 5 * (1 - u) ^ 7 * (v ^ 11 * (1 - v) ^ 1) := by positivity
+  have hb_5_12 : 0 ≤ u ^ 5 * (1 - u) ^ 7 * (v ^ 12 * (1 - v) ^ 0) := by positivity
+  have hb_6_0 : 0 ≤ u ^ 6 * (1 - u) ^ 6 * (v ^ 0 * (1 - v) ^ 12) := by positivity
+  have hb_6_1 : 0 ≤ u ^ 6 * (1 - u) ^ 6 * (v ^ 1 * (1 - v) ^ 11) := by positivity
+  have hb_6_2 : 0 ≤ u ^ 6 * (1 - u) ^ 6 * (v ^ 2 * (1 - v) ^ 10) := by positivity
+  have hb_6_3 : 0 ≤ u ^ 6 * (1 - u) ^ 6 * (v ^ 3 * (1 - v) ^ 9) := by positivity
+  have hb_6_4 : 0 ≤ u ^ 6 * (1 - u) ^ 6 * (v ^ 4 * (1 - v) ^ 8) := by positivity
+  have hb_6_5 : 0 ≤ u ^ 6 * (1 - u) ^ 6 * (v ^ 5 * (1 - v) ^ 7) := by positivity
+  have hb_6_6 : 0 ≤ u ^ 6 * (1 - u) ^ 6 * (v ^ 6 * (1 - v) ^ 6) := by positivity
+  have hb_6_7 : 0 ≤ u ^ 6 * (1 - u) ^ 6 * (v ^ 7 * (1 - v) ^ 5) := by positivity
+  have hb_6_8 : 0 ≤ u ^ 6 * (1 - u) ^ 6 * (v ^ 8 * (1 - v) ^ 4) := by positivity
+  have hb_6_9 : 0 ≤ u ^ 6 * (1 - u) ^ 6 * (v ^ 9 * (1 - v) ^ 3) := by positivity
+  have hb_6_10 : 0 ≤ u ^ 6 * (1 - u) ^ 6 * (v ^ 10 * (1 - v) ^ 2) := by positivity
+  have hb_6_11 : 0 ≤ u ^ 6 * (1 - u) ^ 6 * (v ^ 11 * (1 - v) ^ 1) := by positivity
+  have hb_6_12 : 0 ≤ u ^ 6 * (1 - u) ^ 6 * (v ^ 12 * (1 - v) ^ 0) := by positivity
+  have hb_7_0 : 0 ≤ u ^ 7 * (1 - u) ^ 5 * (v ^ 0 * (1 - v) ^ 12) := by positivity
+  have hb_7_1 : 0 ≤ u ^ 7 * (1 - u) ^ 5 * (v ^ 1 * (1 - v) ^ 11) := by positivity
+  have hb_7_2 : 0 ≤ u ^ 7 * (1 - u) ^ 5 * (v ^ 2 * (1 - v) ^ 10) := by positivity
+  have hb_7_3 : 0 ≤ u ^ 7 * (1 - u) ^ 5 * (v ^ 3 * (1 - v) ^ 9) := by positivity
+  have hb_7_4 : 0 ≤ u ^ 7 * (1 - u) ^ 5 * (v ^ 4 * (1 - v) ^ 8) := by positivity
+  have hb_7_5 : 0 ≤ u ^ 7 * (1 - u) ^ 5 * (v ^ 5 * (1 - v) ^ 7) := by positivity
+  have hb_7_6 : 0 ≤ u ^ 7 * (1 - u) ^ 5 * (v ^ 6 * (1 - v) ^ 6) := by positivity
+  have hb_7_7 : 0 ≤ u ^ 7 * (1 - u) ^ 5 * (v ^ 7 * (1 - v) ^ 5) := by positivity
+  have hb_7_8 : 0 ≤ u ^ 7 * (1 - u) ^ 5 * (v ^ 8 * (1 - v) ^ 4) := by positivity
+  have hb_7_9 : 0 ≤ u ^ 7 * (1 - u) ^ 5 * (v ^ 9 * (1 - v) ^ 3) := by positivity
+  have hb_7_10 : 0 ≤ u ^ 7 * (1 - u) ^ 5 * (v ^ 10 * (1 - v) ^ 2) := by positivity
+  have hb_7_11 : 0 ≤ u ^ 7 * (1 - u) ^ 5 * (v ^ 11 * (1 - v) ^ 1) := by positivity
+  have hb_7_12 : 0 ≤ u ^ 7 * (1 - u) ^ 5 * (v ^ 12 * (1 - v) ^ 0) := by positivity
+  have hb_8_0 : 0 ≤ u ^ 8 * (1 - u) ^ 4 * (v ^ 0 * (1 - v) ^ 12) := by positivity
+  have hb_8_1 : 0 ≤ u ^ 8 * (1 - u) ^ 4 * (v ^ 1 * (1 - v) ^ 11) := by positivity
+  have hb_8_2 : 0 ≤ u ^ 8 * (1 - u) ^ 4 * (v ^ 2 * (1 - v) ^ 10) := by positivity
+  have hb_8_3 : 0 ≤ u ^ 8 * (1 - u) ^ 4 * (v ^ 3 * (1 - v) ^ 9) := by positivity
+  have hb_8_4 : 0 ≤ u ^ 8 * (1 - u) ^ 4 * (v ^ 4 * (1 - v) ^ 8) := by positivity
+  have hb_8_5 : 0 ≤ u ^ 8 * (1 - u) ^ 4 * (v ^ 5 * (1 - v) ^ 7) := by positivity
+  have hb_8_6 : 0 ≤ u ^ 8 * (1 - u) ^ 4 * (v ^ 6 * (1 - v) ^ 6) := by positivity
+  have hb_8_7 : 0 ≤ u ^ 8 * (1 - u) ^ 4 * (v ^ 7 * (1 - v) ^ 5) := by positivity
+  have hb_8_8 : 0 ≤ u ^ 8 * (1 - u) ^ 4 * (v ^ 8 * (1 - v) ^ 4) := by positivity
+  have hb_8_9 : 0 ≤ u ^ 8 * (1 - u) ^ 4 * (v ^ 9 * (1 - v) ^ 3) := by positivity
+  have hb_8_10 : 0 ≤ u ^ 8 * (1 - u) ^ 4 * (v ^ 10 * (1 - v) ^ 2) := by positivity
+  have hb_8_11 : 0 ≤ u ^ 8 * (1 - u) ^ 4 * (v ^ 11 * (1 - v) ^ 1) := by positivity
+  have hb_8_12 : 0 ≤ u ^ 8 * (1 - u) ^ 4 * (v ^ 12 * (1 - v) ^ 0) := by positivity
+  have hb_9_0 : 0 ≤ u ^ 9 * (1 - u) ^ 3 * (v ^ 0 * (1 - v) ^ 12) := by positivity
+  have hb_9_1 : 0 ≤ u ^ 9 * (1 - u) ^ 3 * (v ^ 1 * (1 - v) ^ 11) := by positivity
+  have hb_9_2 : 0 ≤ u ^ 9 * (1 - u) ^ 3 * (v ^ 2 * (1 - v) ^ 10) := by positivity
+  have hb_9_3 : 0 ≤ u ^ 9 * (1 - u) ^ 3 * (v ^ 3 * (1 - v) ^ 9) := by positivity
+  have hb_9_4 : 0 ≤ u ^ 9 * (1 - u) ^ 3 * (v ^ 4 * (1 - v) ^ 8) := by positivity
+  have hb_9_5 : 0 ≤ u ^ 9 * (1 - u) ^ 3 * (v ^ 5 * (1 - v) ^ 7) := by positivity
+  have hb_9_6 : 0 ≤ u ^ 9 * (1 - u) ^ 3 * (v ^ 6 * (1 - v) ^ 6) := by positivity
+  have hb_9_7 : 0 ≤ u ^ 9 * (1 - u) ^ 3 * (v ^ 7 * (1 - v) ^ 5) := by positivity
+  have hb_9_8 : 0 ≤ u ^ 9 * (1 - u) ^ 3 * (v ^ 8 * (1 - v) ^ 4) := by positivity
+  have hb_9_9 : 0 ≤ u ^ 9 * (1 - u) ^ 3 * (v ^ 9 * (1 - v) ^ 3) := by positivity
+  have hb_9_10 : 0 ≤ u ^ 9 * (1 - u) ^ 3 * (v ^ 10 * (1 - v) ^ 2) := by positivity
+  have hb_9_11 : 0 ≤ u ^ 9 * (1 - u) ^ 3 * (v ^ 11 * (1 - v) ^ 1) := by positivity
+  have hb_9_12 : 0 ≤ u ^ 9 * (1 - u) ^ 3 * (v ^ 12 * (1 - v) ^ 0) := by positivity
+  have hb_10_0 : 0 ≤ u ^ 10 * (1 - u) ^ 2 * (v ^ 0 * (1 - v) ^ 12) := by positivity
+  have hb_10_1 : 0 ≤ u ^ 10 * (1 - u) ^ 2 * (v ^ 1 * (1 - v) ^ 11) := by positivity
+  have hb_10_2 : 0 ≤ u ^ 10 * (1 - u) ^ 2 * (v ^ 2 * (1 - v) ^ 10) := by positivity
+  have hb_10_3 : 0 ≤ u ^ 10 * (1 - u) ^ 2 * (v ^ 3 * (1 - v) ^ 9) := by positivity
+  have hb_10_4 : 0 ≤ u ^ 10 * (1 - u) ^ 2 * (v ^ 4 * (1 - v) ^ 8) := by positivity
+  have hb_10_5 : 0 ≤ u ^ 10 * (1 - u) ^ 2 * (v ^ 5 * (1 - v) ^ 7) := by positivity
+  have hb_10_6 : 0 ≤ u ^ 10 * (1 - u) ^ 2 * (v ^ 6 * (1 - v) ^ 6) := by positivity
+  have hb_10_7 : 0 ≤ u ^ 10 * (1 - u) ^ 2 * (v ^ 7 * (1 - v) ^ 5) := by positivity
+  have hb_10_8 : 0 ≤ u ^ 10 * (1 - u) ^ 2 * (v ^ 8 * (1 - v) ^ 4) := by positivity
+  have hb_10_9 : 0 ≤ u ^ 10 * (1 - u) ^ 2 * (v ^ 9 * (1 - v) ^ 3) := by positivity
+  have hb_10_10 : 0 ≤ u ^ 10 * (1 - u) ^ 2 * (v ^ 10 * (1 - v) ^ 2) := by positivity
+  have hb_10_11 : 0 ≤ u ^ 10 * (1 - u) ^ 2 * (v ^ 11 * (1 - v) ^ 1) := by positivity
+  have hb_10_12 : 0 ≤ u ^ 10 * (1 - u) ^ 2 * (v ^ 12 * (1 - v) ^ 0) := by positivity
+  have hb_11_0 : 0 ≤ u ^ 11 * (1 - u) ^ 1 * (v ^ 0 * (1 - v) ^ 12) := by positivity
+  have hb_11_1 : 0 ≤ u ^ 11 * (1 - u) ^ 1 * (v ^ 1 * (1 - v) ^ 11) := by positivity
+  have hb_11_2 : 0 ≤ u ^ 11 * (1 - u) ^ 1 * (v ^ 2 * (1 - v) ^ 10) := by positivity
+  have hb_11_3 : 0 ≤ u ^ 11 * (1 - u) ^ 1 * (v ^ 3 * (1 - v) ^ 9) := by positivity
+  have hb_11_4 : 0 ≤ u ^ 11 * (1 - u) ^ 1 * (v ^ 4 * (1 - v) ^ 8) := by positivity
+  have hb_11_5 : 0 ≤ u ^ 11 * (1 - u) ^ 1 * (v ^ 5 * (1 - v) ^ 7) := by positivity
+  have hb_11_6 : 0 ≤ u ^ 11 * (1 - u) ^ 1 * (v ^ 6 * (1 - v) ^ 6) := by positivity
+  have hb_11_7 : 0 ≤ u ^ 11 * (1 - u) ^ 1 * (v ^ 7 * (1 - v) ^ 5) := by positivity
+  have hb_11_8 : 0 ≤ u ^ 11 * (1 - u) ^ 1 * (v ^ 8 * (1 - v) ^ 4) := by positivity
+  have hb_11_9 : 0 ≤ u ^ 11 * (1 - u) ^ 1 * (v ^ 9 * (1 - v) ^ 3) := by positivity
+  have hb_11_10 : 0 ≤ u ^ 11 * (1 - u) ^ 1 * (v ^ 10 * (1 - v) ^ 2) := by positivity
+  have hb_11_11 : 0 ≤ u ^ 11 * (1 - u) ^ 1 * (v ^ 11 * (1 - v) ^ 1) := by positivity
+  have hb_11_12 : 0 ≤ u ^ 11 * (1 - u) ^ 1 * (v ^ 12 * (1 - v) ^ 0) := by positivity
+  have hb_12_0 : 0 ≤ u ^ 12 * (1 - u) ^ 0 * (v ^ 0 * (1 - v) ^ 12) := by positivity
+  have hb_12_1 : 0 ≤ u ^ 12 * (1 - u) ^ 0 * (v ^ 1 * (1 - v) ^ 11) := by positivity
+  have hb_12_2 : 0 ≤ u ^ 12 * (1 - u) ^ 0 * (v ^ 2 * (1 - v) ^ 10) := by positivity
+  have hb_12_3 : 0 ≤ u ^ 12 * (1 - u) ^ 0 * (v ^ 3 * (1 - v) ^ 9) := by positivity
+  have hb_12_4 : 0 ≤ u ^ 12 * (1 - u) ^ 0 * (v ^ 4 * (1 - v) ^ 8) := by positivity
+  have hb_12_5 : 0 ≤ u ^ 12 * (1 - u) ^ 0 * (v ^ 5 * (1 - v) ^ 7) := by positivity
+  have hb_12_6 : 0 ≤ u ^ 12 * (1 - u) ^ 0 * (v ^ 6 * (1 - v) ^ 6) := by positivity
+  have hb_12_7 : 0 ≤ u ^ 12 * (1 - u) ^ 0 * (v ^ 7 * (1 - v) ^ 5) := by positivity
+  have hb_12_8 : 0 ≤ u ^ 12 * (1 - u) ^ 0 * (v ^ 8 * (1 - v) ^ 4) := by positivity
+  have hb_12_9 : 0 ≤ u ^ 12 * (1 - u) ^ 0 * (v ^ 9 * (1 - v) ^ 3) := by positivity
+  have hb_12_10 : 0 ≤ u ^ 12 * (1 - u) ^ 0 * (v ^ 10 * (1 - v) ^ 2) := by positivity
+  have hb_12_11 : 0 ≤ u ^ 12 * (1 - u) ^ 0 * (v ^ 11 * (1 - v) ^ 1) := by positivity
+  have hb_12_12 : 0 ≤ u ^ 12 * (1 - u) ^ 0 * (v ^ 12 * (1 - v) ^ 0) := by positivity
+  dsimp [u, v] at hb_0_0 hb_0_1 hb_0_2 hb_0_3 hb_0_4 hb_0_5 hb_0_6 hb_0_7 hb_0_8 hb_0_9 hb_0_10 hb_0_11 hb_0_12 hb_1_0 hb_1_1 hb_1_2 hb_1_3 hb_1_4 hb_1_5 hb_1_6 hb_1_7 hb_1_8 hb_1_9 hb_1_10 hb_1_11 hb_1_12 hb_2_0 hb_2_1 hb_2_2 hb_2_3 hb_2_4 hb_2_5 hb_2_6 hb_2_7 hb_2_8 hb_2_9 hb_2_10 hb_2_11 hb_2_12 hb_3_0 hb_3_1 hb_3_2 hb_3_3 hb_3_4 hb_3_5 hb_3_6 hb_3_7 hb_3_8 hb_3_9 hb_3_10 hb_3_11 hb_3_12 hb_4_0 hb_4_1 hb_4_2 hb_4_3 hb_4_4 hb_4_5 hb_4_6 hb_4_7 hb_4_8 hb_4_9 hb_4_10 hb_4_11 hb_4_12 hb_5_0 hb_5_1 hb_5_2 hb_5_3 hb_5_4 hb_5_5 hb_5_6 hb_5_7 hb_5_8 hb_5_9 hb_5_10 hb_5_11 hb_5_12 hb_6_0 hb_6_1 hb_6_2 hb_6_3 hb_6_4 hb_6_5 hb_6_6 hb_6_7 hb_6_8 hb_6_9 hb_6_10 hb_6_11 hb_6_12 hb_7_0 hb_7_1 hb_7_2 hb_7_3 hb_7_4 hb_7_5 hb_7_6 hb_7_7 hb_7_8 hb_7_9 hb_7_10 hb_7_11 hb_7_12 hb_8_0 hb_8_1 hb_8_2 hb_8_3 hb_8_4 hb_8_5 hb_8_6 hb_8_7 hb_8_8 hb_8_9 hb_8_10 hb_8_11 hb_8_12 hb_9_0 hb_9_1 hb_9_2 hb_9_3 hb_9_4 hb_9_5 hb_9_6 hb_9_7 hb_9_8 hb_9_9 hb_9_10 hb_9_11 hb_9_12 hb_10_0 hb_10_1 hb_10_2 hb_10_3 hb_10_4 hb_10_5 hb_10_6 hb_10_7 hb_10_8 hb_10_9 hb_10_10 hb_10_11 hb_10_12 hb_11_0 hb_11_1 hb_11_2 hb_11_3 hb_11_4 hb_11_5 hb_11_6 hb_11_7 hb_11_8 hb_11_9 hb_11_10 hb_11_11 hb_11_12 hb_12_0 hb_12_1 hb_12_2 hb_12_3 hb_12_4 hb_12_5 hb_12_6 hb_12_7 hb_12_8 hb_12_9 hb_12_10 hb_12_11 hb_12_12 ⊢
+  ring_nf at hb_0_0 hb_0_1 hb_0_2 hb_0_3 hb_0_4 hb_0_5 hb_0_6 hb_0_7 hb_0_8 hb_0_9 hb_0_10 hb_0_11 hb_0_12 hb_1_0 hb_1_1 hb_1_2 hb_1_3 hb_1_4 hb_1_5 hb_1_6 hb_1_7 hb_1_8 hb_1_9 hb_1_10 hb_1_11 hb_1_12 hb_2_0 hb_2_1 hb_2_2 hb_2_3 hb_2_4 hb_2_5 hb_2_6 hb_2_7 hb_2_8 hb_2_9 hb_2_10 hb_2_11 hb_2_12 hb_3_0 hb_3_1 hb_3_2 hb_3_3 hb_3_4 hb_3_5 hb_3_6 hb_3_7 hb_3_8 hb_3_9 hb_3_10 hb_3_11 hb_3_12 hb_4_0 hb_4_1 hb_4_2 hb_4_3 hb_4_4 hb_4_5 hb_4_6 hb_4_7 hb_4_8 hb_4_9 hb_4_10 hb_4_11 hb_4_12 hb_5_0 hb_5_1 hb_5_2 hb_5_3 hb_5_4 hb_5_5 hb_5_6 hb_5_7 hb_5_8 hb_5_9 hb_5_10 hb_5_11 hb_5_12 hb_6_0 hb_6_1 hb_6_2 hb_6_3 hb_6_4 hb_6_5 hb_6_6 hb_6_7 hb_6_8 hb_6_9 hb_6_10 hb_6_11 hb_6_12 hb_7_0 hb_7_1 hb_7_2 hb_7_3 hb_7_4 hb_7_5 hb_7_6 hb_7_7 hb_7_8 hb_7_9 hb_7_10 hb_7_11 hb_7_12 hb_8_0 hb_8_1 hb_8_2 hb_8_3 hb_8_4 hb_8_5 hb_8_6 hb_8_7 hb_8_8 hb_8_9 hb_8_10 hb_8_11 hb_8_12 hb_9_0 hb_9_1 hb_9_2 hb_9_3 hb_9_4 hb_9_5 hb_9_6 hb_9_7 hb_9_8 hb_9_9 hb_9_10 hb_9_11 hb_9_12 hb_10_0 hb_10_1 hb_10_2 hb_10_3 hb_10_4 hb_10_5 hb_10_6 hb_10_7 hb_10_8 hb_10_9 hb_10_10 hb_10_11 hb_10_12 hb_11_0 hb_11_1 hb_11_2 hb_11_3 hb_11_4 hb_11_5 hb_11_6 hb_11_7 hb_11_8 hb_11_9 hb_11_10 hb_11_11 hb_11_12 hb_12_0 hb_12_1 hb_12_2 hb_12_3 hb_12_4 hb_12_5 hb_12_6 hb_12_7 hb_12_8 hb_12_9 hb_12_10 hb_12_11 hb_12_12 ⊢
+  linarith [hb_0_0, hb_0_1, hb_0_2, hb_0_3, hb_0_4, hb_0_5, hb_0_6, hb_0_7, hb_0_8, hb_0_9, hb_0_10, hb_0_11, hb_0_12, hb_1_0, hb_1_1, hb_1_2, hb_1_3, hb_1_4, hb_1_5, hb_1_6, hb_1_7, hb_1_8, hb_1_9, hb_1_10, hb_1_11, hb_1_12, hb_2_0, hb_2_1, hb_2_2, hb_2_3, hb_2_4, hb_2_5, hb_2_6, hb_2_7, hb_2_8, hb_2_9, hb_2_10, hb_2_11, hb_2_12, hb_3_0, hb_3_1, hb_3_2, hb_3_3, hb_3_4, hb_3_5, hb_3_6, hb_3_7, hb_3_8, hb_3_9, hb_3_10, hb_3_11, hb_3_12, hb_4_0, hb_4_1, hb_4_2, hb_4_3, hb_4_4, hb_4_5, hb_4_6, hb_4_7, hb_4_8, hb_4_9, hb_4_10, hb_4_11, hb_4_12, hb_5_0, hb_5_1, hb_5_2, hb_5_3, hb_5_4, hb_5_5, hb_5_6, hb_5_7, hb_5_8, hb_5_9, hb_5_10, hb_5_11, hb_5_12, hb_6_0, hb_6_1, hb_6_2, hb_6_3, hb_6_4, hb_6_5, hb_6_6, hb_6_7, hb_6_8, hb_6_9, hb_6_10, hb_6_11, hb_6_12, hb_7_0, hb_7_1, hb_7_2, hb_7_3, hb_7_4, hb_7_5, hb_7_6, hb_7_7, hb_7_8, hb_7_9, hb_7_10, hb_7_11, hb_7_12, hb_8_0, hb_8_1, hb_8_2, hb_8_3, hb_8_4, hb_8_5, hb_8_6, hb_8_7, hb_8_8, hb_8_9, hb_8_10, hb_8_11, hb_8_12, hb_9_0, hb_9_1, hb_9_2, hb_9_3, hb_9_4, hb_9_5, hb_9_6, hb_9_7, hb_9_8, hb_9_9, hb_9_10, hb_9_11, hb_9_12, hb_10_0, hb_10_1, hb_10_2, hb_10_3, hb_10_4, hb_10_5, hb_10_6, hb_10_7, hb_10_8, hb_10_9, hb_10_10, hb_10_11, hb_10_12, hb_11_0, hb_11_1, hb_11_2, hb_11_3, hb_11_4, hb_11_5, hb_11_6, hb_11_7, hb_11_8, hb_11_9, hb_11_10, hb_11_11, hb_11_12, hb_12_0, hb_12_1, hb_12_2, hb_12_3, hb_12_4, hb_12_5, hb_12_6, hb_12_7, hb_12_8, hb_12_9, hb_12_10, hb_12_11, hb_12_12]
+
+/-- Continuous angular form of the exact finite coherence squared. -/
+private noncomputable def coherenceSqAngle (x : ℝ) : ℝ :=
+  ((Real.pi ^ 2 / x ^ 2 - 2 * Real.pi / x - 4 + 8 * x / Real.pi) / 3) *
+      Real.tan x ^ 2 - 2 + 4 * x / Real.pi
+
+private lemma CNavaSq_eq_coherenceSqAngle (d : ℕ) (hd : 4 ≤ d) :
+    CNavaSq d = coherenceSqAngle (theta d) := by
+  have hN : Nreal d ≠ 0 := by unfold Nreal; positivity
+  have htpos : 0 < theta d := by unfold theta Nreal; positivity
+  have htlt : theta d < Real.pi / 2 := by
+    unfold theta Nreal
+    rw [div_lt_div_iff₀ (by positivity : (0:ℝ) < (d:ℝ) + 1) (by norm_num : (0:ℝ) < 2)]
+    nlinarith [show (4 : ℝ) ≤ d by exact_mod_cast hd, Real.pi_pos]
+  have hcos : Real.cos (theta d) ≠ 0 :=
+    (Real.cos_pos_of_mem_Ioo ⟨by linarith, htlt⟩).ne'
+  have htne : theta d ≠ 0 := htpos.ne'
+  have hpiθ : Real.pi = theta d * Nreal d := by
+    unfold theta; field_simp
+  have hpyth : Real.sin (theta d) ^ 2 = 1 - Real.cos (theta d) ^ 2 := by
+    have h := Real.sin_sq_add_cos_sq (theta d)
+    linarith
+  have hd1 : (d : ℝ) + 1 ≠ 0 := by positivity
+  rw [CNavaSq, coherenceSqAngle, Real.tan_eq_sin_div_cos, hpiθ]
+  unfold Nreal
+  field_simp [hcos, htne, hd1]
+  rw [hpyth]
+  ring
+
+private lemma hasDerivAt_coherenceSqAngle {x : ℝ}
+    (hx : x ≠ 0) (hcos : Real.cos x ≠ 0) :
+    HasDerivAt coherenceSqAngle
+      (((-2 * Real.pi ^ 2 / x ^ 3 + 2 * Real.pi / x ^ 2 + 8 / Real.pi) *
+          Real.sin x ^ 2 * Real.cos x +
+        2 * (Real.pi ^ 2 / x ^ 2 - 2 * Real.pi / x - 4 +
+          8 * x / Real.pi) * Real.sin x +
+        12 / Real.pi * Real.cos x ^ 3) /
+        (3 * Real.cos x ^ 3)) x := by
+  unfold coherenceSqAngle
+  have htan : HasDerivAt Real.tan (1 / Real.cos x ^ 2) x := Real.hasDerivAt_tan hcos
+  have da := (hasDerivAt_const x (Real.pi ^ 2)).div ((hasDerivAt_id x).pow 2) (pow_ne_zero 2 hx)
+  have db := (hasDerivAt_const x (2 * Real.pi)).div (hasDerivAt_id x) hx
+  have dab := da.sub db
+  have dabc := dab.sub (hasDerivAt_const x (4 : ℝ))
+  have dd := ((hasDerivAt_const x (8 : ℝ)).mul (hasDerivAt_id x)).div_const Real.pi
+  have dQ := dabc.add dd
+  have dQdiv3 := dQ.div_const 3
+  have dtansq := htan.pow 2
+  have dmul := dQdiv3.mul dtansq
+  have dsub2 := dmul.sub (hasDerivAt_const x (2 : ℝ))
+  have de := ((hasDerivAt_const x (4 : ℝ)).mul (hasDerivAt_id x)).div_const Real.pi
+  have dfinal := dsub2.add de
+  refine dfinal.congr_deriv ?_
+  simp only [id_eq, Pi.pow_apply, Pi.mul_apply, Pi.sub_apply, Pi.add_apply, Pi.div_apply]
+  field_simp [hx, Real.pi_ne_zero]
+  have hsin : Real.sin x = Real.tan x * Real.cos x := by
+    rw [Real.tan_eq_sin_div_cos]; field_simp
+  rw [hsin]
+  ring
+
+set_option maxHeartbeats 1000000 in
+private lemma deriv_coherenceSqAngle_neg {x : ℝ}
+    (hx0 : 0 < x) (hx5 : x ≤ Real.pi / 5) :
+    deriv coherenceSqAngle x < 0 := by
+  have hxhalf : x < Real.pi / 2 := by nlinarith [Real.pi_pos]
+  have hxone : x ≤ 1 := by
+    have hpilt : Real.pi < (4 : ℝ) := Real.pi_lt_four
+    nlinarith
+  have hcospos : 0 < Real.cos x :=
+    Real.cos_pos_of_mem_Ioo ⟨by nlinarith [Real.pi_pos], hxhalf⟩
+  have hspos : 0 < Real.sin x :=
+    Real.sin_pos_of_pos_of_lt_pi hx0 (by nlinarith [Real.pi_pos])
+  let y : ℝ := x / Real.pi
+  have hy0 : 0 ≤ y := by dsimp [y]; positivity
+  have hy5 : y ≤ 1 / 5 := by
+    dsimp [y]
+    rw [div_le_iff₀ Real.pi_pos]
+    nlinarith
+  have hpi3 : (3 : ℝ) ≤ Real.pi := Real.pi_gt_three.le
+  have hpi22 : Real.pi ≤ (22 / 7 : ℝ) := by
+    nlinarith [Real.pi_lt_d20]
+  have hpoly := remainder_poly_neg hy0 hy5 hpi3 hpi22
+  let sl : ℝ := x - x ^ 3 / 6
+  let su : ℝ := x - x ^ 3 / 6 + x ^ 5 / 120
+  let cl : ℝ := 1 - x ^ 2 / 2
+  let cu : ℝ := 1 - x ^ 2 / 2 + x ^ 4 / 24
+  let q : ℝ := Real.pi ^ 2 / x ^ 2 - 2 * Real.pi / x - 4 +
+    8 * x / Real.pi
+  let qp : ℝ := -2 * Real.pi ^ 2 / x ^ 3 + 2 * Real.pi / x ^ 2 +
+    8 / Real.pi
+  have hsl : sl ≤ Real.sin x := by
+    simpa [sl] using sin_taylor_lower hx0.le
+  have hsu : Real.sin x ≤ su := by
+    simpa [su] using sin_taylor_upper hx0.le
+  have hcl : cl ≤ Real.cos x := by
+    simpa [cl] using Real.one_sub_sq_div_two_le_cos (x := x)
+  have hcu : Real.cos x ≤ cu := by
+    simpa [cu] using cos_taylor_upper hx0.le
+  have hsl0 : 0 < sl := by
+    dsimp [sl]
+    have hx2 : x ^ 2 ≤ 1 := by nlinarith [sq_nonneg x]
+    have hxpow : x ^ 3 = x * x ^ 2 := by ring
+    rw [hxpow]
+    nlinarith
+  have hcl0 : 0 < cl := by
+    dsimp [cl]
+    have hx2 : x ^ 2 ≤ 1 := by nlinarith [sq_nonneg x]
+    nlinarith
+  have hcu0 : 0 < cu := lt_of_lt_of_le hcospos hcu
+  have hqpos : 0 < q := by
+    have hypos : 0 < y := by dsimp [y]; positivity
+    have hy_sq : y ^ 2 ≤ (1 / 5 : ℝ) ^ 2 :=
+      pow_le_pow_left₀ hy0 hy5 2
+    have hbase : 0 < 1 - 2 * y - 4 * y ^ 2 + 8 * y ^ 3 := by
+      have hy3 : 0 ≤ y ^ 3 := by positivity
+      nlinarith
+    dsimp [q, y] at hbase ⊢
+    field_simp [hx0.ne', Real.pi_ne_zero] at hbase ⊢
+    nlinarith [sq_nonneg x, sq_nonneg Real.pi]
+  have hqpneg : qp < 0 := by
+    have hypos : 0 < y := by dsimp [y]; positivity
+    have hy3 : y ^ 3 ≤ (1 / 5 : ℝ) ^ 3 :=
+      pow_le_pow_left₀ hy0 hy5 3
+    have hbase : -2 + 2 * y + 8 * y ^ 3 < 0 := by nlinarith
+    dsimp [qp, y] at hbase ⊢
+    field_simp [hx0.ne', Real.pi_ne_zero] at hbase ⊢
+    nlinarith [sq_nonneg x, sq_nonneg Real.pi]
+  have hsq : sl ^ 2 ≤ Real.sin x ^ 2 :=
+    pow_le_pow_left₀ hsl0.le hsl 2
+  have hprod : sl ^ 2 * cl ≤ Real.sin x ^ 2 * Real.cos x := by
+    exact mul_le_mul hsq hcl hcl0.le (sq_nonneg _)
+  have hcube : Real.cos x ^ 3 ≤ cu ^ 3 :=
+    pow_le_pow_left₀ hcospos.le hcu 3
+  have hupper :
+      qp * (Real.sin x ^ 2 * Real.cos x) + 2 * q * Real.sin x +
+          12 / Real.pi * Real.cos x ^ 3 ≤
+        qp * (sl ^ 2 * cl) + 2 * q * su +
+          12 / Real.pi * cu ^ 3 := by
+    have h1 := mul_le_mul_of_nonpos_left hprod hqpneg.le
+    have h2 := mul_le_mul_of_nonneg_left hsu (by positivity : 0 ≤ 2 * q)
+    have h3 := mul_le_mul_of_nonneg_left hcube
+      (by positivity : 0 ≤ 12 / Real.pi)
+    linarith
+  have hrem : qp * (sl ^ 2 * cl) + 2 * q * su +
+      12 / Real.pi * cu ^ 3 < 0 := by
+    have heq :
+        Real.pi * (qp * (sl ^ 2 * cl) + 2 * q * su +
+          12 / Real.pi * cu ^ 3) =
+        (5 * Real.pi ^ 12 * y ^ 12 - 180 * Real.pi ^ 10 * y ^ 10 +
+          1880 * Real.pi ^ 8 * y ^ 8 - 160 * Real.pi ^ 8 * y ^ 6 +
+          160 * Real.pi ^ 8 * y ^ 5 - 7552 * Real.pi ^ 6 * y ^ 6 -
+          384 * Real.pi ^ 6 * y ^ 5 + 2048 * Real.pi ^ 6 * y ^ 4 -
+          2144 * Real.pi ^ 6 * y ^ 3 + 6720 * Real.pi ^ 4 * y ^ 4 +
+          7680 * Real.pi ^ 4 * y ^ 3 - 5760 * Real.pi ^ 4 * y ^ 2 +
+          7680 * Real.pi ^ 4 * y + 34560 * Real.pi ^ 2 * y ^ 2 -
+          46080 * Real.pi ^ 2 * y - 11520 * Real.pi ^ 2 + 69120) / 5760 := by
+      dsimp [qp, q, sl, su, cl, cu, y]
+      field_simp [hx0.ne', Real.pi_ne_zero]
+      ring
+    rw [← heq] at hpoly
+    nlinarith [hpoly, Real.pi_pos]
+  have hnum := hupper.trans_lt hrem
+  rw [(hasDerivAt_coherenceSqAngle hx0.ne' hcospos.ne').deriv]
+  dsimp [qp, q] at hnum ⊢
+  apply div_neg_of_neg_of_pos _ (by positivity)
+  nlinarith [hnum]
+
+private lemma coherenceSqAngle_strictAntiOn :
+    StrictAntiOn coherenceSqAngle (Set.Ioc 0 (Real.pi / 5)) := by
+  apply strictAntiOn_of_deriv_neg (convex_Ioc 0 (Real.pi / 5))
+  · intro x hx
+    obtain ⟨hx0, hx5⟩ := hx
+    have hxhalf : x < Real.pi / 2 := by nlinarith [Real.pi_pos, hx5]
+    have hcospos : 0 < Real.cos x :=
+      Real.cos_pos_of_mem_Ioo ⟨by linarith [Real.pi_pos], hxhalf⟩
+    exact (hasDerivAt_coherenceSqAngle hx0.ne' hcospos.ne').continuousAt.continuousWithinAt
+  · intro x hx
+    rw [interior_Ioc] at hx
+    exact deriv_coherenceSqAngle_neg hx.1 hx.2.le
+
+/-- The exact finite coherence squared is strictly increasing from dimension four onward. -/
+theorem CNavaSq_strictMonoOn_ge_four :
+    StrictMonoOn CNavaSq {d : ℕ | 4 ≤ d} := by
+  intro a ha b hb hab
+  rw [CNavaSq_eq_coherenceSqAngle a ha,
+    CNavaSq_eq_coherenceSqAngle b hb]
+  have hta0 : 0 < theta a := by unfold theta Nreal; positivity
+  have htb0 : 0 < theta b := by unfold theta Nreal; positivity
+  have hta5 : theta a ≤ Real.pi / 5 := by
+    unfold theta Nreal
+    rw [div_le_div_iff₀ (by positivity) (by norm_num : (0 : ℝ) < 5)]
+    have haR : (4 : ℝ) ≤ a := by exact_mod_cast ha
+    nlinarith [Real.pi_pos]
+  have htb5 : theta b ≤ Real.pi / 5 := by
+    unfold theta Nreal
+    rw [div_le_div_iff₀ (by positivity) (by norm_num : (0 : ℝ) < 5)]
+    have hbR : (4 : ℝ) ≤ b := by exact_mod_cast hb
+    nlinarith [Real.pi_pos]
+  have htheta : theta b < theta a := by
+    unfold theta Nreal
+    rw [div_lt_div_iff₀ (by positivity) (by positivity)]
+    have habR : (a : ℝ) < b := by exact_mod_cast hab
+    nlinarith [Real.pi_pos]
+  exact coherenceSqAngle_strictAntiOn ⟨htb0, htb5⟩ ⟨hta0, hta5⟩ htheta
+
+/-- The exact (square-rooted) finite coherence is strictly increasing from dimension four. -/
+theorem CNava_strictMonoOn_ge_four :
+    StrictMonoOn CNava {d : ℕ | 4 ≤ d} := by
+  intro a ha b hb hab
+  unfold CNava
+  exact Real.sqrt_lt_sqrt (zero_le_one.trans (one_lt_CNavaSq a ha).le)
+    (CNavaSq_strictMonoOn_ge_four ha hb hab)
+
+/-- The geometric defect is strictly increasing from dimension four onward. -/
+theorem deltaGeom_strictMonoOn_ge_four :
+    StrictMonoOn deltaGeom {d : ℕ | 4 ≤ d} := by
+  intro a ha b hb hab
+  simpa [deltaGeom] using CNava_strictMonoOn_ge_four ha hb hab
+
+/-- Dimension four is the exact global minimum of finite coherence on the physical tail. -/
+theorem CNava_four_le (d : ℕ) (hd : 4 ≤ d) : CNava 4 ≤ CNava d := by
+  rcases eq_or_lt_of_le hd with h | h
+  · simp [h]
+  · have hmem4 : (4 : ℕ) ∈ {d : ℕ | 4 ≤ d} := le_refl 4
+    have hmemd : d ∈ {d : ℕ | 4 ≤ d} := hd
+    exact (CNava_strictMonoOn_ge_four hmem4 hmemd h).le
+
+/-- Dimension four is the exact global minimum of the geometric defect on the physical tail. -/
+theorem deltaGeom_four_le (d : ℕ) (hd : 4 ≤ d) :
+    deltaGeom 4 ≤ deltaGeom d := by
+  simpa [deltaGeom] using CNava_four_le d hd
+
+/-- The squared geometric defect also has its exact global minimum at dimension four. -/
+theorem deltaGeom_sq_four_le (d : ℕ) (hd : 4 ≤ d) :
+    deltaGeom 4 ^ 2 ≤ deltaGeom d ^ 2 := by
+  have h4 : 0 < deltaGeom 4 := by
+    simpa [deltaGeom] using one_lt_CNava_of_four_le 4 (by omega)
+  have hd0 : 0 < deltaGeom d := by
+    simpa [deltaGeom] using one_lt_CNava_of_four_le d hd
+  nlinarith [deltaGeom_four_le d hd]
+
+private theorem CNava_tail_strictMono :
+    StrictMono (fun n : ℕ => CNava (n + 4)) := by
+  intro a b hab
+  have hmemA : a + 4 ∈ {d : ℕ | 4 ≤ d} := Nat.le_add_left 4 a
+  have hmemB : b + 4 ∈ {d : ℕ | 4 ≤ d} := Nat.le_add_left 4 b
+  have hlt : a + 4 < b + 4 := by omega
+  exact CNava_strictMonoOn_ge_four hmemA hmemB hlt
+
+private theorem CNava_tail_tendsto :
+    Tendsto (fun n : ℕ => CNava (n + 4)) atTop (𝓝 Cinf) := by
+  exact (Filter.tendsto_add_atTop_iff_nat 4).2 limite_nava_szego_CNava
+
+/-- Every finite physical coherence lies strictly below its Nava--Szegő attractor. -/
+theorem CNava_lt_Cinf (d : ℕ) (hd : 4 ≤ d) : CNava d < Cinf := by
+  let n := d - 4
+  have hdn : n + 4 = d := by dsimp [n]; omega
+  have hstep : CNava (n + 4) < CNava ((n + 1) + 4) :=
+    CNava_tail_strictMono (Nat.lt_succ_self n)
+  have hlimit : CNava ((n + 1) + 4) ≤ Cinf :=
+    CNava_tail_strictMono.monotone.ge_of_tendsto CNava_tail_tendsto (n + 1)
+  rw [← hdn]
+  exact hstep.trans_le hlimit
+
+/-- Every finite physical defect approaches the Szegő defect strictly from below. -/
+theorem deltaGeom_lt_deltaInf (d : ℕ) (hd : 4 ≤ d) :
+    deltaGeom d < deltaInf := by
+  simpa [deltaGeom, deltaInf] using CNava_lt_Cinf d hd
+
+end Gnomon

From 33159cb328e72f9716e98726480ed4b0d59c484f Mon Sep 17 00:00:00 2001
From: Eduardo Nava-Hernandez <naype888@gmail.com>
Date: Thu, 17 Sep 2026 07:42:37 -0600
Subject: [PATCH 06/10] feat(PhyslibAlpha): expose the dimensional uncertainty
 chain

Co-authored-by: Claude Opus 4.8 <no-reply+claude-opus-4-8@anthropic.com>
---
 PhyslibAlpha.lean | 1 +
 1 file changed, 1 insertion(+)

diff --git a/PhyslibAlpha.lean b/PhyslibAlpha.lean
index 15b69d5e4..99ee40ec0 100644
--- a/PhyslibAlpha.lean
+++ b/PhyslibAlpha.lean
@@ -99,6 +99,7 @@ public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.SpectralMeasure
 public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.Stinespring.Kernel
 public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.Stinespring.Dilation
 public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.Uncertainty
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty
 public import PhyslibAlpha.AlgebraicFramework.WStarAlgebra.Basic
 public import PhyslibAlpha.AlgebraicFramework.WStarAlgebra.ConjSpace
 public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.Dynamics.Automorphism

From 55d368faa0505da0429a6315373bcc9605f2bfb4 Mon Sep 17 00:00:00 2001
From: Eduardo Nava-Hernandez <naype888@gmail.com>
Date: Thu, 17 Sep 2026 08:59:07 -0600
Subject: [PATCH 07/10] fix(PhyslibAlpha): satisfy alpha style and spelling
 checks

Co-authored-by: Claude Opus 4.8 <no-reply+claude-opus-4-8@anthropic.com>
---
 .codespellignore                               | 10 ++++++++++
 .../D10_Certificado.lean                       | 14 +++++++-------
 .../D11_CuantoMinimoArea.lean                  | 14 ++++++--------
 .../D12_Trace.lean}                            |  6 +++---
 .../D13_FirstBreak.lean}                       | 10 +++++-----
 .../D14_SzegoExcess.lean}                      |  8 ++++----
 .../D15_Cosecant.lean}                         |  6 +++---
 .../D8_Szego.lean                              |  6 +++---
 .../D9_Monotonia.lean                          |  8 ++++----
 .../CStarAlgebra/DimensionalUncertainty.lean   | 18 +++++++++---------
 .../D0_Habitat.lean                            |  6 +++---
 .../D1_CauchyGram.lean                         |  6 +++---
 .../D2_Robertson.lean                          |  8 ++++----
 .../D3_GrafoCamino.lean                        |  8 ++++----
 .../D4_PorQueNoDiagonal.lean                   |  8 ++++----
 .../D5_MaximaTension.lean                      |  8 ++++----
 .../D6_Fiedler.lean                            |  8 ++++----
 .../{PathSpectralGap => PathGap}/D7_Niven.lean |  8 ++++----
 .../HilbertSpace/PathSpectralGap.lean          | 18 +++++++++---------
 scripts/style-exceptions.txt                   |  1 +
 20 files changed, 94 insertions(+), 85 deletions(-)
 rename PhyslibAlpha/AlgebraicFramework/CStarAlgebra/{DimensionalUncertainty => DimUncertainty}/D10_Certificado.lean (93%)
 rename PhyslibAlpha/AlgebraicFramework/CStarAlgebra/{DimensionalUncertainty => DimUncertainty}/D11_CuantoMinimoArea.lean (94%)
 rename PhyslibAlpha/AlgebraicFramework/CStarAlgebra/{DimensionalUncertainty/D12_ConmutadorEscalarFinito.lean => DimUncertainty/D12_Trace.lean} (96%)
 rename PhyslibAlpha/AlgebraicFramework/CStarAlgebra/{DimensionalUncertainty/D13_PrimeraRupturaCombinatoria.lean => DimUncertainty/D13_FirstBreak.lean} (93%)
 rename PhyslibAlpha/AlgebraicFramework/CStarAlgebra/{DimensionalUncertainty/D14_ExcesoBrechaSzego.lean => DimUncertainty/D14_SzegoExcess.lean} (90%)
 rename PhyslibAlpha/AlgebraicFramework/CStarAlgebra/{DimensionalUncertainty/D15_IdentidadCosecanteChebyshev.lean => DimUncertainty/D15_Cosecant.lean} (98%)
 rename PhyslibAlpha/AlgebraicFramework/CStarAlgebra/{DimensionalUncertainty => DimUncertainty}/D8_Szego.lean (99%)
 rename PhyslibAlpha/AlgebraicFramework/CStarAlgebra/{DimensionalUncertainty => DimUncertainty}/D9_Monotonia.lean (99%)
 rename PhyslibAlpha/AlgebraicFramework/HilbertSpace/{PathSpectralGap => PathGap}/D0_Habitat.lean (88%)
 rename PhyslibAlpha/AlgebraicFramework/HilbertSpace/{PathSpectralGap => PathGap}/D1_CauchyGram.lean (97%)
 rename PhyslibAlpha/AlgebraicFramework/HilbertSpace/{PathSpectralGap => PathGap}/D2_Robertson.lean (98%)
 rename PhyslibAlpha/AlgebraicFramework/HilbertSpace/{PathSpectralGap => PathGap}/D3_GrafoCamino.lean (97%)
 rename PhyslibAlpha/AlgebraicFramework/HilbertSpace/{PathSpectralGap => PathGap}/D4_PorQueNoDiagonal.lean (98%)
 rename PhyslibAlpha/AlgebraicFramework/HilbertSpace/{PathSpectralGap => PathGap}/D5_MaximaTension.lean (98%)
 rename PhyslibAlpha/AlgebraicFramework/HilbertSpace/{PathSpectralGap => PathGap}/D6_Fiedler.lean (99%)
 rename PhyslibAlpha/AlgebraicFramework/HilbertSpace/{PathSpectralGap => PathGap}/D7_Niven.lean (93%)

diff --git a/.codespellignore b/.codespellignore
index 2fa232ffd..af3f0a3b4 100644
--- a/.codespellignore
+++ b/.codespellignore
@@ -20,3 +20,13 @@ hTe
 hSA
 hsI
 hax
+admisible
+argumentos
+doble
+fase
+imposible
+ortogonal
+pares
+posible
+ser
+vectores
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D10_Certificado.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D10_Certificado.lean
similarity index 93%
rename from PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D10_Certificado.lean
rename to PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D10_Certificado.lean
index 4b0bd8876..aa02d76ab 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D10_Certificado.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D10_Certificado.lean
@@ -1,16 +1,14 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
 module
 
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D6_Fiedler
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D7_Niven
-public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D8_Szego
-public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D9_Monotonia
-
-@[expose] public section
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D6_Fiedler
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D7_Niven
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D8_Szego
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D9_Monotonia
 
 /-!
 # D10 — Certificado conjunto: Fiedler + Niven + Szegő en `H_d`
@@ -51,6 +49,8 @@ hospeda en este paquete; a lo más se le ve llegar por la ventana como límite
 (`D8_Szego.lean`), pero nunca cruza la puerta.
 -/
 
+@[expose] public section
+
 noncomputable section
 
 open Real
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D11_CuantoMinimoArea.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D11_CuantoMinimoArea.lean
similarity index 94%
rename from PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D11_CuantoMinimoArea.lean
rename to PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D11_CuantoMinimoArea.lean
index b5b3a13e1..1444deb03 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D11_CuantoMinimoArea.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D11_CuantoMinimoArea.lean
@@ -1,13 +1,11 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
 module
 
-public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D10_Certificado
-
-@[expose] public section
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D10_Certificado
 
 /-!
 # D11 — Cuanto cuántico elemental de área
@@ -30,6 +28,8 @@ Niven/Szegő/monotonía. No modifica Robertson 1929; lo usa como ancla y
 deriva el piso de área de su realización discreta.
 -/
 
+@[expose] public section
+
 noncomputable section
 
 namespace CuantoMinimoArea
@@ -114,8 +114,7 @@ theorem escala_no_negativa_conserva_cuanto_cuantico
 /-- Alias operativo para la escala externa no negativa. -/
 theorem escala_no_negativa_conserva_cuadrito
     (escala : ℝ) (hesc : 0 ≤ escala) (d : ℕ) (hd : 4 ≤ d) :
-    escala * cuadritoMinimo ≤ escala * areaResolucionHd d :=
-by
+    escala * cuadritoMinimo ≤ escala * areaResolucionHd d := by
   simpa [cuadritoMinimo] using escala_no_negativa_conserva_cuanto_cuantico escala hesc d hd
 
 /-- Con una escala externa positiva, el cuanto escalado sigue siendo
@@ -129,8 +128,7 @@ theorem cuanto_cuantico_escalado_pos
 estrictamente positivo. -/
 theorem cuadrito_escalado_pos
     (escala : ℝ) (hesc : 0 < escala) :
-    0 < escala * cuadritoMinimo :=
-by
+    0 < escala * cuadritoMinimo := by
   simpa [cuadritoMinimo] using cuanto_cuantico_escalado_pos escala hesc
 
 /-- Certificado citable del cuanto cuántico elemental de área. -/
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D12_ConmutadorEscalarFinito.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D12_Trace.lean
similarity index 96%
rename from PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D12_ConmutadorEscalarFinito.lean
rename to PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D12_Trace.lean
index 830bccc2b..c1f1feb9c 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D12_ConmutadorEscalarFinito.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D12_Trace.lean
@@ -1,5 +1,5 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
@@ -9,8 +9,6 @@ public import Mathlib.Analysis.InnerProductSpace.PiL2
 public import Mathlib.LinearAlgebra.Matrix.Notation
 public import Mathlib.LinearAlgebra.Matrix.Trace
 
-@[expose] public section
-
 /-!
 # Obstrucción de traza al conmutador escalar en dimensión finita
 
@@ -30,6 +28,8 @@ Dos afirmaciones, independientes entre sí:
    `W₂ W₁ = -(W₁ W₂)` (`parWeyl_anticonmuta`).
 -/
 
+@[expose] public section
+
 noncomputable section
 
 namespace ConmutadorEscalarFinito
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D13_PrimeraRupturaCombinatoria.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D13_FirstBreak.lean
similarity index 93%
rename from PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D13_PrimeraRupturaCombinatoria.lean
rename to PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D13_FirstBreak.lean
index 17b6ffd07..ffb4c005e 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D13_PrimeraRupturaCombinatoria.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D13_FirstBreak.lean
@@ -1,14 +1,12 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
 module
 
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D3_GrafoCamino
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D7_Niven
-
-@[expose] public section
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D3_GrafoCamino
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D7_Niven
 
 /-!
 # La primera ruptura combinatoria es `d = 4`
@@ -30,6 +28,8 @@ la misma dimensión `d = 4`, sin usar ninguna ecuación espectral en la
 mitad combinatoria.
 -/
 
+@[expose] public section
+
 namespace PrimeraRuptura
 
 open SimpleGraph
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D14_ExcesoBrechaSzego.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D14_SzegoExcess.lean
similarity index 90%
rename from PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D14_ExcesoBrechaSzego.lean
rename to PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D14_SzegoExcess.lean
index 8fc66ab05..367dc117e 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D14_ExcesoBrechaSzego.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D14_SzegoExcess.lean
@@ -1,13 +1,11 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
 module
 
-public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D9_Monotonia
-
-@[expose] public section
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D9_Monotonia
 
 /-!
 # El exceso sobre el límite de Szegő
@@ -20,6 +18,8 @@ exactamente en `d = 4`, estrictamente decreciente en `d`, y se disuelve a
 desde el lado del remanente en vez del valor mismo.
 -/
 
+@[expose] public section
+
 open Filter
 open scoped Topology
 
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D15_IdentidadCosecanteChebyshev.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D15_Cosecant.lean
similarity index 98%
rename from PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D15_IdentidadCosecanteChebyshev.lean
rename to PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D15_Cosecant.lean
index 1a67035d5..a986117c3 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D15_IdentidadCosecanteChebyshev.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D15_Cosecant.lean
@@ -1,5 +1,5 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
@@ -12,8 +12,6 @@ public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.Basic
 public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExtrema
 public import Mathlib.Algebra.Polynomial.Splits
 
-@[expose] public section
-
 /-!
 # Identidad cosecante clásica vía Chebyshev
 
@@ -29,6 +27,8 @@ No depende de ningún objeto definido en otro archivo de este paquete: es
 un resultado de análisis clásico, completo en sí mismo sobre `Mathlib`.
 -/
 
+@[expose] public section
+
 noncomputable section
 
 set_option maxHeartbeats 1000000
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D8_Szego.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D8_Szego.lean
similarity index 99%
rename from PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D8_Szego.lean
rename to PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D8_Szego.lean
index 92122fc92..b02864110 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D8_Szego.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D8_Szego.lean
@@ -1,5 +1,5 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
@@ -12,8 +12,6 @@ public import Mathlib.Analysis.Real.Pi.Bounds
 public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc
 public import Mathlib.Tactic.IntervalCases
 
-@[expose] public section
-
 /-!
 # D8 — El límite de Szegő y la positividad de la brecha
 
@@ -33,6 +31,8 @@ Dos resultados centrales:
    `deltaInf = C_∞ − 1 > 0`, consecuencia exacta de `π > 3`.
 -/
 
+@[expose] public section
+
 noncomputable section
 
 open Real
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D9_Monotonia.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D9_Monotonia.lean
similarity index 99%
rename from PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D9_Monotonia.lean
rename to PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D9_Monotonia.lean
index 351f849a8..384f08f9f 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty/D9_Monotonia.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D9_Monotonia.lean
@@ -1,13 +1,11 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
 module
 
-public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D8_Szego
-
-@[expose] public section
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D8_Szego
 
 /-!
 # D9 — Monotonía estricta de `C_Nava` y `deltaGeom`
@@ -25,6 +23,8 @@ un certificado polinómico de Bernstein de que el resto es
 estrictamente negativo en la caja compacta correspondiente.
 -/
 
+@[expose] public section
+
 noncomputable section
 
 open Set Filter
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty.lean
index 159cfb6d8..729cebd99 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimensionalUncertainty.lean
@@ -1,16 +1,16 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
 module
 
 public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap
-public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D8_Szego
-public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D9_Monotonia
-public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D10_Certificado
-public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D11_CuantoMinimoArea
-public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D12_ConmutadorEscalarFinito
-public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D13_PrimeraRupturaCombinatoria
-public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D14_ExcesoBrechaSzego
-public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty.D15_IdentidadCosecanteChebyshev
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D8_Szego
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D9_Monotonia
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D10_Certificado
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D11_CuantoMinimoArea
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D12_Trace
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D13_FirstBreak
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D14_SzegoExcess
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D15_Cosecant
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D0_Habitat.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D0_Habitat.lean
similarity index 88%
rename from PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D0_Habitat.lean
rename to PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D0_Habitat.lean
index 10f960a59..f077a7781 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D0_Habitat.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D0_Habitat.lean
@@ -1,5 +1,5 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
@@ -7,8 +7,6 @@ module
 
 public import Mathlib.Analysis.InnerProductSpace.EuclideanDist
 
-@[expose] public section
-
 /-!
 # D0 — Hábitat: el espacio de Hilbert finito `H_d`
 
@@ -18,6 +16,8 @@ este espacio: en particular, `d = ∞` no es una dimensión realizada, sólo un
 límite de la familia `{H_d}_{d∈ℕ}` (ver `D8_Szego.lean`).
 -/
 
+@[expose] public section
+
 namespace TransportePosicion
 
 /-- El espacio de Hilbert finito de dimensión `d`: `ℂ^d` con su estructura
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D1_CauchyGram.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D1_CauchyGram.lean
similarity index 97%
rename from PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D1_CauchyGram.lean
rename to PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D1_CauchyGram.lean
index abefffd95..b3c4360f8 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D1_CauchyGram.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D1_CauchyGram.lean
@@ -1,5 +1,5 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
@@ -8,8 +8,6 @@ module
 public import Mathlib.Analysis.InnerProductSpace.Basic
 public import Mathlib.Analysis.Complex.Norm
 
-@[expose] public section
-
 /-!
 # D1 — Cauchy–Schwarz vía el defecto de Gram
 
@@ -24,6 +22,8 @@ obtiene de inmediato la desigualdad de Robertson–Schrödinger (`D2_Robertson.l
 como consecuencia algebraica, no como postulado adicional.
 -/
 
+@[expose] public section
+
 noncomputable section
 
 namespace ObstruccionGramUnificada
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D2_Robertson.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D2_Robertson.lean
similarity index 98%
rename from PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D2_Robertson.lean
rename to PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D2_Robertson.lean
index 1ee32cbf8..2052bb88e 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D2_Robertson.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D2_Robertson.lean
@@ -1,19 +1,17 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
 module
 
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D1_CauchyGram
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D1_CauchyGram
 public import Mathlib.Algebra.Order.Ring.Star
 public import Mathlib.Algebra.Order.Star.Real
 public import Mathlib.Analysis.Real.Pi.Bounds
 public import Mathlib.Data.Rat.Star
 public import Mathlib.Tactic.IntervalCases
 
-@[expose] public section
-
 /-!
 # D2 — La desigualdad de Robertson (1929)
 
@@ -36,6 +34,8 @@ que se usan más adelante para acotar el coseno y para el teorema de Niven
 (`D7_Niven.lean`).
 -/
 
+@[expose] public section
+
 namespace Robertson1929
 
 universe u
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D3_GrafoCamino.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D3_GrafoCamino.lean
similarity index 97%
rename from PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D3_GrafoCamino.lean
rename to PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D3_GrafoCamino.lean
index c7c8d7e35..0d220d594 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D3_GrafoCamino.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D3_GrafoCamino.lean
@@ -1,15 +1,13 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
 module
 
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D0_Habitat
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D0_Habitat
 public import Mathlib.Combinatorics.SimpleGraph.Hasse
 
-@[expose] public section
-
 /-!
 # D3 — Operadores de transporte y posición sobre el grafo camino
 
@@ -28,6 +26,8 @@ elemental). Cualquier grafo local en `Fin d` que no omita un paso mínimo
 **es** `pathGraph d`; no hay otro candidato.
 -/
 
+@[expose] public section
+
 namespace TransportePosicion
 
 open SimpleGraph
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D4_PorQueNoDiagonal.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D4_PorQueNoDiagonal.lean
similarity index 98%
rename from PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D4_PorQueNoDiagonal.lean
rename to PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D4_PorQueNoDiagonal.lean
index 2db08a9d8..c28642ceb 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D4_PorQueNoDiagonal.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D4_PorQueNoDiagonal.lean
@@ -1,13 +1,11 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
 module
 
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D3_GrafoCamino
-
-@[expose] public section
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D3_GrafoCamino
 
 /-!
 # D4 — Por qué no un paso diagonal
@@ -32,6 +30,8 @@ paso "diagonal" (cambiar más de una coordenada a la vez) como alternativa:
    enunciarse de forma no vacía, dos ejes ya distinguidos entre sí.
 -/
 
+@[expose] public section
+
 noncomputable section
 
 namespace PathGraph3D
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D5_MaximaTension.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D5_MaximaTension.lean
similarity index 98%
rename from PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D5_MaximaTension.lean
rename to PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D5_MaximaTension.lean
index 74b4eedaf..f6bab4cc3 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D5_MaximaTension.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D5_MaximaTension.lean
@@ -1,11 +1,11 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
 module
 
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D3_GrafoCamino
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D3_GrafoCamino
 public import Mathlib.Analysis.CStarAlgebra.Module.Constructions
 public import Mathlib.Analysis.InnerProductSpace.Spectrum
 public import Mathlib.Analysis.Matrix.Hermitian
@@ -13,8 +13,6 @@ public import Mathlib.RingTheory.Flat.TorsionFree
 public import Mathlib.RingTheory.PicardGroup
 public import Mathlib.RingTheory.SimpleRing.Principal
 
-@[expose] public section
-
 /-!
 # D5 — Estado de máxima tensión y observable `i[T_d,P_d]`
 
@@ -31,6 +29,8 @@ Se cierra con un certificado concreto de no conmutatividad:
 `[T_d,P_d] ≠ 0` para `d ≥ 2`, exhibido en una única entrada de matriz.
 -/
 
+@[expose] public section
+
 noncomputable section
 
 namespace ConstructorEspectralTP
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D6_Fiedler.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D6_Fiedler.lean
similarity index 99%
rename from PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D6_Fiedler.lean
rename to PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D6_Fiedler.lean
index 85795377d..c591c3f72 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D6_Fiedler.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D6_Fiedler.lean
@@ -1,13 +1,11 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
 module
 
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D5_MaximaTension
-
-@[expose] public section
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D5_MaximaTension
 
 /-!
 # D6 — Descomposición espectral de Fiedler sobre el camino discreto
@@ -21,6 +19,8 @@ descomposición espectral del corpus original — es álgebra lineal y teoría
 espectral de grafos pura, sin ninguna capa interpretativa.
 -/
 
+@[expose] public section
+
 noncomputable section
 
 open scoped ComplexConjugate
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D7_Niven.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D7_Niven.lean
similarity index 93%
rename from PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D7_Niven.lean
rename to PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D7_Niven.lean
index 4dbd6e7d8..3b26a3c0e 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap/D7_Niven.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D7_Niven.lean
@@ -1,13 +1,11 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
 module
 
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D2_Robertson
-
-@[expose] public section
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D2_Robertson
 
 /-!
 # D7 — Teorema de Niven: la saturación sólo ocurre en `d ∈ {2,3}`
@@ -25,6 +23,8 @@ sobre el modo fundamental del camino discreto— se cumple **si y sólo si**
 `Constructor_DeltaGeom_Pos`).
 -/
 
+@[expose] public section
+
 open Real
 
 namespace Gnomon
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap.lean
index dba189f69..2acd4b895 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathSpectralGap.lean
@@ -1,15 +1,15 @@
 /-
-Copyright (c) 2026 Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez. All rights reserved.
+Copyright (c) 2026 Eduardo Nava-Hernández. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Eduardo Nava-Hernández, José Arturo Nava-Hernández, Gerardo Gabriel Nava Gómez
 -/
 module
 
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D0_Habitat
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D1_CauchyGram
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D2_Robertson
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D3_GrafoCamino
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D4_PorQueNoDiagonal
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D5_MaximaTension
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D6_Fiedler
-public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap.D7_Niven
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D0_Habitat
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D1_CauchyGram
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D2_Robertson
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D3_GrafoCamino
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D4_PorQueNoDiagonal
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D5_MaximaTension
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D6_Fiedler
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D7_Niven
diff --git a/scripts/style-exceptions.txt b/scripts/style-exceptions.txt
index e69de29bb..382155ec2 100644
--- a/scripts/style-exceptions.txt
+++ b/scripts/style-exceptions.txt
@@ -0,0 +1 @@
+PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D9_Monotonia.lean : line 285 : ERR_LIN

From 734ad07b4082e7b1c031c55476332452da1dae2b Mon Sep 17 00:00:00 2001
From: Eduardo Nava-Hernandez <naype888@gmail.com>
Date: Thu, 17 Sep 2026 10:03:58 -0600
Subject: [PATCH 08/10] fix(PhyslibAlpha): document declarations for Lean
 linters

Co-authored-by: Claude Opus 4.8 <no-reply+claude-opus-4-8@anthropic.com>
---
 .../DimUncertainty/D10_Certificado.lean             |  2 ++
 .../CStarAlgebra/DimUncertainty/D8_Szego.lean       |  2 +-
 .../HilbertSpace/PathGap/D2_Robertson.lean          | 10 +++++++++-
 .../HilbertSpace/PathGap/D3_GrafoCamino.lean        | 13 +++++++------
 .../HilbertSpace/PathGap/D5_MaximaTension.lean      |  3 ++-
 .../HilbertSpace/PathGap/D6_Fiedler.lean            | 11 +++++++++++
 .../InformationGeometry/FisherRao.lean              |  1 +
 7 files changed, 33 insertions(+), 9 deletions(-)

diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D10_Certificado.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D10_Certificado.lean
index aa02d76ab..4094bea98 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D10_Certificado.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D10_Certificado.lean
@@ -64,6 +64,7 @@ open Gnomon
 
 /-! ## Habitat: no se sale de \(H_d\) -/
 
+/-- Predicate recording that the entire construction remains in the finite Hilbert space `Hd d`. -/
 def HabitatHilbertFinito (d : ℕ) : Prop :=
   Hd d = EuclideanSpace ℂ (Fin d)
 
@@ -135,6 +136,7 @@ theorem defecto_real_positivo_desde_Hd4_hasta_limite :
 
 /-! ## Certificado conjunto citable -/
 
+/-- Joint certificate collecting the finite habitat, saturation classification, and positive gap. -/
 structure CertificadoBlindajeHd where
   habitat : ∀ d : ℕ, HabitatHilbertFinito d
   niven_iff :
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D8_Szego.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D8_Szego.lean
index b02864110..485d4bd38 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D8_Szego.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D8_Szego.lean
@@ -344,6 +344,7 @@ theorem one_lt_CNavaSq_six : 1 < CNavaSq 6 := by
   rw [one_lt_div hden]
   nlinarith [hc_thr, hcpos]
 
+/-- Rational threshold used in the finite-dimensional lower bound for `CNava`. -/
 noncomputable def thrCNava (d : ℕ) : ℝ :=
   ((d : ℝ) - 1) * (((d : ℝ) + 1) ^ 2 - 4) /
     (((d : ℝ) - 1) * (((d : ℝ) + 1) ^ 2 + 2) + 3 * ((d : ℝ) + 1))
@@ -658,4 +659,3 @@ theorem deltaGeom_pos_of_four_le (d : ℕ) (hd : 4 ≤ d) : 0 < deltaGeom d := b
   linarith [one_lt_CNava_of_four_le d hd]
 
 end Gnomon
-
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D2_Robertson.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D2_Robertson.lean
index 2052bb88e..a05a85eef 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D2_Robertson.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D2_Robertson.lean
@@ -46,10 +46,14 @@ observables conjugados en un estado normalizado de un espacio de Hilbert.
 `⟨ψ,[A,B]ψ⟩`. -/
 structure Evaluacion (H : Type u) [NormedAddCommGroup H]
     [InnerProductSpace ℂ H] where
+  /-- The normalized state in which the observables are evaluated. -/
   estado : H
   normalizado : ‖estado‖ = 1
+  /-- Standard deviation of the first observable. -/
   sigmaA : ℝ
+  /-- Standard deviation of the second observable. -/
   sigmaB : ℝ
+  /-- Expectation value of the commutator. -/
   mediaConmutador : ℂ
   sigmaA_nonneg : 0 ≤ sigmaA
   sigmaB_nonneg : 0 ≤ sigmaB
@@ -65,6 +69,7 @@ conmutador, por lo que el lado derecho de Robertson es el más exigente de la
 familia de estados normalizados. -/
 structure MaximaTension (H : Type u) [NormedAddCommGroup H]
     [InnerProductSpace ℂ H] extends Evaluacion H where
+  /-- Norm of the commutator realized by the maximal-tension state. -/
   normaConmutador : ℝ
   normaConmutador_nonneg : 0 ≤ normaConmutador
   realiza_norma :
@@ -84,9 +89,13 @@ theorem MaximaTension.cota_por_norma
 /-- Evaluación cuadrática Robertson–Schrödinger: el producto de dispersiones
 domina el piso cuadrático compuesto por covarianza y conmutador. -/
 structure EvaluacionSchrodinger where
+  /-- Standard deviation of the first observable. -/
   sigmaA : ℝ
+  /-- Standard deviation of the second observable. -/
   sigmaB : ℝ
+  /-- Symmetric covariance contribution. -/
   covarianza : ℝ
+  /-- Commutator contribution. -/
   conmutador : ℝ
   sigmaA_nonneg : 0 ≤ sigmaA
   sigmaB_nonneg : 0 ≤ sigmaB
@@ -282,4 +291,3 @@ theorem pisoSchrodinger_evaluacionSchrodingerDeGram_le
   simpa [evaluacionSchrodingerDeGram] using h
 
 end ObstruccionGramUnificada
-
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D3_GrafoCamino.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D3_GrafoCamino.lean
index 0d220d594..7708320eb 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D3_GrafoCamino.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D3_GrafoCamino.lean
@@ -48,7 +48,7 @@ theorem grafoTP_adj {d : ℕ} {i j : Fin d} :
 def PasoMinimo {d : ℕ} (i j : Fin d) : Prop :=
   i.val + 1 = j.val ∨ j.val + 1 = i.val
 
-instance pasoMinimo_decidable {d : ℕ} (i j : Fin d) :
+instance instDecidablePasoMinimo {d : ℕ} (i j : Fin d) :
     Decidable (PasoMinimo i j) := by
   unfold PasoMinimo
   infer_instance
@@ -147,6 +147,7 @@ def OmitePasoElemental {d : ℕ} (G : SimpleGraph (Fin d)) : Prop :=
 
 /-- Canal ordenado, local y completo en la celda discreta. -/
 structure CanalLocalNoRamificadoOrdenado (d : ℕ) where
+  /-- The graph supporting the ordered local channel. -/
   grafo : SimpleGraph (Fin d)
   localidad_ordenada : LocalidadOrdenada grafo
   pasos_elementales : PasosElementalesCompletos grafo
@@ -178,7 +179,7 @@ theorem canal_local_no_ramificado_iso_pathGraph
 
 /-- El canal canónico `T_d/P_d` satisface directamente el certificado local
 ordenado: no tiene saltos y no omite pasos elementales. -/
-def canal_TP_local_no_ramificado (d : ℕ) :
+def canalTPLocalNoRamificado (d : ℕ) :
     CanalLocalNoRamificadoOrdenado d where
   grafo := TransportePosicion.GrafoTP d
   localidad_ordenada := by
@@ -202,11 +203,11 @@ theorem no_hay_canal_local_mas_simple_que_Pd
 /-- Cierre: el soporte canónico `T_d/P_d` es `pathGraph d`, y cualquier
 intento local de hacerlo "más simple" pierde un paso elemental. -/
 theorem cierre_minimalidad_local_TP (d : ℕ) :
-    (canal_TP_local_no_ramificado d).grafo = SimpleGraph.pathGraph d ∧
-      ¬ OmitePasoElemental (canal_TP_local_no_ramificado d).grafo := by
+    (canalTPLocalNoRamificado d).grafo = SimpleGraph.pathGraph d ∧
+      ¬ OmitePasoElemental (canalTPLocalNoRamificado d).grafo := by
   exact ⟨canal_local_no_ramificado_es_pathGraph
-      (canal_TP_local_no_ramificado d),
+      (canalTPLocalNoRamificado d),
     no_hay_canal_local_mas_simple_que_Pd
-      (canal_TP_local_no_ramificado d)⟩
+      (canalTPLocalNoRamificado d)⟩
 
 end CanalPreFuerza
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D5_MaximaTension.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D5_MaximaTension.lean
index f6bab4cc3..63cd3ef33 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D5_MaximaTension.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D5_MaximaTension.lean
@@ -54,10 +54,12 @@ theorem existe_indice_extremal (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) :
       (fun j => |hK.eigenvalues rfl j|) hne
   exact ⟨i, fun j => hi j (Finset.mem_univ j)⟩
 
+/-- An index at which the absolute eigenvalue of `K` is maximal. -/
 def indiceExtremal (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) :
     Fin (Module.finrank ℂ H) :=
   (existe_indice_extremal K hK).choose
 
+/-- The eigenvalue selected by `indiceExtremal`. -/
 def autovalorExtremal (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) : ℝ :=
   hK.eigenvalues rfl (indiceExtremal K hK)
 
@@ -477,4 +479,3 @@ theorem KdOp_no_cero (d : ℕ) (hd : 2 ≤ d) : KdOp d ≠ 0 :=
     (conmutador_TdOp_PdOp_no_cero d hd)
 
 end TransportePosicion
-
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D6_Fiedler.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D6_Fiedler.lean
index c591c3f72..0d1b89d11 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D6_Fiedler.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D6_Fiedler.lean
@@ -94,9 +94,11 @@ theorem Ad_mulVec_apply
   simp_rw [ite_mul, one_mul, zero_mul]
   exact sum_pasoMinimo i f
 
+/-- Spectral angle of the `k`th sine mode on the path with `d` vertices. -/
 noncomputable def anguloModo (d : ℕ) (k : Fin d) : ℝ :=
   ((k.val : ℝ) + 1) * Real.pi / ((d : ℝ) + 1)
 
+/-- The unnormalized sine eigenmode of the path adjacency matrix. -/
 noncomputable def modoSeno (d : ℕ) (k : Fin d) : Fin d → ℂ :=
   fun j => (Real.sin (((j.val : ℝ) + 1) * anguloModo d k) : ℂ)
 
@@ -166,6 +168,7 @@ theorem Ad_modoSeno
       subst i
       norm_num [modoSeno, anguloModo]
 
+/-- Adjacency eigenvalue associated with `modoSeno d k`. -/
 noncomputable def autovalorAd (d : ℕ) (k : Fin d) : ℂ :=
   (2 * Real.cos (anguloModo d k) : ℝ)
 
@@ -242,6 +245,7 @@ theorem modosSeno_linearIndependent
     (autovalorAd d) autovalorAd_injective (modoSeno d)
     (modoSeno_hasEigenvector hd)
 
+/-- Basis of sine eigenmodes for the finite path adjacency operator. -/
 noncomputable def baseModosSeno
     {d : ℕ} (hd : 1 ≤ d) : Module.Basis (Fin d) ℂ (Fin d → ℂ) := by
   classical
@@ -364,11 +368,14 @@ theorem norma_autovalorAd_le_rho
       abs_cos_anguloModo_le_cos_fiedler hd k)
     (by norm_num)
 
+/-- Matrix of the Hermitian commutator observable `i[T_d,P_d]`. -/
 noncomputable def Kmat (d : ℕ) : Matrix (Fin d) (Fin d) ℂ :=
   Complex.I • ((Td d * Pd d) - (Pd d * Td d))
 
+/-- Phase factor converting sine modes into commutator eigenmodes. -/
 noncomputable def fase (j : ℕ) : ℂ := (-Complex.I) ^ j
 
+/-- Phase-twisted sine mode for the commutator matrix. -/
 noncomputable def modoFase (d : ℕ) (k : Fin d) : Fin d → ℂ :=
   fun j => fase j.val * modoSeno d k j
 
@@ -479,6 +486,7 @@ theorem Kmat_mulVec_modoFase
   push_cast
   ring
 
+/-- Eigenvalue of `Kmat d` associated with `modoFase d k`. -/
 noncomputable def autovalorK (d : ℕ) (k : Fin d) : ℂ :=
   ((((2 / ((d : ℝ) - 1)) / rho d : ℝ) : ℂ) * autovalorAd d k)
 
@@ -538,6 +546,7 @@ theorem modosFase_linearIndependent
     (autovalorK d) (autovalorK_injective hd) (modoFase d)
     (modoFase_hasEigenvector hd)
 
+/-- Basis of phase-twisted eigenmodes for `Kmat`. -/
 noncomputable def baseModosFase
     {d : ℕ} (hd : 2 ≤ d) : Module.Basis (Fin d) ℂ (Fin d → ℂ) := by
   classical
@@ -627,6 +636,7 @@ theorem KdOp_eq_Kmat (d : ℕ) :
     Matrix (Fin d) (Fin d) ℂ ≃ₗ[ℂ] (Hd d →ₗ[ℂ] Hd d)).map_smul
       Complex.I ((Td d * Pd d) - (Pd d * Td d)) |>.symm
 
+/-- The phase-twisted commutator eigenmode represented in `Hd d`. -/
 noncomputable def modoFaseHd (d : ℕ) (k : Fin d) : Hd d :=
   WithLp.toLp 2 (modoFase d k)
 
@@ -663,6 +673,7 @@ theorem modosFaseHd_linearIndependent
     (autovalorK d) (autovalorK_injective hd) (modoFaseHd d)
     (modoFaseHd_hasEigenvector hd)
 
+/-- Basis of commutator eigenmodes in the finite Hilbert space `Hd d`. -/
 noncomputable def baseModosFaseHd
     {d : ℕ} (hd : 2 ≤ d) : Module.Basis (Fin d) ℂ (Hd d) :=
   (baseModosFase hd).map (WithLp.linearEquiv 2 ℂ (Fin d → ℂ)).symm
diff --git a/PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean b/PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean
index 396bad8d8..4a22a1893 100644
--- a/PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean
+++ b/PhyslibAlpha/AlgebraicFramework/InformationGeometry/FisherRao.lean
@@ -48,6 +48,7 @@ namespace PhyslibAlpha
 
 /-- A point on the open probability simplex: strictly positive weights summing to 1. -/
 structure OpenSimplex (α : Type*) [Fintype α] where
+  /-- The strictly positive probability weight at each outcome. -/
   val : α → ℝ
   pos : ∀ i, 0 < val i
   sum_one : ∑ i : α, val i = 1

From 50ad2ce9fc3eec887a4d1df87c9be7f860aa4967 Mon Sep 17 00:00:00 2001
From: Eduardo Nava Hernandez <naype888@gmail.com>
Date: Thu, 17 Sep 2026 10:25:19 -0600
Subject: [PATCH 09/10] fix(PhyslibAlpha): register all PathGap and
 DimUncertainty modules in import file
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

The alphaFileImports CI check requires every .lean file under
PhyslibAlpha/ to have a corresponding public import line.
Adds the 17 missing entries (D0–D7 PathGap, PathSpectralGap,
D8–D15 DimUncertainty).

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>
---
 PhyslibAlpha.lean | 17 +++++++++++++++++
 1 file changed, 17 insertions(+)

diff --git a/PhyslibAlpha.lean b/PhyslibAlpha.lean
index 99ee40ec0..81fa02f5f 100644
--- a/PhyslibAlpha.lean
+++ b/PhyslibAlpha.lean
@@ -100,6 +100,23 @@ public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.Stinespring.Kernel
 public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.Stinespring.Dilation
 public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.Uncertainty
 public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimensionalUncertainty
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D8_Szego
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D9_Monotonia
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D10_Certificado
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D11_CuantoMinimoArea
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D12_Trace
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D13_FirstBreak
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D14_SzegoExcess
+public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D15_Cosecant
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D0_Habitat
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D1_CauchyGram
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D2_Robertson
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D3_GrafoCamino
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D4_PorQueNoDiagonal
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D5_MaximaTension
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D6_Fiedler
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D7_Niven
+public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathSpectralGap
 public import PhyslibAlpha.AlgebraicFramework.WStarAlgebra.Basic
 public import PhyslibAlpha.AlgebraicFramework.WStarAlgebra.ConjSpace
 public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.Dynamics.Automorphism

From d9fde6a6107447baa0c80ef9593951ed17e4734e Mon Sep 17 00:00:00 2001
From: Eduardo Nava Hernandez <naype888@gmail.com>
Date: Fri, 18 Sep 2026 04:30:55 -0600
Subject: [PATCH 10/10] fix(PhyslibAlpha): translate all Spanish comments and
 docstrings to English

Addresses reviewer request from TomOleDiem on PR #1658 to make all
comments in the files English. Translates all /-! module headers,
/-- docstrings, and inline comments across D0-D15, preserving Lean
identifier names unchanged.

Co-Authored-By: Claude Opus 4.6 <noreply@anthropic.com>
---
 .../DimUncertainty/D10_Certificado.lean       |  72 +++++------
 .../DimUncertainty/D11_CuantoMinimoArea.lean  |  72 +++++------
 .../DimUncertainty/D12_Trace.lean             |  48 +++----
 .../DimUncertainty/D13_FirstBreak.lean        |  60 ++++-----
 .../DimUncertainty/D14_SzegoExcess.lean       |  36 +++---
 .../DimUncertainty/D15_Cosecant.lean          |  33 ++---
 .../CStarAlgebra/DimUncertainty/D8_Szego.lean | 112 ++++++++--------
 .../DimUncertainty/D9_Monotonia.lean          |  22 ++--
 .../HilbertSpace/PathGap/D0_Habitat.lean      |  18 +--
 .../HilbertSpace/PathGap/D1_CauchyGram.lean   |  43 ++++---
 .../HilbertSpace/PathGap/D2_Robertson.lean    | 120 +++++++++---------
 .../HilbertSpace/PathGap/D3_GrafoCamino.lean  | 109 ++++++++--------
 .../PathGap/D4_PorQueNoDiagonal.lean          | 115 +++++++++--------
 .../PathGap/D5_MaximaTension.lean             |  89 ++++++-------
 .../HilbertSpace/PathGap/D6_Fiedler.lean      |  18 +--
 .../HilbertSpace/PathGap/D7_Niven.lean        |  30 ++---
 16 files changed, 502 insertions(+), 495 deletions(-)

diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D10_Certificado.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D10_Certificado.lean
index 4094bea98..b0f8d5f82 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D10_Certificado.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D10_Certificado.lean
@@ -11,42 +11,42 @@ public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D8_Sze
 public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D9_Monotonia
 
 /-!
-# D10 — Certificado conjunto: Fiedler + Niven + Szegő en `H_d`
+# D10 — Joint certificate: Fiedler + Niven + Szegő in `H_d`
 
-Reúne, en un único certificado citable, los tres pilares que se apoyan sobre
-el hábitat común `H_d = ℂ^d` (`D0_Habitat.lean`): la descomposición
-espectral de Fiedler (`D6_Fiedler.lean`), el teorema de Niven
-(`D7_Niven.lean`) y el límite de Szegő con la positividad de la brecha
-(`D8_Szego.lean`). Este es el teorema terminal del paquete: aquí se acaba
-la matemática que se demuestra en este repositorio.
+Gathers, in a single citable certificate, the three pillars that rest on
+the common habitat `H_d = ℂ^d` (`D0_Habitat.lean`): the Fiedler spectral
+decomposition (`D6_Fiedler.lean`), the Niven theorem (`D7_Niven.lean`),
+and the Szegő limit with gap positivity (`D8_Szego.lean`). This is the
+terminal theorem of the package: the mathematics proved in this
+repository ends here.
 
-# Blindaje de `δ_geom(d)` en el Hilbert finito `H_d`
+# Shielding `δ_geom(d)` in the finite Hilbert space `H_d`
 
-**Hábitat:** \(H_d=\mathtt{EuclideanSpace}\,\mathbb{C}\,(\mathtt{Fin}\,d)\).
-No se abandona ese espacio: es el que acoge la derivación del marco.
+**Habitat:** \(H_d=\mathtt{EuclideanSpace}\,\mathbb{C}\,(\mathtt{Fin}\,d)\).
+This space is never left: it is where the framework derivation lives.
 
-**Terna de escudos** (todo sobre el discreto):
+**Shield triad** (all on the discrete setting):
 
-| Escudo | Contenido Lean |
-|--------|----------------|
-| **Fiedler** | modo fundamental / `KdOp` / radio espectral en \(H_d\) |
+| Shield | Lean content |
+|--------|-------------|
+| **Fiedler** | fundamental mode / `KdOp` / spectral radius in \(H_d\) |
 | **Niven** | `saturacion_iff` + `no_reposición_saturacion_camino` + `deltaGeom_pos_of_four_le` |
-| **Szegő** | `limite_szego_CNava` + `deltaInf_pos` + ∞ no es dimensión |
-| **Monotonía** | `deltaGeom_four_le`: `δ_geom(4)` es el piso global para todo `d ≥ 4` |
-
-Lectura: los productos trigonométricos simultáneamente racionales de la
-saturación del camino **solo** existen en \(d\in\{2,3\}\). No hay más
-semillas; por eso **nada repone la cota unitaria** después de \(d=4\).
-Además, la monotonía certificada fija a \(d=4\) como el menor defecto
-realizado: cualquier medición en un \(H_d\) físico con \(d\ge4\) queda
-separada del cero por al menos \(\delta_{\rm geom}(4)\). Al crecer la
-familia finita, el defecto no se apaga: converge a
+| **Szegő** | `limite_szego_CNava` + `deltaInf_pos` + ∞ is not a dimension |
+| **Monotonicity** | `deltaGeom_four_le`: `δ_geom(4)` is the global floor for all `d ≥ 4` |
+
+Reading: the simultaneously rational trigonometric products of path
+saturation **only** exist at \(d\in\{2,3\}\). There are no more seeds;
+that is why **nothing restores the unit bound** after \(d=4\).
+Moreover, certified monotonicity pins \(d=4\) as the smallest realized
+defect: any measurement in a physical \(H_d\) with \(d\ge4\) is
+separated from zero by at least \(\delta_{\rm geom}(4)\). As the finite
+family grows, the defect does not vanish: it converges to
 \(\delta_\infty>0\).
 
-**Cierre del hábitat:** \(H_d = \mathbb{C}^d \cong \mathbb{R}^{2d}\), finito.
-Punto. Si quieren continuo infinito, aquí no es hotel — \(d=\infty\) no se
-hospeda en este paquete; a lo más se le ve llegar por la ventana como límite
-(`D8_Szego.lean`), pero nunca cruza la puerta.
+**Habitat closure:** \(H_d = \mathbb{C}^d \cong \mathbb{R}^{2d}\), finite.
+Period. For continuous infinite, this is not a hotel — \(d=\infty\) is not
+hosted in this package; at most one sees it arriving through the window as
+a limit (`D8_Szego.lean`), but it never crosses the door.
 -/
 
 @[expose] public section
@@ -62,7 +62,7 @@ namespace BlindajeHd
 open TransportePosicion
 open Gnomon
 
-/-! ## Habitat: no se sale de \(H_d\) -/
+/-! ## Habitat: stays within \(H_d\) -/
 
 /-- Predicate recording that the entire construction remains in the finite Hilbert space `Hd d`. -/
 def HabitatHilbertFinito (d : ℕ) : Prop :=
@@ -77,13 +77,13 @@ theorem infinito_no_es_habitat :
       0 < deltaInf :=
   infinito_no_es_dimension_sino_limite
 
-/-! ## Niven: cota unitaria no se repone -/
+/-! ## Niven: the unit bound does not recover -/
 
 theorem niven_saturacion_solo_semillas (d : ℕ) (hd : 2 ≤ d) :
     cos (π / (d + 1)) ^ 2 = ((d : ℝ) - 1) / 4 ↔ d = 2 ∨ d = 3 :=
   saturacion_iff d hd
 
-/-- **Nada repone la cota** tras \(d=4\). -/
+/-- **Nothing restores the bound** after \(d=4\). -/
 theorem niven_cota_unitaria_no_se_repone (d : ℕ) (hd : 4 ≤ d) :
     cos (π / (d + 1)) ^ 2 ≠ ((d : ℝ) - 1) / 4 :=
   no_reposición_saturacion_camino d hd
@@ -96,14 +96,14 @@ theorem piso_precision_deltaGeom_d4_en_Hd (d : ℕ) (hd : 4 ≤ d) :
     deltaGeom 4 ≤ deltaGeom d :=
   deltaGeom_four_le d hd
 
-/-- En el régimen físico finito `H_d`, `d ≥ 4`, no existe lectura con defecto
-por debajo del piso elemental `δ_geom(4)`. -/
+/-- In the finite physical regime `H_d`, `d ≥ 4`, no reading has defect
+below the elementary floor `δ_geom(4)`. -/
 theorem no_medicion_absoluta_bajo_piso_d4_en_Hd
     (d : ℕ) (hd : 4 ≤ d) (ε : ℝ) (hε : ε < deltaGeom 4) :
     ε < deltaGeom d :=
   lt_of_lt_of_le hε (piso_precision_deltaGeom_d4_en_Hd d hd)
 
-/-! ## Fiedler: espectro y banda en \(H_d\) -/
+/-! ## Fiedler: spectrum and band in \(H_d\) -/
 
 theorem fiedler_autovector_en_Hd (d : ℕ) (hd : 2 ≤ d) :
     KdOp d (vectorFiedlerExplicito d) =
@@ -119,7 +119,7 @@ theorem fiedler_radio_banda (d : ℕ) (hd : 2 ≤ d) :
   letI : Nontrivial (Hd d) := inferInstance
   exact radioEspectral_KdOp_eq_paso d hd
 
-/-! ## Szegő: asintótica de la familia finita -/
+/-! ## Szegő: asymptotics of the finite family -/
 
 theorem szego_limite_familia_finita :
     Tendsto CNava atTop (𝓝 Cinf) :=
@@ -134,7 +134,7 @@ theorem defecto_real_positivo_desde_Hd4_hasta_limite :
       0 < deltaInf :=
   ⟨niven_deltaGeom_pos_en_Hd, limite_defecto_geometrico, szego_deltaInf_pos⟩
 
-/-! ## Certificado conjunto citable -/
+/-! ## Joint citable certificate -/
 
 /-- Joint certificate collecting the finite habitat, saturation classification, and positive gap. -/
 structure CertificadoBlindajeHd where
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D11_CuantoMinimoArea.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D11_CuantoMinimoArea.lean
index 1444deb03..8624b16a4 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D11_CuantoMinimoArea.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D11_CuantoMinimoArea.lean
@@ -8,24 +8,24 @@ module
 public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D10_Certificado
 
 /-!
-# D11 — Cuanto cuántico elemental de área
+# D11 — Elementary quantum of area
 
-Este módulo baja a Lean la lectura estrictamente matemática del cuanto
-cuántico elemental:
+This module brings to Lean the strictly mathematical reading of the
+elementary quantum:
 
-* `deltaGeom 4` es el primer defecto lineal positivo de la cola `d ≥ 4`.
-* `deltaGeom 4 ^ 2` es el primer cuanto cuántico elemental de área.
-* Por monotonía, ninguna resolución de área realizada en `H_d`, `d ≥ 4`,
-  queda por debajo de ese cuanto.
+* `deltaGeom 4` is the first positive linear defect of the `d ≥ 4` tail.
+* `deltaGeom 4 ^ 2` is the first elementary quantum of area.
+* By monotonicity, no area resolution realized in `H_d`, `d ≥ 4`,
+  falls below that quantum.
 
-No introduce unidades físicas, bariones ni escala de Planck. Si luego se
-quiere poner una unidad externa, basta multiplicar por una escala no negativa:
-la cota sobrevive por orden.
+No physical units, baryons, or Planck scale are introduced. To attach
+an external unit later, it suffices to multiply by a nonnegative scale:
+the bound survives by order.
 
-La dependencia matemática es la cadena del paquete:
-Cauchy--Gram → Robertson--Schrödinger → instancia `T_d/P_d` →
-Niven/Szegő/monotonía. No modifica Robertson 1929; lo usa como ancla y
-deriva el piso de área de su realización discreta.
+The mathematical dependency is the package chain:
+Cauchy--Gram → Robertson--Schrödinger → `T_d/P_d` instance →
+Niven/Szegő/monotonicity. It does not modify Robertson 1929; it uses
+it as an anchor and derives the area floor from its discrete realization.
 -/
 
 @[expose] public section
@@ -36,102 +36,102 @@ namespace CuantoMinimoArea
 
 open Gnomon
 
-/-- Cuanto cuántico elemental de área de la cola `H_d`, `d ≥ 4`. -/
+/-- Elementary quantum of area of the `H_d` tail, `d ≥ 4`. -/
 def cuantoCuanticoElemental : ℝ :=
   deltaGeom 4 ^ 2
 
-/-- Alias operativo: el "cuadrito" mínimo es el cuanto cuántico elemental. -/
+/-- Operational alias: the minimum "small square" is the elementary quantum. -/
 def cuadritoMinimo : ℝ :=
   cuantoCuanticoElemental
 
-/-- Área de resolución inducida por el defecto geométrico en `H_d`. -/
+/-- Resolution area induced by the geometric defect in `H_d`. -/
 def areaResolucionHd (d : ℕ) : ℝ :=
   deltaGeom d ^ 2
 
-/-- El nombre citable y el alias operativo son la misma cantidad. -/
+/-- The citable name and the operational alias are the same quantity. -/
 theorem cuadritoMinimo_eq_cuantoCuanticoElemental :
     cuadritoMinimo = cuantoCuanticoElemental := by
   rfl
 
-/-- El "cuadrito" es exactamente la resolución de área en `H_4`. -/
+/-- The "small square" is exactly the area resolution in `H_4`. -/
 theorem cuadritoMinimo_eq_areaResolucionH4 :
     cuadritoMinimo = areaResolucionHd 4 := by
   rfl
 
-/-- El cuanto cuántico elemental es exactamente la resolución de área en `H_4`. -/
+/-- The elementary quantum is exactly the area resolution in `H_4`. -/
 theorem cuantoCuanticoElemental_eq_areaResolucionH4 :
     cuantoCuanticoElemental = areaResolucionHd 4 := by
   rfl
 
-/-- El cuanto cuántico elemental de área es estrictamente positivo. -/
+/-- The elementary quantum of area is strictly positive. -/
 theorem cuantoCuanticoElemental_pos : 0 < cuantoCuanticoElemental := by
   unfold cuantoCuanticoElemental
   have hδ : 0 < deltaGeom 4 :=
     deltaGeom_pos_of_four_le 4 (by omega)
   positivity
 
-/-- Alias de positividad para el nombre operativo. -/
+/-- Positivity alias for the operational name. -/
 theorem cuadritoMinimo_pos : 0 < cuadritoMinimo := by
   simpa [cuadritoMinimo] using cuantoCuanticoElemental_pos
 
-/-- Toda área de resolución en `H_d`, `d ≥ 4`, está por encima del cuanto. -/
+/-- Every resolution area in `H_d`, `d ≥ 4`, lies above the quantum. -/
 theorem cuantoCuanticoElemental_le_areaResolucionHd (d : ℕ) (hd : 4 ≤ d) :
     cuantoCuanticoElemental ≤ areaResolucionHd d := by
   unfold cuantoCuanticoElemental areaResolucionHd
   exact deltaGeom_sq_four_le d hd
 
-/-- Toda área de resolución en `H_d`, `d ≥ 4`, está por encima del cuadrito. -/
+/-- Every resolution area in `H_d`, `d ≥ 4`, lies above the small square. -/
 theorem cuadritoMinimo_le_areaResolucionHd (d : ℕ) (hd : 4 ≤ d) :
     cuadritoMinimo ≤ areaResolucionHd d := by
   simpa [cuadritoMinimo] using cuantoCuanticoElemental_le_areaResolucionHd d hd
 
-/-- No existe una resolución realizada en `H_d`, `d ≥ 4`, estrictamente menor
-que el cuanto cuántico elemental. -/
+/-- No realized resolution in `H_d`, `d ≥ 4`, is strictly less than the
+elementary quantum. -/
 theorem no_hay_resolucion_menor_que_cuanto_cuantico
     (d : ℕ) (hd : 4 ≤ d) :
     ¬ areaResolucionHd d < cuantoCuanticoElemental := by
   exact not_lt.mpr (cuantoCuanticoElemental_le_areaResolucionHd d hd)
 
-/-- Alias operativo: no hay resolución menor que el cuadrito mínimo. -/
+/-- Operational alias: no resolution is less than the minimum small square. -/
 theorem no_hay_resolucion_menor_que_cuadrito
     (d : ℕ) (hd : 4 ≤ d) :
     ¬ areaResolucionHd d < cuadritoMinimo := by
   simpa [cuadritoMinimo] using no_hay_resolucion_menor_que_cuanto_cuantico d hd
 
-/-- Cualquier umbral por debajo del cuadrito queda por debajo de toda
-resolución realizada en la cola `d ≥ 4`. -/
+/-- Any threshold below the small square falls below every realized
+resolution in the `d ≥ 4` tail. -/
 theorem umbral_bajo_cuadrito_no_alcanza_Hd
     (d : ℕ) (hd : 4 ≤ d) (ε : ℝ) (hε : ε < cuadritoMinimo) :
     ε < areaResolucionHd d :=
   lt_of_lt_of_le hε (cuadritoMinimo_le_areaResolucionHd d hd)
 
-/-- Poner una escala externa no negativa conserva la cota mínima. -/
+/-- Applying a nonnegative external scale preserves the minimum bound. -/
 theorem escala_no_negativa_conserva_cuanto_cuantico
     (escala : ℝ) (hesc : 0 ≤ escala) (d : ℕ) (hd : 4 ≤ d) :
     escala * cuantoCuanticoElemental ≤ escala * areaResolucionHd d :=
   mul_le_mul_of_nonneg_left (cuantoCuanticoElemental_le_areaResolucionHd d hd) hesc
 
-/-- Alias operativo para la escala externa no negativa. -/
+/-- Operational alias for the nonnegative external scale. -/
 theorem escala_no_negativa_conserva_cuadrito
     (escala : ℝ) (hesc : 0 ≤ escala) (d : ℕ) (hd : 4 ≤ d) :
     escala * cuadritoMinimo ≤ escala * areaResolucionHd d := by
   simpa [cuadritoMinimo] using escala_no_negativa_conserva_cuanto_cuantico escala hesc d hd
 
-/-- Con una escala externa positiva, el cuanto escalado sigue siendo
-estrictamente positivo. -/
+/-- With a positive external scale, the scaled quantum remains strictly
+positive. -/
 theorem cuanto_cuantico_escalado_pos
     (escala : ℝ) (hesc : 0 < escala) :
     0 < escala * cuantoCuanticoElemental :=
   mul_pos hesc cuantoCuanticoElemental_pos
 
-/-- Alias operativo: con escala positiva, el cuadrito escalado sigue siendo
-estrictamente positivo. -/
+/-- Operational alias: with a positive scale, the scaled small square
+remains strictly positive. -/
 theorem cuadrito_escalado_pos
     (escala : ℝ) (hesc : 0 < escala) :
     0 < escala * cuadritoMinimo := by
   simpa [cuadritoMinimo] using cuanto_cuantico_escalado_pos escala hesc
 
-/-- Certificado citable del cuanto cuántico elemental de área. -/
+/-- Citable certificate for the elementary quantum of area. -/
 structure CertificadoCuantoMinimoArea where
   cuanto_pos : 0 < cuantoCuanticoElemental
   area_minima : ∀ d : ℕ, 4 ≤ d → cuantoCuanticoElemental ≤ areaResolucionHd d
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D12_Trace.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D12_Trace.lean
index c1f1feb9c..7734fcd95 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D12_Trace.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D12_Trace.lean
@@ -10,21 +10,21 @@ public import Mathlib.LinearAlgebra.Matrix.Notation
 public import Mathlib.LinearAlgebra.Matrix.Trace
 
 /-!
-# Obstrucción de traza al conmutador escalar en dimensión finita
+# Trace obstruction to a scalar commutator in finite dimension
 
-Complemento algebraico a la construcción de `T_d`, `P_d` en
-`D3_GrafoCamino.lean`: en dimensión finita ningún conmutador matricial
-puede ser un múltiplo escalar no nulo de la identidad, mientras que sí
-existen pares de matrices unitarias que anticonmutan exactamente.
+Algebraic complement to the construction of `T_d`, `P_d` in
+`D3_GrafoCamino.lean`: in finite dimension no matrix commutator can
+equal a nonzero scalar multiple of the identity, while exact
+anticommuting unitary pairs do exist.
 
-Dos afirmaciones, independientes entre sí:
+Two independent statements:
 
-1. Todo conmutador matricial `[Q,P] = QP - PQ` tiene traza cero
-   (`Matrix.trace_mul_comm`), así que nunca puede igualar `c • 1` para
+1. Every matrix commutator `[Q,P] = QP - PQ` has trace zero
+   (`Matrix.trace_mul_comm`), so it can never equal `c • 1` for
    `c ≠ 0` (`no_nonzero_scalar_exact_commutator`).
-2. Esa obstrucción es sobre el conmutador aditivo; no impide la
-   no-conmutatividad multiplicativa: el par de matrices `2×2`
-   `W₁ = !![0,1;1,0]`, `W₂ = !![1,0;0,-1]` satisface exactamente
+2. That obstruction is on the additive commutator; it does not prevent
+   multiplicative non-commutativity: the `2×2` matrix pair
+   `W₁ = !![0,1;1,0]`, `W₂ = !![1,0;0,-1]` satisfies exactly
    `W₂ W₁ = -(W₁ W₂)` (`parWeyl_anticonmuta`).
 -/
 
@@ -34,19 +34,19 @@ noncomputable section
 
 namespace ConmutadorEscalarFinito
 
-/-- Conmutador matricial. -/
+/-- Matrix commutator. -/
 def commutator {d : ℕ}
     (Q P : Matrix (Fin d) (Fin d) ℂ) : Matrix (Fin d) (Fin d) ℂ :=
   Q * P - P * Q
 
-/-- La traza de todo conmutador matricial finito es cero. -/
+/-- The trace of every finite matrix commutator is zero. -/
 theorem trace_commutator_zero {d : ℕ}
     (Q P : Matrix (Fin d) (Fin d) ℂ) :
     Matrix.trace (commutator Q P) = 0 := by
   rw [commutator, Matrix.trace_sub, Matrix.trace_mul_comm Q P, sub_self]
 
-/-- En dimensión finita positiva, un conmutador no puede ser un múltiplo
-escalar no nulo de la identidad. -/
+/-- In positive finite dimension, a commutator cannot be a nonzero
+scalar multiple of the identity. -/
 theorem no_nonzero_scalar_exact_commutator {d : ℕ} (hd : 0 < d)
     (Q P : Matrix (Fin d) (Fin d) ℂ) (c : ℂ) (hc : c ≠ 0) :
     commutator Q P ≠ c • (1 : Matrix (Fin d) (Fin d) ℂ) := by
@@ -61,33 +61,33 @@ theorem no_nonzero_scalar_exact_commutator {d : ℕ} (hd : 0 < d)
     exact_mod_cast (Nat.ne_of_gt hd)
   exact (mul_ne_zero hc hd0) ht.symm
 
-/-- Corolario: en particular, el conmutador tampoco puede igualar un
-múltiplo imaginario `i·c` de la identidad para ningún real `c ≠ 0`. -/
+/-- Corollary: in particular, the commutator cannot equal an imaginary
+multiple `i·c` of the identity for any real `c ≠ 0`. -/
 theorem commutador_ne_escalar_imaginario {d : ℕ} (hd : 0 < d)
     (Q P : Matrix (Fin d) (Fin d) ℂ) (c : ℝ) (hc : c ≠ 0) :
     commutator Q P ≠ (Complex.I * (c : ℂ)) • (1 : Matrix (Fin d) (Fin d) ℂ) := by
   apply no_nonzero_scalar_exact_commutator hd Q P
   exact mul_ne_zero Complex.I_ne_zero (Complex.ofReal_ne_zero.mpr hc)
 
-/-! ## Un par de Weyl exacto en dimensión dos -/
+/-! ## An exact Weyl pair in dimension two -/
 
-/-- Primera matriz del par de Weyl `2×2`. -/
+/-- First matrix of the `2×2` Weyl pair. -/
 def W1 : Matrix (Fin 2) (Fin 2) ℂ :=
   !![0, 1; 1, 0]
 
-/-- Segunda matriz del par de Weyl `2×2`. -/
+/-- Second matrix of the `2×2` Weyl pair. -/
 def W2 : Matrix (Fin 2) (Fin 2) ℂ :=
   !![1, 0; 0, -1]
 
-/-- Relación de Weyl exacta: `W₂ W₁ = -(W₁ W₂)`. La obstrucción de traza de
-arriba es sobre el conmutador *aditivo*; no impide esta anticonmutación
-*multiplicativa* exacta en dimensión finita. -/
+/-- Exact Weyl relation: `W₂ W₁ = -(W₁ W₂)`. The trace obstruction
+above is on the *additive* commutator; it does not prevent this exact
+*multiplicative* anticommutation in finite dimension. -/
 theorem parWeyl_anticonmuta : W2 * W1 = -(W1 * W2) := by
   ext i j
   fin_cases i <;> fin_cases j <;>
     norm_num [W1, W2, Matrix.mul_apply, Fin.sum_univ_two]
 
-/-- En particular, `W₁` y `W₂` no conmutan. -/
+/-- In particular, `W₁` and `W₂` do not commute. -/
 theorem parWeyl_no_conmuta : W2 * W1 ≠ W1 * W2 := by
   intro h
   have hij := congrFun (congrFun h (0 : Fin 2)) (1 : Fin 2)
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D13_FirstBreak.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D13_FirstBreak.lean
index ffb4c005e..d993f5a8a 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D13_FirstBreak.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D13_FirstBreak.lean
@@ -9,23 +9,23 @@ public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D3_GrafoCamin
 public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D7_Niven
 
 /-!
-# La primera ruptura combinatoria es `d = 4`
+# The first combinatorial break is `d = 4`
 
-Complemento combinatorio al teorema de Niven (`D7_Niven.lean`): la
-saturación espectral de Robertson–Schrödinger sobre `P_d` deja de
-cumplirse exactamente cuando `P_d` adquiere su primera arista *interior*
-(una arista entre dos vértices que no son extremos del camino), y esa
-coincidencia ocurre exactamente en `d = 4`.
+Combinatorial complement to the Niven theorem (`D7_Niven.lean`):
+Robertson–Schrödinger spectral saturation on `P_d` fails exactly when
+`P_d` acquires its first *interior* edge (an edge between two vertices
+that are not endpoints of the path), and that coincidence occurs
+exactly at `d = 4`.
 
-Dos rutas independientes hacia la misma dimensión:
+Two independent routes to the same dimension:
 
-* **espectral** (`D7_Niven.lean`): `cos²(π/(d+1)) = (d-1)/4 ↔ d ∈ {2,3}`;
-* **combinatoria** (aquí): `P_d` tiene una arista entre dos vértices
-  interiores si y sólo si `4 ≤ d`.
+* **spectral** (`D7_Niven.lean`): `cos²(π/(d+1)) = (d-1)/4 ↔ d ∈ {2,3}`;
+* **combinatorial** (here): `P_d` has an edge between two interior
+  vertices if and only if `4 ≤ d`.
 
-`primera_ruptura_iff_dimension_cuatro` certifica que ambas rutas señalan
-la misma dimensión `d = 4`, sin usar ninguna ecuación espectral en la
-mitad combinatoria.
+`primera_ruptura_iff_dimension_cuatro` certifies that both routes point
+to the same dimension `d = 4`, without using any spectral equation in
+the combinatorial half.
 -/
 
 @[expose] public section
@@ -34,39 +34,39 @@ namespace PrimeraRuptura
 
 open SimpleGraph
 
-/-- Ecuación aritmético-espectral que representa la saturación del camino
-(la misma de `Gnomon.saturacion_iff`, escrita como predicado). -/
+/-- Arithmetic-spectral equation representing path saturation
+(the same as `Gnomon.saturacion_iff`, written as a predicate). -/
 def SaturacionCamino (d : ℕ) : Prop :=
   Real.cos (Real.pi / (d + 1)) ^ 2 = ((d : ℝ) - 1) / 4
 
-/-- Ruptura: negación de la igualdad de saturación del camino. -/
+/-- Break: negation of the path saturation equality. -/
 def RupturaCamino (d : ℕ) : Prop := ¬ SaturacionCamino d
 
-/-- Desde la dimensión mínima `2`, la ruptura ocurre exactamente desde `4`. -/
+/-- From the minimum dimension `2`, the break occurs exactly from `4`. -/
 theorem ruptura_camino_iff_cuatro_le (d : ℕ) (hd : 2 ≤ d) :
     RupturaCamino d ↔ 4 ≤ d := by
   unfold RupturaCamino SaturacionCamino
   rw [Gnomon.saturacion_iff d hd]
   omega
 
-/-- La dimensión cuatro ya está en ruptura. -/
+/-- Dimension four is already in break. -/
 theorem ruptura_camino_cuatro : RupturaCamino 4 := by
   exact (ruptura_camino_iff_cuatro_le 4 (by norm_num)).2 (by norm_num)
 
-/-- Predicado puramente combinatorio de vértice no terminal del camino. -/
+/-- Purely combinatorial predicate for non-terminal path vertices. -/
 def VerticeInterior {d : ℕ} (i : Fin d) : Prop :=
   0 < i.val ∧ i.val + 1 < d
 
-/-- Existe una arista genuinamente interior cuando dos vértices no
-terminales del camino son adyacentes. -/
+/-- A genuinely interior edge exists when two non-terminal
+vertices of the path are adjacent. -/
 def TieneAristaInterior (d : ℕ) : Prop :=
   ∃ i j : Fin d,
     VerticeInterior i ∧ VerticeInterior j ∧
       (SimpleGraph.pathGraph d).Adj i j
 
-/-- El camino tiene una arista interior si y sólo si posee al menos cuatro
-vértices. Esta equivalencia no usa Robertson ni la ecuación de saturación:
-es pura combinatoria del camino. -/
+/-- The path has an interior edge iff it has at least four
+vertices. This equivalence uses neither Robertson nor the saturation
+equation: it is pure path combinatorics. -/
 theorem tiene_arista_interior_iff_cuatro_le (d : ℕ) :
     TieneAristaInterior d ↔ 4 ≤ d := by
   constructor
@@ -86,21 +86,21 @@ theorem tiene_arista_interior_iff_cuatro_le (d : ℕ) :
     · rw [SimpleGraph.pathGraph_adj]
       exact Or.inl rfl
 
-/-- Coincidencia central: dentro del régimen `d ≥ 2`, tener una arista entre
-dos vértices interiores equivale exactamente a romper la saturación. Las dos
-caras se demuestran por rutas independientes: combinatoria y espectral. -/
+/-- Central coincidence: within the `d ≥ 2` regime, having an edge between
+two interior vertices is exactly equivalent to breaking saturation. The two
+faces are proved by independent routes: combinatorial and spectral. -/
 theorem transporte_interior_iff_ruptura (d : ℕ) (hd : 2 ≤ d) :
     TieneAristaInterior d ↔ RupturaCamino d := by
   exact (tiene_arista_interior_iff_cuatro_le d).trans
     (ruptura_camino_iff_cuatro_le d hd).symm
 
-/-- La primera ruptura es una propiedad de orden: hay ruptura en `d`, y `d`
-es menor o igual que cualquier otra dimensión admisible que también rompa. -/
+/-- The first break is an order property: there is a break at `d`, and `d`
+is less than or equal to every other admissible dimension that also breaks. -/
 def EsPrimeraRuptura (d : ℕ) : Prop :=
   2 ≤ d ∧ RupturaCamino d ∧
     ∀ n : ℕ, 2 ≤ n → RupturaCamino n → d ≤ n
 
-/-- Caracterización dimensional: la primera ruptura es exactamente `d = 4`. -/
+/-- Dimensional characterization: the first break is exactly `d = 4`. -/
 theorem primera_ruptura_iff_dimension_cuatro (d : ℕ) :
     EsPrimeraRuptura d ↔ d = 4 := by
   constructor
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D14_SzegoExcess.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D14_SzegoExcess.lean
index 367dc117e..0e77b1233 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D14_SzegoExcess.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D14_SzegoExcess.lean
@@ -8,14 +8,14 @@ module
 public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D9_Monotonia
 
 /-!
-# El exceso sobre el límite de Szegő
-
-Corolario aritmético directo de la monotonía (`D9_Monotonia.lean`) y el
-límite de Szegő (`D8_Szego.lean`): el "exceso" `Cinf - CNava(d)` —cuánto
-le falta a `CNava(d)` para alcanzar el límite `C∞`— es positivo, máximo
-exactamente en `d = 4`, estrictamente decreciente en `d`, y se disuelve a
-`0`. No es un pilar nuevo: es la misma cadena de `D9_Monotonia.lean` leída
-desde el lado del remanente en vez del valor mismo.
+# The excess over the Szegő limit
+
+Direct arithmetic corollary of monotonicity (`D9_Monotonia.lean`) and
+the Szegő limit (`D8_Szego.lean`): the "excess" `Cinf - CNava(d)` — how
+far `CNava(d)` is from reaching the limit `C∞` — is positive, maximal
+exactly at `d = 4`, strictly decreasing in `d`, and dissolves to `0`.
+Not a new pillar: it is the same chain from `D9_Monotonia.lean` read
+from the remainder side instead of the value itself.
 -/
 
 @[expose] public section
@@ -25,32 +25,32 @@ open scoped Topology
 
 namespace Gnomon
 
-/-- El exceso de coherencia: cuánto le falta a `CNava(d)` para alcanzar el
-límite de Szegő `C∞`. -/
+/-- The coherence excess: how far `CNava(d)` is from reaching the
+Szegő limit `C∞`. -/
 noncomputable def excesoBrecha (d : ℕ) : ℝ := Cinf - CNava d
 
-/-- El exceso es siempre positivo: `CNava(d)` nunca alcanza `C∞` a `d`
-finito. -/
+/-- The excess is always positive: `CNava(d)` never reaches `C∞` at
+finite `d`. -/
 theorem excesoBrecha_pos (d : ℕ) (hd : 4 ≤ d) : 0 < excesoBrecha d := by
   unfold excesoBrecha
   linarith [CNava_lt_Cinf d hd]
 
-/-- El exceso es estrictamente decreciente en `d`, heredado de la
-monotonía de `CNava`. -/
+/-- The excess is strictly decreasing in `d`, inherited from the
+monotonicity of `CNava`. -/
 theorem excesoBrecha_strictAnti {a b : ℕ} (ha : 4 ≤ a) (hb : 4 ≤ b) (hab : a < b) :
     excesoBrecha b < excesoBrecha a := by
   unfold excesoBrecha
   linarith [CNava_strictMonoOn_ge_four ha hb hab]
 
-/-- El exceso máximo de toda la cola `d ≥ 4` se alcanza exactamente en
-`d = 4`: el mínimo global de `CNava` es el techo del exceso. -/
+/-- The maximum excess over the entire `d ≥ 4` tail is attained exactly at
+`d = 4`: the global minimum of `CNava` is the ceiling of the excess. -/
 theorem excesoBrecha_le_four (d : ℕ) (hd : 4 ≤ d) :
     excesoBrecha d ≤ excesoBrecha 4 := by
   unfold excesoBrecha
   linarith [CNava_four_le d hd]
 
-/-- El exceso se apaga por completo: `Cinf − CNava(d) → 0`, acotado arriba
-por `excesoBrecha 4` y llevado a `0` por el límite de Szegő. -/
+/-- The excess vanishes completely: `Cinf − CNava(d) → 0`, bounded above
+by `excesoBrecha 4` and driven to `0` by the Szegő limit. -/
 theorem excesoBrecha_tendsto_zero :
     Tendsto (fun d : ℕ => excesoBrecha d) atTop (𝓝 0) := by
   unfold excesoBrecha
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D15_Cosecant.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D15_Cosecant.lean
index a986117c3..713d67822 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D15_Cosecant.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D15_Cosecant.lean
@@ -13,18 +13,18 @@ public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Chebyshev.RootsExt
 public import Mathlib.Algebra.Polynomial.Splits
 
 /-!
-# Identidad cosecante clásica vía Chebyshev
+# Classical cosecant identity via Chebyshev
 
-Identidad clásica de análisis, autocontenida:
+Self-contained classical analysis identity:
 \[
 \sum_{k=1}^{N-1}\csc^2(k\pi/N)=(N^2-1)/3.
 \]
-Prueba vía el polinomio de Chebyshev de segunda especie `U_{N-1}`: sus
-raíces son `cos(kπ/N)`, y la derivada logarítmica evaluada en `±1`
-(descomponiendo `1/(1-x²) = ½(1/(1-x) + 1/(1+x))`) da la suma cerrada.
+Proof via the Chebyshev polynomial of the second kind `U_{N-1}`: its
+roots are `cos(kπ/N)`, and the logarithmic derivative evaluated at `±1`
+(decomposing `1/(1-x²) = ½(1/(1-x) + 1/(1+x))`) gives the closed sum.
 
-No depende de ningún objeto definido en otro archivo de este paquete: es
-un resultado de análisis clásico, completo en sí mismo sobre `Mathlib`.
+Does not depend on any object defined in another file of this package:
+it is a classical analysis result, self-contained over `Mathlib`.
 -/
 
 @[expose] public section
@@ -38,7 +38,7 @@ open scoped BigOperators
 
 namespace IdentidadCosecanteChebyshev
 
-/-! ## Chebyshev: `U_n` se parte en lineales reales -/
+/-! ## Chebyshev: `U_n` factors into real linears -/
 
 theorem U_splits_real (n : ℕ) : (U ℝ n).Splits := by
   rw [splits_iff_card_roots]
@@ -60,13 +60,13 @@ theorem U_splits_real (n : ℕ) : (U ℝ n).Splits := by
     rw [hdeg]
     exact hcard
 
-/-! ## Derivadas de `U_n` en `±1` -/
+/-! ## Derivatives of `U_n` at `±1` -/
 
 theorem U_deriv_eval_one (n : ℕ) :
     (derivative (U ℝ (n : ℤ))).eval (1 : ℝ) =
       ((n : ℝ) + 2) * ((n : ℝ) + 1) * (n : ℝ) / 3 := by
   have h := derivative_U_eval_one (R := ℝ) (n : ℤ)
-  -- `3 * U'(1) = (n+2)(n+1)n` con coerciones enteras
+  -- `3 * U'(1) = (n+2)(n+1)n` with integer coercions
   push_cast at h
   linarith
 
@@ -74,7 +74,7 @@ theorem U_deriv_eval_neg_one (n : ℕ) :
     (derivative (U ℝ (n : ℤ))).eval (-1 : ℝ) =
       -((-1 : ℝ) ^ n) *
         (((n : ℝ) + 2) * ((n : ℝ) + 1) * (n : ℝ) / 3) := by
-  -- Paridad: U_n(-x) = (-1)^n U_n(x)
+  -- Parity: U_n(-x) = (-1)^n U_n(x)
   have hfun :
       (fun x : ℝ ↦ (U ℝ (n : ℤ)).eval (-x)) =
         fun x ↦ (-1 : ℝ) ^ n * (U ℝ (n : ℤ)).eval x := by
@@ -99,7 +99,8 @@ theorem U_deriv_eval_neg_one (n : ℕ) :
   rw [hUd1] at hder
   linarith
 
-/-! ## Sumas sobre raíces -/
+/-! ## Sums over roots -/
+
 
 theorem sum_one_div_one_sub_roots (n : ℕ) (hn : 1 ≤ n) :
     ((U ℝ (n : ℤ)).roots.map fun z : ℝ ↦ (1 : ℝ) / (1 - z)).sum =
@@ -198,7 +199,7 @@ theorem sum_one_div_one_sub_sq_roots (n : ℕ) (hn : 1 ≤ n) :
       intro h; exact hzm1 (by linarith)
     field_simp [hz1', hzm, hden]
     ring
-  -- levantar la identidad puntual a la suma sobre el multiconjunto
+  -- lift the pointwise identity to the sum over the multiset
   have hdecomp :
       ((U ℝ (n : ℤ)).roots.map fun z : ℝ ↦ (1 : ℝ) / (1 - z ^ 2)).sum =
         (1 / 2 : ℝ) *
@@ -221,7 +222,7 @@ theorem sum_one_div_one_sub_sq_roots (n : ℕ) (hn : 1 ≤ n) :
   rw [hdecomp, h1, h2]
   ring
 
-/-- **Identidad cosecante clásica.** `∑_{k=1}^{N-1} csc²(kπ/N) = (N²-1)/3`. -/
+/-- **Classical cosecant identity.** `∑_{k=1}^{N-1} csc²(kπ/N) = (N²-1)/3`. -/
 theorem sum_csc_sq (N : ℕ) (hN : 2 ≤ N) :
     ∑ k ∈ Finset.Ico 1 N, (sin ((k : ℝ) * π / N))⁻¹ ^ 2 =
       ((N : ℝ) ^ 2 - 1) / 3 := by
@@ -232,7 +233,7 @@ theorem sum_csc_sq (N : ℕ) (hN : 2 ≤ N) :
   have hinj :
       Set.InjOn (fun k : ℕ ↦ cos ((k + 1) * π / (n + 1))) (Finset.range n) :=
     (Finset.range n).nodup_map_iff_injOn.mp (roots_U_real_nodup n)
-  -- suma sobre raíces vista como multiconjunto → suma sobre `range n`
+  -- sum over roots as multiset → sum over `range n`
   have hfin :
       ((U ℝ (n : ℤ)).roots.map fun z : ℝ ↦ (1 : ℝ) / (1 - z ^ 2)).sum =
         ∑ k ∈ Finset.range n,
@@ -249,7 +250,7 @@ theorem sum_csc_sq (N : ℕ) (hN : 2 ≤ N) :
         sin ((k + 1 : ℝ) * π / (n + 1)) ^ 2 := by
       linarith [sin_sq_add_cos_sq ((k + 1 : ℝ) * π / (n + 1))]
     rw [h1, one_div, inv_pow]
-  -- reindexar `Ico 1 (n+1)` como imagen de `range n` bajo `·+1`
+  -- reindex `Ico 1 (n+1)` as image of `range n` under `·+1`
   have himg :
       Finset.Ico 1 (n + 1) = (Finset.range n).image (fun k ↦ k + 1) := by
     ext k
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D8_Szego.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D8_Szego.lean
index 485d4bd38..ef6e95512 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D8_Szego.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D8_Szego.lean
@@ -13,22 +13,22 @@ public import Mathlib.Analysis.SpecialFunctions.Trigonometric.Sinc
 public import Mathlib.Tactic.IntervalCases
 
 /-!
-# D8 — El límite de Szegő y la positividad de la brecha
-
-Define la constante de coherencia finita `C_Nava(d)` (forma cerrada exacta
-en `cos`/`sin` de `π/(d+1)`) y su brecha `deltaGeom(d) = C_Nava(d) − 1`.
-Dos resultados centrales:
-
-1. **Positividad** (`deltaGeom_pos_of_four_le`): `deltaGeom(d) > 0` para
-   todo `d ≥ 4`. Se verifica `d = 4, 5` en forma cerrada exacta y `d ≥ 6`
-   mediante cotas de Taylor certificadas para seno y coseno — sin apelar a
-   ningún cálculo numérico externo, sólo álgebra racional y las cotas
-   `Real.pi_gt_d2`/`Real.pi_lt_d2` de Mathlib.
-2. **Límite de Szegő** (`limite_szego_CNava`): `C_Nava(d) → C_∞ = √(π²/3−2)`
-   cuando `d → ∞` (como filtro `atTop` sobre la sucesión de espacios
-   finitos, no como un nuevo espacio de Hilbert en `d = ∞`; ver
-   `infinito_no_es_dimension_sino_limite`). En particular
-   `deltaInf = C_∞ − 1 > 0`, consecuencia exacta de `π > 3`.
+# D8 — The Szegő limit and gap positivity
+
+Defines the finite coherence constant `C_Nava(d)` (exact closed form
+in `cos`/`sin` of `π/(d+1)`) and its gap `deltaGeom(d) = C_Nava(d) − 1`.
+Two central results:
+
+1. **Positivity** (`deltaGeom_pos_of_four_le`): `deltaGeom(d) > 0` for
+   all `d ≥ 4`. Verified at `d = 4, 5` in exact closed form and `d ≥ 6`
+   via certified Taylor bounds on sine and cosine — without appealing to
+   any external numerical computation, only rational algebra and the
+   `Real.pi_gt_d2`/`Real.pi_lt_d2` bounds from Mathlib.
+2. **Szegő limit** (`limite_szego_CNava`): `C_Nava(d) → C_∞ = √(π²/3−2)`
+   as `d → ∞` (as an `atTop` filter over the sequence of finite spaces,
+   not as a new Hilbert space at `d = ∞`; see
+   `infinito_no_es_dimension_sino_limite`). In particular
+   `deltaInf = C_∞ − 1 > 0`, an exact consequence of `π > 3`.
 -/
 
 @[expose] public section
@@ -41,32 +41,32 @@ open scoped Topology
 
 namespace Gnomon
 
-/-! ## Forma cerrada y límite asintótico -/
+/-! ## Closed form and asymptotic limit -/
 
-/-- `N = d + 1`, notación para el grafo camino `pathGraph d`. -/
+/-- `N = d + 1`, notation for the path graph `pathGraph d`. -/
 noncomputable def Nreal (d : ℕ) : ℝ := (d : ℝ) + 1
 
-/-- Ángulo espectral fundamental `θ_d = π/(d+1)`. -/
+/-- Fundamental spectral angle `θ_d = π/(d+1)`. -/
 noncomputable def theta (d : ℕ) : ℝ := π / Nreal d
 
-/-- Forma cerrada exacta de `C_Nava(d)²`. -/
+/-- Exact closed form of `C_Nava(d)²`. -/
 noncomputable def CNavaSq (d : ℕ) : ℝ :=
   2 * ((d : ℝ) - 1) / (Nreal d * Real.cos (theta d) ^ 2) *
     (((Nreal d ^ 2 + 2) / 6) * Real.sin (theta d) ^ 2 - 1)
 
-/-- Constante de coherencia finita. -/
+/-- Finite coherence constant. -/
 noncomputable def CNava (d : ℕ) : ℝ := Real.sqrt (CNavaSq d)
 
-/-- Límite universal de Szegő. -/
+/-- Universal Szegő limit. -/
 noncomputable def Cinf : ℝ := Real.sqrt (π ^ 2 / 3 - 2)
 
-/-- Defecto geométrico finito `δ_geom(d) = C_Nava(d) - 1`. -/
+/-- Finite geometric defect `δ_geom(d) = C_Nava(d) - 1`. -/
 noncomputable def deltaGeom (d : ℕ) : ℝ := CNava d - 1
 
-/-- Defecto asintótico `δ_∞ = C_∞ - 1`. -/
+/-- Asymptotic defect `δ_∞ = C_∞ - 1`. -/
 noncomputable def deltaInf : ℝ := Cinf - 1
 
-/-- Término principal de la expansión de Szegő. -/
+/-- Leading term of the Szegő expansion. -/
 noncomputable def deltaSzegoPrincipal (d : ℕ) : ℝ :=
   deltaInf - Cinf / Nreal d
 
@@ -75,8 +75,8 @@ theorem CNavaSq_forma_cerrada (d : ℕ) :
       2 * ((d : ℝ) - 1) / (Nreal d * Real.cos (theta d) ^ 2) *
         (((Nreal d ^ 2 + 2) / 6) * Real.sin (theta d) ^ 2 - 1) := rfl
 
-/-- La forma cerrada reescrita mediante `sinc`. Elimina la singularidad
-aparente y permite tomar el límite en Lean. -/
+/-- The closed form rewritten via `sinc`. Removes the apparent
+singularity and allows taking the limit in Lean. -/
 theorem CNavaSq_forma_regularizada (d : ℕ) :
     CNavaSq d =
       2 * (1 - 2 / Nreal d) / Real.cos (theta d) ^ 2 *
@@ -147,32 +147,32 @@ private theorem CNavaSq_regularizada_tendsto :
   · ext d
     ring_nf
 
-/-- TEOREMA DE SZEGŐ, forma cuadrática: `C_Nava(d)² → (π²-6)/3`. -/
+/-- SZEGŐ THEOREM, quadratic form: `C_Nava(d)² → (π²-6)/3`. -/
 theorem limite_szego_CNavaSq :
     Tendsto CNavaSq atTop (𝓝 (π ^ 2 / 3 - 2)) := by
   apply CNavaSq_regularizada_tendsto.congr'
   filter_upwards with d
   exact (CNavaSq_forma_regularizada d).symm
 
-/-- LÍMITE DE SZEGŐ: `C_Nava(d) → C_∞ = √((π²-6)/3)`, construido sobre la
-teoría clásica de distribución espectral de Szegő. -/
+/-- SZEGŐ LIMIT: `C_Nava(d) → C_∞ = √((π²-6)/3)`, built on the
+classical Szegő spectral distribution theory. -/
 theorem limite_szego_CNava : Tendsto CNava atTop (𝓝 Cinf) := by
   unfold CNava Cinf
   exact Real.continuous_sqrt.continuousAt.tendsto.comp limite_szego_CNavaSq
 
-/-- Nombre citable de la especialización. Es un alias del resultado ya
-demostrado, no una rederivación de la teoría clásica de Toeplitz/Szegő. -/
+/-- Citable name for the specialization. An alias of the already proved
+result, not a re-derivation of the classical Toeplitz/Szegő theory. -/
 theorem limite_nava_szego_CNava : Tendsto CNava atTop (𝓝 Cinf) :=
   limite_szego_CNava
 
-/-- El defecto geométrico converge al defecto universal asintótico. -/
+/-- The geometric defect converges to the universal asymptotic defect. -/
 theorem limite_defecto_geometrico :
     Tendsto deltaGeom atTop (𝓝 deltaInf) := by
   unfold deltaGeom deltaInf
   exact limite_szego_CNava.sub_const 1
 
-/-- El defecto universal jamás se anula: `δ_∞ > 0`, consecuencia exacta de
-`π > 3`. -/
+/-- The universal defect never vanishes: `δ_∞ > 0`, an exact consequence
+of `π > 3`. -/
 theorem deltaInf_pos : 0 < deltaInf := by
   unfold deltaInf Cinf
   have hpi : (3 : ℝ) < π := Real.pi_gt_three
@@ -182,8 +182,8 @@ theorem deltaInf_pos : 0 < deltaInf := by
   rw [Real.sqrt_one] at hs
   linarith
 
-/-- El infinito no se añade como una dimensión realizada: la sucesión de
-defectos finitos sólo converge al valor límite estricto `δ∞ = C∞ - 1`. -/
+/-- Infinity is not added as a realized dimension: the sequence of
+finite defects only converges to the strict limit value `δ∞ = C∞ - 1`. -/
 theorem infinito_no_es_dimension_sino_limite :
     Tendsto deltaGeom atTop (𝓝 deltaInf) ∧ deltaInf = Cinf - 1 ∧ 0 < deltaInf :=
   ⟨limite_defecto_geometrico, rfl, deltaInf_pos⟩
@@ -193,8 +193,8 @@ theorem Cinf_pos : 0 < Cinf := by
   unfold deltaInf at h
   linarith
 
-/-- Monotonía asintótica: el término principal `δ_∞ - C_∞/(d+1)` de la
-expansión de Szegő es estrictamente creciente. -/
+/-- Asymptotic monotonicity: the leading term `δ_∞ - C_∞/(d+1)` of the
+Szegő expansion is strictly increasing. -/
 theorem deltaSzegoPrincipal_strictMono : StrictMono deltaSzegoPrincipal := by
   intro a b hab
   have hNa : 0 < Nreal a := by
@@ -210,23 +210,23 @@ theorem deltaSzegoPrincipal_strictMono : StrictMono deltaSzegoPrincipal := by
   have hmul := mul_lt_mul_of_pos_left hinv Cinf_pos
   simpa [div_eq_mul_inv] using sub_lt_sub_left hmul deltaInf
 
-/-! ## Positividad de la brecha para `d ≥ 4`
+/-! ## Gap positivity for `d ≥ 4`
 
-Niven cierra la ecuación de saturación en `d ∈ {2,3}` (`D7_Niven.lean`).
-Lo que sigue traduce ese hecho trigonométrico a la desigualdad algebraica
-`1 < CNava(d)` (equivalentemente `0 < deltaGeom d`) para todo `d ≥ 4`,
-verificando los casos `d = 4, 5` en forma cerrada exacta y `d ≥ 6` mediante
-las mismas cotas de Taylor certificadas usadas arriba para el límite. -/
+Niven closes the saturation equation at `d ∈ {2,3}` (`D7_Niven.lean`).
+What follows translates that trigonometric fact into the algebraic
+inequality `1 < CNava(d)` (equivalently `0 < deltaGeom d`) for all
+`d ≥ 4`, verifying `d = 4, 5` in exact closed form and `d ≥ 6` via the
+same certified Taylor bounds used above for the limit. -/
 
-/-! ## Consecuencia: `δ_geom(d) > 0` para todo `d ≥ 4`
+/-! ## Consequence: `δ_geom(d) > 0` for all `d ≥ 4`
 
-Niven cierra la ecuación de saturación en `d ∈ {2,3}`. El resto de esta
-sección traduce ese hecho trigonométrico a la desigualdad algebraica
-`1 < CNava(d)` (equivalentemente `0 < deltaGeom d`) para todo `d ≥ 4`,
-verificando los casos `d = 4, 5` en forma cerrada exacta y `d ≥ 6` mediante
-cotas de Taylor certificadas para seno y coseno (sin apelar a ningún
-resultado numérico externo: sólo `Real.pi_gt_d2`/`pi_lt_d2` de Mathlib y
-álgebra racional). -/
+Niven closes the saturation equation at `d ∈ {2,3}`. The rest of this
+section translates that trigonometric fact into the algebraic inequality
+`1 < CNava(d)` (equivalently `0 < deltaGeom d`) for all `d ≥ 4`,
+verifying `d = 4, 5` in exact closed form and `d ≥ 6` via certified
+Taylor bounds on sine and cosine (without appealing to any external
+numerical result: only `Real.pi_gt_d2`/`pi_lt_d2` from Mathlib and
+rational algebra). -/
 
 theorem CNavaSq_two : CNavaSq 2 = 1 := by
   simp only [CNavaSq, Nreal, theta, Nat.cast_ofNat]
@@ -639,8 +639,8 @@ theorem one_lt_CNavaSq_of_six_le (d : ℕ) (hd : 6 ≤ d) : 1 < CNavaSq d := by
     (by simpa [N] using hcos_pos)
     (by simpa [N] using hcos_thr)
 
-/-- `CNava(d)² > 1` para todo `d ≥ 4`: casos `4, 5` exactos, `d ≥ 6` vía cotas
-de Taylor certificadas. -/
+/-- `CNava(d)² > 1` for all `d ≥ 4`: exact cases `4, 5`, `d ≥ 6` via
+certified Taylor bounds. -/
 theorem one_lt_CNavaSq (d : ℕ) (hd : 4 ≤ d) : 1 < CNavaSq d := by
   match d with
   | 0 | 1 | 2 | 3 => omega
@@ -652,7 +652,7 @@ theorem one_lt_CNava_of_four_le (d : ℕ) (hd : 4 ≤ d) : 1 < CNava d := by
   rw [CNava, ← sqrt_one]
   exact sqrt_lt_sqrt (by norm_num) (one_lt_CNavaSq d hd)
 
-/-- TEOREMA CENTRAL DE NIVEN → POSITIVIDAD: `δ_geom(d) > 0` para todo
+/-- CENTRAL NIVEN THEOREM → POSITIVITY: `δ_geom(d) > 0` for all
 `d ≥ 4`. -/
 theorem deltaGeom_pos_of_four_le (d : ℕ) (hd : 4 ≤ d) : 0 < deltaGeom d := by
   unfold deltaGeom
diff --git a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D9_Monotonia.lean b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D9_Monotonia.lean
index 384f08f9f..028ad6c48 100644
--- a/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D9_Monotonia.lean
+++ b/PhyslibAlpha/AlgebraicFramework/CStarAlgebra/DimUncertainty/D9_Monotonia.lean
@@ -8,19 +8,19 @@ module
 public import PhyslibAlpha.AlgebraicFramework.CStarAlgebra.DimUncertainty.D8_Szego
 
 /-!
-# D9 — Monotonía estricta de `C_Nava` y `deltaGeom`
+# D9 — Strict monotonicity of `C_Nava` and `deltaGeom`
 
-Sin barrido numérico: `CNava` (y por tanto `deltaGeom`) crece
-estrictamente para toda dimensión `d ≥ 4`. En particular, `d = 4` es
-el mínimo global de la cola `d ≥ 4` y cada valor finito se aproxima a
-`Cinf` estrictamente por debajo (`CNava_lt_Cinf`, `deltaGeom_lt_deltaInf`).
+Without numerical sweep: `CNava` (and hence `deltaGeom`) is strictly
+increasing for every dimension `d ≥ 4`. In particular, `d = 4` is the
+global minimum of the `d ≥ 4` tail and each finite value approaches
+`Cinf` strictly from below (`CNava_lt_Cinf`, `deltaGeom_lt_deltaInf`).
 
-La prueba no supone que `π` sea racional. Las llamadas a `ring`
-certifican únicamente identidades algebraicas formales con `π` como
-elemento real simbólico. El signo estricto de la derivada se obtiene
-mediante cotas formales `3 < π < 22/7`, cotas de Taylor verificadas y
-un certificado polinómico de Bernstein de que el resto es
-estrictamente negativo en la caja compacta correspondiente.
+The proof does not assume `π` is rational. The `ring` calls certify
+only formal algebraic identities with `π` as a symbolic real element.
+The strict sign of the derivative is obtained via formal bounds
+`3 < π < 22/7`, verified Taylor bounds, and a Bernstein polynomial
+certificate that the remainder is strictly negative in the
+corresponding compact box.
 -/
 
 @[expose] public section
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D0_Habitat.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D0_Habitat.lean
index f077a7781..7818641cf 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D0_Habitat.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D0_Habitat.lean
@@ -8,23 +8,25 @@ module
 public import Mathlib.Analysis.InnerProductSpace.EuclideanDist
 
 /-!
-# D0 — Hábitat: el espacio de Hilbert finito `H_d`
+# D0 — Habitat: the finite-dimensional Hilbert space `H_d`
 
-Todo el argumento vive en un único espacio de Hilbert complejo de dimensión
-finita, `H_d = ℂ^d` con su producto interno estándar. No se sale nunca de
-este espacio: en particular, `d = ∞` no es una dimensión realizada, sólo un
-límite de la familia `{H_d}_{d∈ℕ}` (ver `D8_Szego.lean`).
+The entire argument lives in a single complex finite-dimensional
+Hilbert space, `H_d = ℂ^d` with its standard inner product.
+We never leave this space: in particular `d = ∞` is not a
+realized dimension, only a limit of the family `{H_d}_{d∈ℕ}`
+(see `D8_Szego.lean`).
 -/
 
 @[expose] public section
 
 namespace TransportePosicion
 
-/-- El espacio de Hilbert finito de dimensión `d`: `ℂ^d` con su estructura
-euclidiana estándar. -/
+/-- The finite-dimensional Hilbert space of dimension `d`:
+`ℂ^d` with its standard Euclidean structure. -/
 abbrev Hd (d : ℕ) := EuclideanSpace ℂ (Fin d)
 
-/-- Identidad definicional: `H_d` es, literalmente, `EuclideanSpace ℂ (Fin d)`. -/
+/-- Definitional identity: `H_d` is literally
+`EuclideanSpace ℂ (Fin d)`. -/
 theorem Hd_eq_euclidean (d : ℕ) :
     Hd d = EuclideanSpace ℂ (Fin d) := rfl
 
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D1_CauchyGram.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D1_CauchyGram.lean
index b3c4360f8..191675a42 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D1_CauchyGram.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D1_CauchyGram.lean
@@ -9,17 +9,17 @@ public import Mathlib.Analysis.InnerProductSpace.Basic
 public import Mathlib.Analysis.Complex.Norm
 
 /-!
-# D1 — Cauchy–Schwarz vía el defecto de Gram
+# D1 — Cauchy–Schwarz via the Gram defect
 
-Núcleo puramente algebraico: para dos vectores `x, y` de un espacio de
-Hilbert complejo, el determinante de la matriz de Gram hermitiana
+Purely algebraic kernel: for two vectors `x, y` in a complex
+Hilbert space, the determinant of the Hermitian Gram matrix
 
 `gramDefectC x y = ‖x‖² ‖y‖² − |⟨x,y⟩|²`
 
-nunca es negativo. Esa es, palabra por palabra, la desigualdad de
-Cauchy–Schwarz. Escribiendo `⟨x,y⟩` en sus partes real e imaginaria se
-obtiene de inmediato la desigualdad de Robertson–Schrödinger (`D2_Robertson.lean`)
-como consecuencia algebraica, no como postulado adicional.
+is never negative. That is, word for word, the Cauchy–Schwarz
+inequality. Writing `⟨x,y⟩` in real and imaginary parts gives
+the Robertson–Schrödinger inequality (`D2_Robertson.lean`)
+as an algebraic consequence, not an additional postulate.
 -/
 
 @[expose] public section
@@ -32,16 +32,16 @@ universe u
 
 variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
 
-/-- Cuadrado de la dispersión representada por un vector centrado. -/
+/-- Squared dispersion of a centered vector. -/
 def varianceC (x : H) : ℝ := ‖x‖ ^ 2
 
-/-- Determinante de la matriz de Gram hermitiana de dos vectores. -/
+/-- Determinant of the Hermitian Gram matrix of two vectors. -/
 def gramDefectC (x y : H) : ℝ :=
   varianceC x * varianceC y - ‖@inner ℂ H _ x y‖ ^ 2
 
-/-- La obstrucción universal: el determinante de Gram nunca es negativo.
-Esto ES la desigualdad de Cauchy–Schwarz, reescrita como positividad de un
-determinante 2×2. -/
+/-- The universal obstruction: the Gram determinant is never
+negative. This IS the Cauchy–Schwarz inequality, rewritten as
+positivity of a 2×2 determinant. -/
 theorem gramDefectC_nonneg (x y : H) : 0 ≤ gramDefectC x y := by
   have hxy : ‖@inner ℂ H _ x y‖ ≤ ‖x‖ * ‖y‖ := norm_inner_le_norm x y
   have hleft : 0 ≤ ‖x‖ * ‖y‖ - ‖@inner ℂ H _ x y‖ := sub_nonneg.mpr hxy
@@ -52,8 +52,8 @@ theorem gramDefectC_nonneg (x y : H) : 0 ≤ gramDefectC x y := by
         (‖x‖ * ‖y‖ + ‖@inner ℂ H _ x y‖) := mul_nonneg hleft hright
     _ = gramDefectC x y := by simp [gramDefectC, varianceC, pow_two]; ring
 
-/-- Saturar Cauchy–Schwarz equivale a anular, no las dispersiones, sino el
-determinante de Gram. -/
+/-- Saturating Cauchy–Schwarz amounts to zeroing
+the Gram determinant, not the dispersions. -/
 theorem gramDefectC_eq_zero_iff (x y : H) :
     gramDefectC x y = 0 ↔ ‖@inner ℂ H _ x y‖ = ‖x‖ * ‖y‖ := by
   have hxy : ‖@inner ℂ H _ x y‖ ≤ ‖x‖ * ‖y‖ := norm_inner_le_norm x y
@@ -68,15 +68,15 @@ theorem gramDefectC_eq_zero_iff (x y : H) :
     rw [heq]
     ring
 
-/-- Parte simétrica del producto interno de las fluctuaciones. -/
+/-- Symmetric part of the inner product of the fluctuations. -/
 def covarianceC (x y : H) : ℝ := (@inner ℂ H _ x y).re
 
-/-- Coordenada antisimétrica real: para fluctuaciones operatoriales es la
-coordenada real de la esperanza del conmutador. -/
+/-- Real antisymmetric coordinate: for operator fluctuations
+this is the real coordinate of the commutator expectation. -/
 def commutatorCoordinateC (x y : H) : ℝ := 2 * (@inner ℂ H _ x y).im
 
-/-- Robertson–Schrödinger es exactamente la positividad de Gram escrita en
-coordenadas real e imaginaria. -/
+/-- Robertson–Schrödinger is exactly Gram positivity
+written in real and imaginary coordinates. -/
 theorem robertsonSchrodinger_from_gram (x y : H) :
     covarianceC x y ^ 2 + (commutatorCoordinateC x y / 2) ^ 2 ≤
       varianceC x * varianceC y := by
@@ -91,12 +91,13 @@ theorem robertsonSchrodinger_from_gram (x y : H) :
   rw [hnorm] at hbase
   simpa [covarianceC, commutatorCoordinateC] using hbase
 
-/-- Saturación Robertson–Schrödinger abstracta. -/
+/-- Abstract Robertson–Schrödinger saturation. -/
 def RSSaturated (x y : H) : Prop :=
   covarianceC x y ^ 2 + (commutatorCoordinateC x y / 2) ^ 2 =
     varianceC x * varianceC y
 
-/-- La saturación Robertson–Schrödinger es exactamente defecto de Gram cero. -/
+/-- Robertson–Schrödinger saturation is exactly
+zero Gram defect. -/
 theorem robertsonSchrodinger_saturated_iff_gram_zero (x y : H) :
     RSSaturated x y ↔ gramDefectC x y = 0 := by
   have hnorm :
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D2_Robertson.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D2_Robertson.lean
index a05a85eef..a8b84c736 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D2_Robertson.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D2_Robertson.lean
@@ -13,25 +13,26 @@ public import Mathlib.Data.Rat.Star
 public import Mathlib.Tactic.IntervalCases
 
 /-!
-# D2 — La desigualdad de Robertson (1929)
-
-Formalización abstracta de la desigualdad de Robertson (1929) para un par de
-observables conjugados evaluados en un estado normalizado de un espacio de
-Hilbert. Se incluyen dos formas: la forma lineal clásica (`Evaluacion`,
-con la cota `|⟨[A,B]⟩|/2 ≤ σ_A σ_B`) y la forma cuadrática de
-Robertson–Schrödinger (`EvaluacionSchrodinger`, con covarianza).
-
-Punto central de este archivo: la hipótesis `cota_cuadratica` que
-`EvaluacionSchrodinger` exige como dato **no se postula** — al final del
-archivo (`ObstruccionGramUnificada.evaluacionSchrodingerDeGram`) se prueba
-que todo par de vectores de un espacio de Hilbert produce automáticamente una
-`EvaluacionSchrodinger` válida, con esa cota derivada directamente de
-`D1_CauchyGram.gramDefectC_nonneg`. Cauchy–Schwarz ⇒ Gram ⇒
-Robertson–Schrödinger, como teorema, no como axioma adicional.
-
-También se incluyen aquí cinco lemas aritméticos elementales (`Blindaje`)
-que se usan más adelante para acotar el coseno y para el teorema de Niven
-(`D7_Niven.lean`).
+# D2 — The Robertson inequality (1929)
+
+Abstract formalization of Robertson's (1929) inequality for a pair
+of conjugate observables evaluated in a normalized Hilbert-space
+state. Two forms are included: the classical linear form
+(`Evaluacion`, with `|⟨[A,B]⟩|/2 ≤ σ_A σ_B`) and the quadratic
+Robertson–Schrödinger form (`EvaluacionSchrodinger`, with
+covariance).
+
+Key point: the hypothesis `cota_cuadratica` required by
+`EvaluacionSchrodinger` is **not postulated** — at the end of
+this file (`ObstruccionGramUnificada.evaluacionSchrodingerDeGram`)
+we prove that every pair of Hilbert-space vectors automatically
+yields a valid `EvaluacionSchrodinger`, with the bound derived
+from `D1_CauchyGram.gramDefectC_nonneg`. Cauchy–Schwarz ⇒ Gram ⇒
+Robertson–Schrödinger, as a theorem, not an additional axiom.
+
+Five elementary arithmetic lemmas (`Blindaje` / Shielding) used
+later for cosine bounds and the Niven theorem (`D7_Niven.lean`)
+are also included here.
 -/
 
 @[expose] public section
@@ -40,10 +41,9 @@ namespace Robertson1929
 
 universe u
 
-/-- Evaluación exacta del teorema de Robertson (1929) tras evaluar dos
-observables conjugados en un estado normalizado de un espacio de Hilbert.
-`sigmaA`, `sigmaB` son las desviaciones y `mediaConmutador` es
-`⟨ψ,[A,B]ψ⟩`. -/
+/-- Exact evaluation of Robertson's (1929) theorem for two conjugate
+observables in a normalized Hilbert-space state. `sigmaA`, `sigmaB`
+are the standard deviations; `mediaConmutador` is `⟨ψ,[A,B]ψ⟩`. -/
 structure Evaluacion (H : Type u) [NormedAddCommGroup H]
     [InnerProductSpace ℂ H] where
   /-- The normalized state in which the observables are evaluated. -/
@@ -59,14 +59,14 @@ structure Evaluacion (H : Type u) [NormedAddCommGroup H]
   sigmaB_nonneg : 0 ≤ sigmaB
   cota : ‖mediaConmutador‖ / 2 ≤ sigmaA * sigmaB
 
-/-- Saturación exacta de la cota de Robertson en la evaluación dada. -/
+/-- Exact saturation of the Robertson bound for the given evaluation. -/
 def Saturada {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
     (R : Evaluacion H) : Prop :=
   R.sigmaA * R.sigmaB = ‖R.mediaConmutador‖ / 2
 
-/-- Evaluación en máxima tensión: el estado normalizado realiza la norma del
-conmutador, por lo que el lado derecho de Robertson es el más exigente de la
-familia de estados normalizados. -/
+/-- Maximal-tension evaluation: the normalized state realizes the
+commutator norm, so Robertson's right-hand side is the tightest
+among all normalized states. -/
 structure MaximaTension (H : Type u) [NormedAddCommGroup H]
     [InnerProductSpace ℂ H] extends Evaluacion H where
   /-- Norm of the commutator realized by the maximal-tension state. -/
@@ -75,8 +75,8 @@ structure MaximaTension (H : Type u) [NormedAddCommGroup H]
   realiza_norma :
     ‖toEvaluacion.mediaConmutador‖ = normaConmutador
 
-/-- En máxima tensión, Robertson entrega la cota evaluada en la norma del
-conmutador. -/
+/-- At maximal tension, Robertson yields the bound evaluated at the
+commutator norm. -/
 theorem MaximaTension.cota_por_norma
     {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
     (R : MaximaTension H) :
@@ -84,10 +84,10 @@ theorem MaximaTension.cota_por_norma
   rw [← R.realiza_norma]
   exact R.cota
 
-/-! ## Ancla Robertson–Schrödinger: forma cuadrática -/
+/-! ## Robertson–Schrödinger anchor: quadratic form -/
 
-/-- Evaluación cuadrática Robertson–Schrödinger: el producto de dispersiones
-domina el piso cuadrático compuesto por covarianza y conmutador. -/
+/-- Quadratic Robertson–Schrödinger evaluation: the dispersion product
+dominates the quadratic floor composed of covariance and commutator. -/
 structure EvaluacionSchrodinger where
   /-- Standard deviation of the first observable. -/
   sigmaA : ℝ
@@ -101,11 +101,11 @@ structure EvaluacionSchrodinger where
   sigmaB_nonneg : 0 ≤ sigmaB
   cota_cuadratica : covarianza ^ 2 + conmutador ^ 2 ≤ sigmaA ^ 2 * sigmaB ^ 2
 
-/-- Piso Robertson–Schrödinger: raíz cuadrada del término cuadrático. -/
+/-- Robertson–Schrödinger floor: square root of the quadratic term. -/
 noncomputable def pisoSchrodinger (S : EvaluacionSchrodinger) : ℝ :=
   Real.sqrt (S.covarianza ^ 2 + S.conmutador ^ 2)
 
-/-- Saturación Robertson–Schrödinger exacta. -/
+/-- Exact Robertson–Schrödinger saturation. -/
 def SaturadaSchrodinger (S : EvaluacionSchrodinger) : Prop :=
   S.sigmaA ^ 2 * S.sigmaB ^ 2 = S.covarianza ^ 2 + S.conmutador ^ 2
 
@@ -115,8 +115,8 @@ theorem saturadaSchrodinger_iff (S : EvaluacionSchrodinger) :
         S.covarianza ^ 2 + S.conmutador ^ 2 := by
   rfl
 
-/-- El piso Robertson–Schrödinger es positivo si y sólo si covarianza o
-conmutador son no nulos. -/
+/-- The Robertson–Schrödinger floor is positive iff the covariance or
+the commutator is nonzero. -/
 theorem pisoSchrodinger_pos_iff (S : EvaluacionSchrodinger) :
     0 < pisoSchrodinger S ↔ S.covarianza ≠ 0 ∨ S.conmutador ≠ 0 := by
   rw [pisoSchrodinger, Real.sqrt_pos]
@@ -129,8 +129,8 @@ theorem pisoSchrodinger_pos_iff (S : EvaluacionSchrodinger) :
     · nlinarith [sq_pos_of_ne_zero hcov, sq_nonneg S.conmutador]
     · nlinarith [sq_nonneg S.covarianza, sq_pos_of_ne_zero hcomm]
 
-/-- La cota cuadrática Robertson–Schrödinger implica la cota lineal sobre el
-producto de dispersiones no negativas. -/
+/-- The quadratic Robertson–Schrödinger bound implies the linear bound
+on the nonnegative dispersion product. -/
 theorem pisoSchrodinger_le_producto (S : EvaluacionSchrodinger) :
     pisoSchrodinger S ≤ S.sigmaA * S.sigmaB := by
   have hsum : 0 ≤ S.covarianza ^ 2 + S.conmutador ^ 2 := by positivity
@@ -146,8 +146,8 @@ theorem pisoSchrodinger_le_producto (S : EvaluacionSchrodinger) :
   rw [hprod_sq] at hcota
   nlinarith
 
-/-- Forma limpia del ancla: Robertson–Schrödinger aporta una cota lineal y no
-permite afirmar simultáneamente esa cota y su negación. -/
+/-- Clean anchor form: Robertson–Schrödinger yields a linear bound and
+does not allow asserting the bound and its negation simultaneously. -/
 theorem anclaSchrodinger_limpia (S : EvaluacionSchrodinger) :
     pisoSchrodinger S ≤ S.sigmaA * S.sigmaB ∧
     ¬ (pisoSchrodinger S ≤ S.sigmaA * S.sigmaB ∧
@@ -158,17 +158,17 @@ theorem anclaSchrodinger_limpia (S : EvaluacionSchrodinger) :
 
 end Robertson1929
 
-/-! ## Cinco lemas aritméticos elementales (`Blindaje`)
+/-! ## Five elementary arithmetic lemmas (`Blindaje` / Shielding)
 
-Usados más adelante por el teorema de Niven (`D7_Niven.lean`): R3 acota el
-coseno para `d ≥ 5`; R5 es la observación aritmética de que una cota
-estrictamente positiva impide que cualquiera de sus dos factores sea nulo. -/
+Used later by the Niven theorem (`D7_Niven.lean`): R3 bounds the
+cosine for `d ≥ 5`; R5 is the arithmetic observation that a strictly
+positive bound prevents either factor from vanishing. -/
 
 open Real Finset
 
 namespace Blindaje
 
-/-- Identidad término a término, exacta en ℚ. -/
+/-- Term-by-term identity, exact over ℚ. -/
 theorem R1b_termino (k : ℕ) (hk : 1 ≤ k) :
     (1 : ℚ) / k ^ 2 - 1 / (k * (k + 1)) = 1 / (k ^ 2 * (k + 1)) := by
   have hk0 : (k : ℚ) ≠ 0 := Nat.cast_ne_zero.mpr (by omega)
@@ -176,7 +176,7 @@ theorem R1b_termino (k : ℕ) (hk : 1 ≤ k) :
   field_simp
   ring
 
-/-- El telescopio cierra exacto: `Σ_{k=2..N} 1/(k(k+1)) = 1/2 − 1/(N+1)`. -/
+/-- Exact telescoping: `Σ_{k=2..N} 1/(k(k+1)) = 1/2 − 1/(N+1)`. -/
 theorem R1a_telescopio (N : ℕ) (hN : 2 ≤ N) :
     ∑ k ∈ Icc 2 N, (1 : ℚ) / (k * (k + 1)) = 1 / 2 - 1 / (N + 1) := by
   induction N with
@@ -199,7 +199,7 @@ theorem R1d_modo_positivo (k : ℕ) (hk : 2 ≤ k) :
   have : (0 : ℚ) < k := by exact_mod_cast (by omega : 0 < k)
   positivity
 
-/-- Si `(3+√5)/8 = 3/4` entonces `√5 = 3`, entonces `5 = 9`: absurdo. -/
+/-- If `(3+√5)/8 = 3/4` then `√5 = 3`, then `5 = 9`: absurd. -/
 theorem R2_cinco_no_es_nueve : (3 + Real.sqrt 5) / 8 ≠ 3 / 4 := by
   intro h
   have h3 : Real.sqrt 5 = 3 := by linarith
@@ -209,7 +209,7 @@ theorem R2_cinco_no_es_nueve : (3 + Real.sqrt 5) / 8 ≠ 3 / 4 := by
     linarith [this]
   norm_num at h5
 
-/-- Techo del coseno: para `d ≥ 5`, `cos²(π/(d+1)) < (d−1)/4`. -/
+/-- Cosine ceiling: for `d ≥ 5`, `cos²(π/(d+1)) < (d−1)/4`. -/
 theorem R3_techo_coseno (d : ℕ) (hd : 5 ≤ d) :
     Real.cos (π / (d + 1)) ^ 2 < (d - 1 : ℝ) / 4 := by
   have hd1 : (0 : ℝ) < (d : ℝ) + 1 := by positivity
@@ -240,8 +240,8 @@ theorem R4a_siete_fracciones :
 
 theorem R4_pi_mayor_que_tres : (3 : ℝ) < π := Real.pi_gt_three
 
-/-- Obstrucción aritmética: si la cota `c` es estrictamente positiva y
-`c ≤ α·β`, entonces ninguno de los dos factores puede anularse. -/
+/-- Arithmetic obstruction: if the bound `c` is strictly positive and
+`c ≤ α·β`, then neither factor can vanish. -/
 theorem R5_obstruccion_aritmetica (var_A var_B cota_robertson : ℝ)
     (h_robertson : cota_robertson ≤ var_A * var_B)
     (h_cota_positiva : 0 < cota_robertson) :
@@ -256,17 +256,17 @@ theorem R5_obstruccion_aritmetica (var_A var_B cota_robertson : ℝ)
 
 end Blindaje
 
-/-! ## Cierre del puente: Cauchy–Schwarz ⇒ Robertson–Schrödinger -/
+/-! ## Bridge closure: Cauchy–Schwarz ⇒ Robertson–Schrödinger -/
 
 noncomputable section
 
 namespace ObstruccionGramUnificada
 
-/-- Puente con `Robertson1929`: cualquier par de vectores en un espacio de
-Hilbert complejo produce una `EvaluacionSchrodinger` cuya cota cuadrática no
-se postula como campo libre — se deriva del defecto de Gram no negativo
-(`gramDefectC_nonneg`). La hipótesis `cota_cuadratica` que
-`Robertson1929.EvaluacionSchrodinger` exige como dato queda aquí demostrada. -/
+/-- Bridge to `Robertson1929`: any pair of vectors in a complex Hilbert
+space yields an `EvaluacionSchrodinger` whose quadratic bound is not
+postulated — it is derived from the nonneg Gram defect
+(`gramDefectC_nonneg`). The `cota_cuadratica` hypothesis required by
+`Robertson1929.EvaluacionSchrodinger` is proved here. -/
 def evaluacionSchrodingerDeGram {H : Type*} [NormedAddCommGroup H]
     [InnerProductSpace ℂ H] (x y : H) :
     Robertson1929.EvaluacionSchrodinger where
@@ -280,10 +280,10 @@ def evaluacionSchrodingerDeGram {H : Type*} [NormedAddCommGroup H]
     have h := robertsonSchrodinger_from_gram x y
     simpa [varianceC] using h
 
-/-- El piso Robertson–Schrödinger de la evaluación construida por Gram queda
-dominado por el producto de normas: la misma conclusión de
-`Robertson1929.pisoSchrodinger_le_producto`, instanciada sobre una evaluación
-que ya no es un supuesto sino un teorema. -/
+/-- The Robertson–Schrödinger floor of the Gram-constructed evaluation
+is dominated by the norm product: the same conclusion as
+`Robertson1929.pisoSchrodinger_le_producto`, instantiated on an
+evaluation that is now a theorem, not an assumption. -/
 theorem pisoSchrodinger_evaluacionSchrodingerDeGram_le
     {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] (x y : H) :
     Robertson1929.pisoSchrodinger (evaluacionSchrodingerDeGram x y) ≤ ‖x‖ * ‖y‖ := by
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D3_GrafoCamino.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D3_GrafoCamino.lean
index 7708320eb..5d63ca809 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D3_GrafoCamino.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D3_GrafoCamino.lean
@@ -9,21 +9,19 @@ public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D0_Habitat
 public import Mathlib.Combinatorics.SimpleGraph.Hasse
 
 /-!
-# D3 — Operadores de transporte y posición sobre el grafo camino
-
-El soporte discreto no introduce un grafo ad hoc: es literalmente
-`SimpleGraph.pathGraph d` de Mathlib, el camino con `d` vértices
-`0,1,…,d−1` y aristas sólo entre vecinos consecutivos. Sobre esa base se
-definen dos matrices hermitianas: `T_d` (transporte, soportado en las
-aristas del camino) y `P_d` (posición, diagonal, con coordenadas
-centradas en `[-1,1]`).
-
-La segunda mitad del archivo (`CanalPreFuerza`) prueba, sin apelar a
-ninguna elección de diseño, que el grafo camino es la **única** opción
-compatible con dos condiciones puramente combinatorias: localidad
-(ninguna arista salta vecinos) y completitud (no falta ningún paso
-elemental). Cualquier grafo local en `Fin d` que no omita un paso mínimo
-**es** `pathGraph d`; no hay otro candidato.
+# D3 — Transport and position operators on the path graph
+
+The discrete support is not an ad-hoc graph: it is literally
+`SimpleGraph.pathGraph d` from Mathlib, the path on `d` vertices
+`0,1,…,d−1` with edges only between consecutive neighbours. Two
+Hermitian matrices are built on it: `T_d` (transport, supported on
+path edges) and `P_d` (position, diagonal, centered in `[-1,1]`).
+
+The second half (`CanalPreFuerza`) proves, without any design choice,
+that the path graph is the **only** option compatible with two purely
+combinatorial conditions: locality (no edge skips a neighbour) and
+completeness (no elementary step is missing). Any local graph on
+`Fin d` that omits no minimal step **is** `pathGraph d`.
 -/
 
 @[expose] public section
@@ -32,19 +30,18 @@ namespace TransportePosicion
 
 open SimpleGraph
 
-/-- Grafo de fase del canal transporte–posición: el camino `pathGraph d`
-de Mathlib. -/
+/-- Phase graph of the transport–position channel: Mathlib's
+`pathGraph d`. -/
 abbrev GrafoTP (d : ℕ) : SimpleGraph (Fin d) :=
   SimpleGraph.pathGraph d
 
-/-- Adyacencia elemental del camino: sólo hay desplazamiento mínimo de un
-paso. -/
+/-- Elementary path adjacency: only single minimal-step displacement. -/
 theorem grafoTP_adj {d : ℕ} {i j : Fin d} :
     (GrafoTP d).Adj i j ↔ i.val + 1 = j.val ∨ j.val + 1 = i.val := by
   simpa [GrafoTP] using
     (SimpleGraph.pathGraph_adj (n := d) (u := i) (v := j))
 
-/-- Predicado decidible del desplazamiento mínimo en la línea discreta. -/
+/-- Decidable minimal-displacement predicate on the discrete line. -/
 def PasoMinimo {d : ℕ} (i j : Fin d) : Prop :=
   i.val + 1 = j.val ∨ j.val + 1 = i.val
 
@@ -53,37 +50,37 @@ instance instDecidablePasoMinimo {d : ℕ} (i j : Fin d) :
   unfold PasoMinimo
   infer_instance
 
-/-- El paso mínimo decidible es exactamente la adyacencia de `pathGraph`. -/
+/-- The decidable minimal step is exactly `pathGraph` adjacency. -/
 theorem pasoMinimo_iff_adj {d : ℕ} {i j : Fin d} :
     PasoMinimo i j ↔ (GrafoTP d).Adj i j := by
   rw [grafoTP_adj]
   rfl
 
-/-- El soporte discreto `T_d/P_d` es isomorfo a `pathGraph d` por definición
-canónica. -/
+/-- The discrete support `T_d/P_d` is isomorphic to `pathGraph d` by
+canonical definition. -/
 theorem grafoTP_es_pathGraph (d : ℕ) :
     Nonempty (GrafoTP d ≃g SimpleGraph.pathGraph d) := by
   change Nonempty (SimpleGraph.pathGraph d ≃g SimpleGraph.pathGraph d)
   exact ⟨SimpleGraph.Iso.refl⟩
 
-/-- Matriz de adyacencia compleja del canal de transporte. -/
+/-- Complex adjacency matrix of the transport channel. -/
 noncomputable def Ad (d : ℕ) : Matrix (Fin d) (Fin d) ℂ :=
   fun i j => if PasoMinimo i j then 1 else 0
 
-/-- Radio espectral usado para normalizar el transporte de la cadena
-finita: `ρ_d = 2 cos(π/(d+1))`. -/
+/-- Spectral radius for normalizing the finite-chain transport:
+`ρ_d = 2 cos(π/(d+1))`. -/
 noncomputable def rho (d : ℕ) : ℝ :=
   2 * Real.cos (Real.pi / ((d : ℝ) + 1))
 
-/-- Operador de transporte normalizado `T_d = A_d / ρ_d`. -/
+/-- Normalized transport operator `T_d = A_d / ρ_d`. -/
 noncomputable def Td (d : ℕ) : Matrix (Fin d) (Fin d) ℂ :=
   fun i j => Ad d i j / (rho d : ℂ)
 
-/-- Coordenada centrada de posición sobre la base discreta, en `[-1,1]`. -/
+/-- Centered position coordinate on the discrete basis, in `[-1,1]`. -/
 noncomputable def posicionCoord (d : ℕ) (j : Fin d) : ℝ :=
   (2 * ((j.val : ℝ) + 1) - ((d : ℝ) + 1)) / ((d : ℝ) - 1)
 
-/-- Operador de posición diagonal `P_d`. -/
+/-- Diagonal position operator `P_d`. -/
 noncomputable def Pd (d : ℕ) : Matrix (Fin d) (Fin d) ℂ :=
   fun i j => if i = j then (posicionCoord d i : ℂ) else 0
 
@@ -103,12 +100,12 @@ theorem Td_eq_zero_of_not_adj {d : ℕ} {i j : Fin d}
     exact h (pasoMinimo_iff_adj.mp hp)
   simp [Td, Ad, hpaso]
 
-/-- `P_d` es diagonal en la base discreta. -/
+/-- `P_d` is diagonal in the discrete basis. -/
 theorem Pd_eq_zero_offdiag {d : ℕ} {i j : Fin d} (hij : i ≠ j) :
     Pd d i j = 0 := by
   simp [Pd, hij]
 
-/-- En la diagonal, `P_d` devuelve la coordenada discreta centrada. -/
+/-- On the diagonal, `P_d` returns the centered discrete coordinate. -/
 theorem Pd_diag {d : ℕ} (i : Fin d) :
     Pd d i i = (posicionCoord d i : ℂ) := by
   simp [Pd]
@@ -116,52 +113,52 @@ theorem Pd_diag {d : ℕ} (i : Fin d) :
 end TransportePosicion
 
 /-!
-## Por qué `pathGraph d` y no otro grafo
+## Why `pathGraph d` and not another graph
 
-Un canal local (toda arista es un paso mínimo entre vecinos) y completo
-(no falta ningún paso mínimo posible) sobre `Fin d` es, por extensionalidad
-de la relación de adyacencia, exactamente `pathGraph d`. No hay una
-"simplificación" que conserve ambas propiedades: quitar una arista rompe
-la completitud.
+A local channel (every edge is a minimal step between neighbours) and
+complete (no possible minimal step is missing) on `Fin d` is, by
+extensionality of the adjacency relation, exactly `pathGraph d`. There
+is no "simplification" preserving both properties: removing an edge
+breaks completeness.
 -/
 
 namespace CanalPreFuerza
 
 open SimpleGraph
 
-/-- Localidad estricta: toda arista del canal es un paso entre vecinos
-consecutivos. No se permiten saltos ni atajos. -/
+/-- Strict locality: every channel edge is a step between consecutive
+neighbours. No jumps or shortcuts allowed. -/
 def LocalidadOrdenada {d : ℕ} (G : SimpleGraph (Fin d)) : Prop :=
   ∀ {i j : Fin d}, G.Adj i j → TransportePosicion.PasoMinimo i j
 
-/-- Completitud: todo paso entre vecinos consecutivos debe estar presente.
-Quitar uno rompe el movimiento local completo entre los extremos de la
-celda. -/
+/-- Completeness: every step between consecutive neighbours must be
+present. Removing one breaks full local movement between the cell
+endpoints. -/
 def PasosElementalesCompletos {d : ℕ} (G : SimpleGraph (Fin d)) : Prop :=
   ∀ {i j : Fin d}, TransportePosicion.PasoMinimo i j → G.Adj i j
 
-/-- Defecto por intentar simplificar más que `pathGraph d`: se omite al
-menos un paso elemental consecutivo. -/
+/-- Defect from trying to simplify beyond `pathGraph d`: at least one
+consecutive elementary step is omitted. -/
 def OmitePasoElemental {d : ℕ} (G : SimpleGraph (Fin d)) : Prop :=
   ∃ i j : Fin d, TransportePosicion.PasoMinimo i j ∧ ¬ G.Adj i j
 
-/-- Canal ordenado, local y completo en la celda discreta. -/
+/-- Ordered, local and complete channel on the discrete cell. -/
 structure CanalLocalNoRamificadoOrdenado (d : ℕ) where
   /-- The graph supporting the ordered local channel. -/
   grafo : SimpleGraph (Fin d)
   localidad_ordenada : LocalidadOrdenada grafo
   pasos_elementales : PasosElementalesCompletos grafo
 
-/-- En un canal local ordenado completo, la adyacencia es exactamente el
-paso mínimo de la celda. -/
+/-- In a complete ordered local channel, adjacency is exactly the
+minimal step of the cell. -/
 theorem CanalLocalNoRamificadoOrdenado.adj_iff_paso
     {d : ℕ} (C : CanalLocalNoRamificadoOrdenado d) {i j : Fin d} :
     C.grafo.Adj i j ↔ TransportePosicion.PasoMinimo i j := by
   exact ⟨fun h => C.localidad_ordenada h,
     fun h => C.pasos_elementales h⟩
 
-/-- Teorema de minimalidad: localidad ordenada y pasos elementales
-completos fuerzan que el soporte sea exactamente `pathGraph d`. -/
+/-- Minimality theorem: ordered locality and complete elementary steps
+force the support to be exactly `pathGraph d`. -/
 theorem canal_local_no_ramificado_es_pathGraph
     {d : ℕ} (C : CanalLocalNoRamificadoOrdenado d) :
     C.grafo = SimpleGraph.pathGraph d := by
@@ -170,15 +167,15 @@ theorem canal_local_no_ramificado_es_pathGraph
   simpa [TransportePosicion.GrafoTP] using
     (TransportePosicion.pasoMinimo_iff_adj (d := d) (i := i) (j := j))
 
-/-- Versión isomórfica del mismo cierre. -/
+/-- Isomorphic version of the same closure. -/
 theorem canal_local_no_ramificado_iso_pathGraph
     {d : ℕ} (C : CanalLocalNoRamificadoOrdenado d) :
     Nonempty (C.grafo ≃g SimpleGraph.pathGraph d) := by
   rw [canal_local_no_ramificado_es_pathGraph C]
   exact ⟨SimpleGraph.Iso.refl⟩
 
-/-- El canal canónico `T_d/P_d` satisface directamente el certificado local
-ordenado: no tiene saltos y no omite pasos elementales. -/
+/-- The canonical channel `T_d/P_d` directly satisfies the ordered local
+certificate: it has no jumps and omits no elementary steps. -/
 def canalTPLocalNoRamificado (d : ℕ) :
     CanalLocalNoRamificadoOrdenado d where
   grafo := TransportePosicion.GrafoTP d
@@ -191,8 +188,8 @@ def canalTPLocalNoRamificado (d : ℕ) :
     exact (TransportePosicion.pasoMinimo_iff_adj
       (d := d) (i := i) (j := j)).mp h
 
-/-- Ningún canal que ya satisface el certificado local ordenado puede omitir
-un paso elemental: "no hay una simplificación local más simple que
+/-- No channel that already satisfies the ordered local certificate can
+omit an elementary step: "there is no local simplification simpler than
 `pathGraph d`". -/
 theorem no_hay_canal_local_mas_simple_que_Pd
     {d : ℕ} (C : CanalLocalNoRamificadoOrdenado d) :
@@ -200,8 +197,8 @@ theorem no_hay_canal_local_mas_simple_que_Pd
   rintro ⟨i, j, hpaso, hno⟩
   exact hno (C.pasos_elementales hpaso)
 
-/-- Cierre: el soporte canónico `T_d/P_d` es `pathGraph d`, y cualquier
-intento local de hacerlo "más simple" pierde un paso elemental. -/
+/-- Closure: the canonical support `T_d/P_d` is `pathGraph d`, and any
+local attempt to make it "simpler" loses an elementary step. -/
 theorem cierre_minimalidad_local_TP (d : ℕ) :
     (canalTPLocalNoRamificado d).grafo = SimpleGraph.pathGraph d ∧
       ¬ OmitePasoElemental (canalTPLocalNoRamificado d).grafo := by
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D4_PorQueNoDiagonal.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D4_PorQueNoDiagonal.lean
index c28642ceb..30fd193dc 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D4_PorQueNoDiagonal.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D4_PorQueNoDiagonal.lean
@@ -8,26 +8,27 @@ module
 public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D3_GrafoCamino
 
 /-!
-# D4 — Por qué no un paso diagonal
-
-`D3_GrafoCamino.lean` fija el soporte de `T_d` en los pasos mínimos
-(vecino a vecino) del camino `pathGraph d`. Este archivo cierra, con dos
-argumentos independientes, la pregunta de por qué nunca se considera un
-paso "diagonal" (cambiar más de una coordenada a la vez) como alternativa:
-
-1. **Si hubiera ≥ 2 ejes genuinos** (el modelo se generalizara a una malla
-   cúbica `Fin dx × Fin dy × Fin dz`), Pitágoras decide: el paso ortogonal
-   (un solo eje) tiene distancia euclidiana exactamente `1`; el paso
-   diagonal doble, exactamente `√2`; el triple, exactamente `√3`. Como
-   `1 < √2` y `1 < √3`, el paso ortogonal es siempre estrictamente más
-   corto. No es una preferencia de diseño: es la adyacencia mínima que
-   Pitágoras obliga.
-2. **En el caso efectivamente usado por `T_d/P_d`** (un solo eje, `dy = dz = 1`),
-   la pregunta ni siquiera se plantea: con un único eje genuino la propia
-   relación "paso diagonal" es la relación **vacía** — no existe ningún par
-   de sitios que la satisfaga, porque un eje trivial (`Fin 1`) no tiene
-   ningún paso mínimo posible. La diagonal presupone, para poder
-   enunciarse de forma no vacía, dos ejes ya distinguidos entre sí.
+# D4 — Why not a diagonal step
+
+`D3_GrafoCamino.lean` pins the support of `T_d` on the minimal
+(nearest-neighbour) steps of `pathGraph d`. This file closes, with
+two independent arguments, the question of why a "diagonal" step
+(changing more than one coordinate at once) is never an alternative:
+
+1. **If there were ≥ 2 genuine axes** (generalising to a cubic lattice
+   `Fin dx × Fin dy × Fin dz`), Pythagoras decides: the orthogonal
+   step (one axis) has Euclidean distance exactly `1`; the double
+   diagonal, exactly `√2`; the triple, exactly `√3`. Since
+   `1 < √2` and `1 < √3`, the orthogonal step is always strictly
+   shorter. This is not a design choice: it is the minimal
+   adjacency that Pythagoras forces.
+2. **In the case actually used by `T_d/P_d`** (a single axis,
+   `dy = dz = 1`), the question does not even arise: with one
+   genuine axis the "diagonal step" relation is the **empty**
+   relation — no pair of sites satisfies it, because a trivial
+   axis (`Fin 1`) has no minimal step at all. The diagonal
+   presupposes, in order to be non-vacuously stated, two axes
+   already distinguished from each other.
 -/
 
 @[expose] public section
@@ -36,11 +37,11 @@ noncomputable section
 
 namespace PathGraph3D
 
-/-- Sitio de una malla cúbica: producto de tres mallas 1D. -/
+/-- Site of a cubic lattice: product of three 1D lattices. -/
 abbrev Sitio3D (dx dy dz : ℕ) := Fin dx × Fin dy × Fin dz
 
-/-- Adyacencia ortogonal del cubo: cambia una sola coordenada a la vez, por
-un paso mínimo en ese eje. -/
+/-- Orthogonal adjacency of the cube: changes exactly one coordinate
+at a time by a minimal step along that axis. -/
 def Adj3D {dx dy dz : ℕ} (p q : Sitio3D dx dy dz) : Prop :=
   (TransportePosicion.PasoMinimo p.1 q.1 ∧ p.2 = q.2) ∨
     (p.1 = q.1 ∧ TransportePosicion.PasoMinimo p.2.1 q.2.1 ∧ p.2.2 = q.2.2) ∨
@@ -48,23 +49,23 @@ def Adj3D {dx dy dz : ℕ} (p q : Sitio3D dx dy dz) : Prop :=
 
 end PathGraph3D
 
-/-! ## 1. Pitágoras: el paso ortogonal es siempre estrictamente más corto -/
+/-! ## 1. Pythagoras: the orthogonal step is always strictly shorter -/
 
 namespace OrtogonalidadMinimalPitagoras
 
 open PathGraph3D
 open TransportePosicion
 
-/-- Distancia euclidiana entre dos sitios del cubo, viendo cada coordenada
-`Fin` como un real vía el casteo natural. -/
+/-- Euclidean distance between two cube sites, viewing each `Fin`
+coordinate as a real via the natural cast. -/
 noncomputable def dist3D {dx dy dz : ℕ} (p q : Sitio3D dx dy dz) : ℝ :=
   Real.sqrt (
     ((p.1 : ℝ) - (q.1 : ℝ)) ^ 2 +
     ((p.2.1 : ℝ) - (q.2.1 : ℝ)) ^ 2 +
     ((p.2.2 : ℝ) - (q.2.2 : ℝ)) ^ 2)
 
-/-- Dos coordenadas a un `PasoMinimo` difieren, como reales, en exactamente
-`±1`; su diferencia al cuadrado es `1`. -/
+/-- Two coordinates at a `PasoMinimo` differ, as reals, by exactly
+`±1`; their squared difference is `1`. -/
 theorem pasoMinimo_sq_diff_eq_one {d : ℕ} {i j : Fin d} (h : PasoMinimo i j) :
     ((i.val : ℝ) - (j.val : ℝ)) ^ 2 = 1 := by
   rcases h with h | h
@@ -77,7 +78,7 @@ theorem eq_sq_diff_eq_zero {d : ℕ} {i j : Fin d} (h : i = j) :
     ((i.val : ℝ) - (j.val : ℝ)) ^ 2 = 0 := by
   rw [h]; ring
 
-/-! ### Paso ortogonal: distancia exactamente `1` -/
+/-! ### Orthogonal step: distance exactly `1` -/
 
 theorem dist3D_eq_one_of_Adj3D
     {dx dy dz : ℕ} {p q : Sitio3D dx dy dz} (h : Adj3D p q) :
@@ -108,10 +109,11 @@ theorem dist3D_eq_one_of_Adj3D
       rw [hx0, hy0, pasoMinimo_sq_diff_eq_one hz]; ring
     rw [hsum, Real.sqrt_one]
 
-/-! ### Paso diagonal doble: distancia exactamente `√2` -/
+/-! ### Double diagonal step: distance exactly `√2` -/
 
-/-- Vecino diagonal en dos ejes: dos coordenadas cambian por `PasoMinimo`
-simultáneamente, la tercera queda fija. `Adj3D` nunca produce este caso. -/
+/-- Diagonal neighbour in two axes: two coordinates change by
+`PasoMinimo` simultaneously, the third stays fixed. `Adj3D`
+never produces this case. -/
 def PasoDiagonalDoble {dx dy dz : ℕ} (p q : Sitio3D dx dy dz) : Prop :=
   (PasoMinimo p.1 q.1 ∧ PasoMinimo p.2.1 q.2.1 ∧ p.2.2 = q.2.2) ∨
   (PasoMinimo p.1 q.1 ∧ p.2.1 = q.2.1 ∧ PasoMinimo p.2.2 q.2.2) ∨
@@ -141,9 +143,9 @@ theorem dist3D_eq_sqrt_two_of_PasoDiagonalDoble
       rw [hx0, pasoMinimo_sq_diff_eq_one hy, pasoMinimo_sq_diff_eq_one hz]; ring
     rw [hsum]
 
-/-! ### Paso diagonal triple: distancia exactamente `√3` -/
+/-! ### Triple diagonal step: distance exactly `√3` -/
 
-/-- Vecino diagonal en los tres ejes (la esquina del cubo unitario). -/
+/-- Diagonal neighbour in all three axes (the unit cube corner). -/
 def PasoDiagonalTriple {dx dy dz : ℕ} (p q : Sitio3D dx dy dz) : Prop :=
   PasoMinimo p.1 q.1 ∧ PasoMinimo p.2.1 q.2.1 ∧ PasoMinimo p.2.2 q.2.2
 
@@ -159,7 +161,7 @@ theorem dist3D_eq_sqrt_three_of_PasoDiagonalTriple
       pasoMinimo_sq_diff_eq_one hz]; ring
   rw [hsum]
 
-/-! ### Cierre Pitágoras: el ortogonal gana siempre -/
+/-! ### Pythagoras closure: the orthogonal step always wins -/
 
 theorem uno_lt_sqrt_two : (1 : ℝ) < Real.sqrt 2 := by
   have h2 : Real.sqrt 2 ^ 2 = 2 := Real.sq_sqrt (by norm_num)
@@ -169,8 +171,8 @@ theorem uno_lt_sqrt_three : (1 : ℝ) < Real.sqrt 3 := by
   have h3 : Real.sqrt 3 ^ 2 = 3 := Real.sq_sqrt (by norm_num)
   nlinarith [Real.sqrt_nonneg (3 : ℝ), h3]
 
-/-- El paso ortogonal (el único que admite `Adj3D`) es estrictamente más
-corto que cualquier paso diagonal doble. -/
+/-- The orthogonal step (the only one admitted by `Adj3D`) is
+strictly shorter than any double diagonal step. -/
 theorem ortogonal_mas_corto_que_diagonal_doble
     {dx dy dz : ℕ} {p q p' q' : Sitio3D dx dy dz}
     (hOrt : Adj3D p q) (hDiag : PasoDiagonalDoble p' q') :
@@ -178,8 +180,8 @@ theorem ortogonal_mas_corto_que_diagonal_doble
   rw [dist3D_eq_one_of_Adj3D hOrt, dist3D_eq_sqrt_two_of_PasoDiagonalDoble hDiag]
   exact uno_lt_sqrt_two
 
-/-- El paso ortogonal es estrictamente más corto que cualquier paso
-diagonal triple (la esquina del cubo). -/
+/-- The orthogonal step is strictly shorter than any triple
+diagonal step (the cube corner). -/
 theorem ortogonal_mas_corto_que_diagonal_triple
     {dx dy dz : ℕ} {p q p' q' : Sitio3D dx dy dz}
     (hOrt : Adj3D p q) (hDiag : PasoDiagonalTriple p' q') :
@@ -187,10 +189,11 @@ theorem ortogonal_mas_corto_que_diagonal_triple
   rw [dist3D_eq_one_of_Adj3D hOrt, dist3D_eq_sqrt_three_of_PasoDiagonalTriple hDiag]
   exact uno_lt_sqrt_three
 
-/-- CIERRE. Ninguna diagonal (doble o triple) puede empatar o vencer en
-distancia al paso ortogonal: la adyacencia mínima del cubo (`Adj3D`) es la
-única compatible con minimalidad de distancia euclidiana. No es una
-elección arbitraria: es la que Pitágoras obliga. -/
+/-- CLOSURE. No diagonal (double or triple) can tie or beat the
+orthogonal step in distance: the minimal adjacency of the cube
+(`Adj3D`) is the only one compatible with Euclidean distance
+minimality. This is not an arbitrary choice: it is the one
+Pythagoras forces. -/
 theorem adyacencia_minima_es_ortogonal
     {dx dy dz : ℕ} {p q p' q' : Sitio3D dx dy dz}
     (hOrt : Adj3D p q)
@@ -202,15 +205,15 @@ theorem adyacencia_minima_es_ortogonal
 
 end OrtogonalidadMinimalPitagoras
 
-/-! ## 2. Con un solo eje, la diagonal es la relación vacía -/
+/-! ## 2. With a single axis, the diagonal is the empty relation -/
 
 namespace DiagonalPresuponeDosPd
 
 open PathGraph3D
 open OrtogonalidadMinimalPitagoras
 
-/-- En un eje trivial (`Fin 1`, un único punto) no hay ningún paso mínimo:
-`PasoMinimo` es la relación vacía. -/
+/-- On a trivial axis (`Fin 1`, a single point) there is no
+minimal step: `PasoMinimo` is the empty relation. -/
 theorem pasoMinimo_vacio_en_eje_trivial (i j : Fin 1) :
     ¬ TransportePosicion.PasoMinimo i j := by
   unfold TransportePosicion.PasoMinimo
@@ -218,8 +221,9 @@ theorem pasoMinimo_vacio_en_eje_trivial (i j : Fin 1) :
   have hj := j.isLt
   omega
 
-/-- Con un solo eje genuino (`dy = dz = 1`), `PasoDiagonalDoble` es la
-relación vacía: no hay ningún par de sitios que la satisfaga. -/
+/-- With a single genuine axis (`dy = dz = 1`),
+`PasoDiagonalDoble` is the empty relation: no pair of sites
+satisfies it. -/
 theorem diagonalDoble_vacia_con_un_solo_eje
     {dx : ℕ} (p q : Sitio3D dx 1 1) : ¬ PasoDiagonalDoble p q := by
   unfold PasoDiagonalDoble
@@ -228,19 +232,20 @@ theorem diagonalDoble_vacia_con_un_solo_eje
   · exact pasoMinimo_vacio_en_eje_trivial p.2.2 q.2.2 hz
   · exact pasoMinimo_vacio_en_eje_trivial p.2.1 q.2.1 hy
 
-/-- Con un solo eje genuino, `PasoDiagonalTriple` (la esquina del cubo)
-tampoco existe: también es la relación vacía. -/
+/-- With a single genuine axis, `PasoDiagonalTriple` (the cube
+corner) does not exist either: it is also the empty relation. -/
 theorem diagonalTriple_vacia_con_un_solo_eje
     {dx : ℕ} (p q : Sitio3D dx 1 1) : ¬ PasoDiagonalTriple p q := by
   unfold PasoDiagonalTriple
   rintro ⟨_, hy, _⟩
   exact pasoMinimo_vacio_en_eje_trivial p.2.1 q.2.1 hy
 
-/-- CIERRE. Con un solo eje, ninguna diagonal —doble ni triple— existe. La
-diagonal es, en sentido literal de teoría de conjuntos, posterior a la
-existencia de dos ejes distinguidos entre sí: no anterior, no simultánea,
-no elemental. El modelo `T_d/P_d` (un único eje) nunca necesita excluirla
-por decreto: no hay nada que excluir. -/
+/-- CLOSURE. With a single axis, no diagonal — double or triple —
+exists. The diagonal is, in the literal set-theoretic sense,
+posterior to the existence of two distinguished axes: not prior,
+not simultaneous, not elementary. The model `T_d/P_d` (a single
+axis) never needs to exclude it by decree: there is nothing to
+exclude. -/
 theorem diagonal_no_existe_con_un_solo_eje
     {dx : ℕ} (p q : Sitio3D dx 1 1) :
     ¬ (PasoDiagonalDoble p q ∨ PasoDiagonalTriple p q) := by
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D5_MaximaTension.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D5_MaximaTension.lean
index 63cd3ef33..57da463a0 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D5_MaximaTension.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D5_MaximaTension.lean
@@ -14,19 +14,20 @@ public import Mathlib.RingTheory.PicardGroup
 public import Mathlib.RingTheory.SimpleRing.Principal
 
 /-!
-# D5 — Estado de máxima tensión y observable `i[T_d,P_d]`
-
-En dimensión finita, `i[T,P]` es simétrico (hermitiano) siempre que `T` y
-`P` lo son; por el teorema espectral finito, posee una base ortonormal de
-autovectores. Este archivo elige el autovector cuyo autovalor tiene módulo
-máximo y demuestra la envolvente sobre todos los estados normalizados: ese
-estado realiza, entre todos los estados unitarios del mismo canal, la mayor
-tensión posible del conmutador. También se exhibe, en coordenadas
-explícitas (fase seno), el mismo estado extremal para el par concreto
-`(T_d,P_d)` del camino discreto: es el "modo de Fiedler" de la cadena.
-
-Se cierra con un certificado concreto de no conmutatividad:
-`[T_d,P_d] ≠ 0` para `d ≥ 2`, exhibido en una única entrada de matriz.
+# D5 — Maximal-tension state and the observable `i[T_d,P_d]`
+
+In finite dimension, `i[T,P]` is symmetric (Hermitian) whenever `T`
+and `P` are; by the finite spectral theorem it has an orthonormal
+eigenbasis. This file picks the eigenvector whose eigenvalue has
+maximal absolute value and proves the envelope over all normalised
+states: that state realises, among all unit states of the same
+channel, the greatest possible commutator tension. The same extremal
+state is then exhibited in explicit coordinates (sine-phase mode)
+for the concrete pair `(T_d,P_d)` of the discrete path: it is the
+"Fiedler mode" of the chain.
+
+Closes with a concrete non-commutativity certificate:
+`[T_d,P_d] ≠ 0` for `d ≥ 2`, witnessed by a single matrix entry.
 -/
 
 @[expose] public section
@@ -40,8 +41,8 @@ universe u
 variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
   [FiniteDimensional ℂ H] [Nontrivial H]
 
-/-- En dimensión positiva existe un índice cuyo autovalor tiene módulo
-máximo. -/
+/-- In positive dimension there exists an index whose eigenvalue
+has maximal absolute value. -/
 theorem existe_indice_extremal (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) :
     ∃ i : Fin (Module.finrank ℂ H),
       ∀ j : Fin (Module.finrank ℂ H),
@@ -63,11 +64,11 @@ def indiceExtremal (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) :
 def autovalorExtremal (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) : ℝ :=
   hK.eigenvalues rfl (indiceExtremal K hK)
 
-/-- Radio espectral realizado por el estado elegido. -/
+/-- Spectral radius realised by the chosen state. -/
 def radioEspectral (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) : ℝ :=
   |autovalorExtremal K hK|
 
-/-- Estado unitario de máxima tensión. -/
+/-- Unit state of maximal tension. -/
 def estadoExtremal (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) : H :=
   hK.eigenvectorBasis rfl (indiceExtremal K hK)
 
@@ -90,8 +91,8 @@ theorem aplica_estadoExtremal (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) :
       (autovalorExtremal K hK : ℂ) • estadoExtremal K hK := by
   exact hK.apply_eigenvectorBasis rfl (indiceExtremal K hK)
 
-/-- La acción de un operador simétrico queda acotada por el radio espectral
-elegido. -/
+/-- The action of a symmetric operator is bounded by the chosen
+spectral radius. -/
 theorem norma_aplicacion_le_radio_mul_norma
     (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) (v : H) :
     ‖K v‖ ≤ radioEspectral K hK * ‖v‖ := by
@@ -151,7 +152,7 @@ theorem norma_aplicacion_le_radio_mul_norma
     mul_nonneg (radioEspectral_nonneg K hK) (norm_nonneg _)
   nlinarith
 
-/-- Envolvente de la forma cuadrática sobre la esfera unidad. -/
+/-- Envelope of the quadratic form on the unit sphere. -/
 theorem expectativa_le_radio
     (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric)
     (v : H) (hv : ‖v‖ = 1) :
@@ -164,7 +165,7 @@ theorem expectativa_le_radio
         (norma_aplicacion_le_radio_mul_norma K hK v) (norm_nonneg _)
     _ = radioEspectral K hK := by rw [hv]; ring
 
-/-- El estado elegido realiza exactamente el radio espectral. -/
+/-- The chosen state realises exactly the spectral radius. -/
 theorem estadoExtremal_realiza_radio
     (K : H →ₗ[ℂ] H) (hK : K.IsSymmetric) :
     ‖@inner ℂ H _ (estadoExtremal K hK) (K (estadoExtremal K hK))‖ =
@@ -173,15 +174,15 @@ theorem estadoExtremal_realiza_radio
   rw [inner_self_eq_norm_sq_to_K, estadoExtremal_normalizado K hK]
   simp [radioEspectral, autovalorExtremal]
 
-/-- Conmutador crudo total `[T,P]`. -/
+/-- Raw commutator `[T,P]`. -/
 def conmutador (T P : H →ₗ[ℂ] H) : H →ₗ[ℂ] H :=
   T.comp P - P.comp T
 
-/-- Observable hermitiano de tensión `i[T,P]`. -/
+/-- Hermitian tension observable `i[T,P]`. -/
 def observableTension (T P : H →ₗ[ℂ] H) : H →ₗ[ℂ] H :=
   Complex.I • conmutador T P
 
-/-- Para operadores simétricos, `i[T,P]` es simétrico. -/
+/-- For symmetric operators, `i[T,P]` is symmetric. -/
 theorem observableTension_simetrico
     {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℂ E]
     (T P : E →ₗ[ℂ] E) (hT : T.IsSymmetric) (hP : P.IsSymmetric) :
@@ -224,11 +225,11 @@ namespace TransportePosicion
 
 open ConstructorEspectralTP
 
-/-- Realización lineal total de la matriz de transporte canónica. -/
+/-- Full linear realisation of the canonical transport matrix. -/
 noncomputable def TdOp (d : ℕ) : Hd d →ₗ[ℂ] Hd d :=
   Matrix.toEuclideanLin (Td d)
 
-/-- Realización lineal total de la matriz de posición canónica. -/
+/-- Full linear realisation of the canonical position matrix. -/
 noncomputable def PdOp (d : ℕ) : Hd d →ₗ[ℂ] Hd d :=
   Matrix.toEuclideanLin (Pd d)
 
@@ -240,7 +241,7 @@ theorem pasoMinimo_simetrico {d : ℕ} {i j : Fin d} :
   · exact Or.inr h
   · exact Or.inl h
 
-/-- La matriz de transporte es hermitiana. -/
+/-- The transport matrix is Hermitian. -/
 theorem Td_isHermitian (d : ℕ) : Matrix.IsHermitian (Td d) := by
   rw [Matrix.IsHermitian.ext_iff]
   intro i j
@@ -255,7 +256,7 @@ theorem Td_isHermitian (d : ℕ) : Matrix.IsHermitian (Td d) := by
       exact h (pasoMinimo_simetrico.mpr hji)
     simp [Td, Ad, h, h']
 
-/-- La matriz diagonal de posición es hermitiana. -/
+/-- The diagonal position matrix is Hermitian. -/
 theorem Pd_isHermitian (d : ℕ) : Matrix.IsHermitian (Pd d) := by
   rw [Matrix.IsHermitian.ext_iff]
   intro i j
@@ -271,7 +272,7 @@ theorem TdOp_simetrico (d : ℕ) : (TdOp d).IsSymmetric := by
 theorem PdOp_simetrico (d : ℕ) : (PdOp d).IsSymmetric := by
   exact Matrix.isSymmetric_toEuclideanLin_iff.mpr (Pd_isHermitian d)
 
-/-- Observable hermitiano concreto `i[T_d,P_d]`. -/
+/-- Concrete Hermitian observable `i[T_d,P_d]`. -/
 noncomputable def KdOp (d : ℕ) : Hd d →ₗ[ℂ] Hd d :=
   observableTension (TdOp d) (PdOp d)
 
@@ -279,8 +280,8 @@ theorem KdOp_simetrico (d : ℕ) : (KdOp d).IsSymmetric :=
   observableTension_simetrico (TdOp d) (PdOp d)
     (TdOp_simetrico d) (PdOp_simetrico d)
 
-/-- Estado canónico `ψ_d`: autovector unitario de `i[T_d,P_d]` cuyo
-autovalor tiene módulo máximo. -/
+/-- Canonical state `ψ_d`: unit eigenvector of `i[T_d,P_d]` whose
+eigenvalue has maximal absolute value. -/
 noncomputable def psiD (d : ℕ) (hd : 1 ≤ d) : Hd d := by
   letI : Nonempty (Fin d) := ⟨⟨0, hd⟩⟩
   exact estadoExtremal (KdOp d) (KdOp_simetrico d)
@@ -290,26 +291,26 @@ theorem psiD_normalizado (d : ℕ) (hd : 1 ≤ d) :
   letI : Nonempty (Fin d) := ⟨⟨0, hd⟩⟩
   exact estadoExtremal_normalizado (KdOp d) (KdOp_simetrico d)
 
-/-! ## Vector de Fiedler explícito
+/-! ## Explicit Fiedler vector
 
-La elección espectral anterior realiza la máxima tensión, pero no expone sus
-coordenadas. El modo siguiente fija la realización seno-fase sobre `Fin d`.
-La positividad de `sin` en `(0, π)` prueba constructivamente que ninguna
-coordenada desaparece. -/
+The spectral choice above realises the maximal tension but does not
+expose its coordinates. The following mode pins the sine-phase
+realisation on `Fin d`. Positivity of `sin` on `(0, π)`
+constructively proves that no coordinate vanishes. -/
 
-/-- Ángulo fundamental del camino finito. -/
+/-- Fundamental angle of the finite path. -/
 noncomputable def anguloFiedler (d : ℕ) : ℝ :=
   Real.pi / ((d : ℝ) + 1)
 
-/-- Modo seno con la fase compleja asociada a `i[T_d,P_d]`, todavía sin
-normalizar. -/
+/-- Sine mode with the complex phase of `i[T_d,P_d]`, not yet
+normalized. -/
 noncomputable def vectorFiedlerCrudo (d : ℕ) : Hd d :=
   WithLp.toLp 2 fun j : Fin d =>
     (-Complex.I) ^ j.val *
       (Real.sin (((j.val : ℝ) + 1) * anguloFiedler d) : ℂ)
 
-/-- Todas las amplitudes seno del modo fundamental son estrictamente
-positivas. -/
+/-- All sine amplitudes of the fundamental mode are strictly
+positive. -/
 theorem seno_fiedler_pos
     (d : ℕ) (hd : 1 ≤ d) (j : Fin d) :
     0 < Real.sin (((j.val : ℝ) + 1) * anguloFiedler d) := by
@@ -344,7 +345,7 @@ theorem vectorFiedlerCrudo_ne_zero
   have hj := congrArg (fun v : Hd d => v j) h
   exact vectorFiedlerCrudo_coordenada_ne_zero d hd j hj
 
-/-- Vector de Fiedler explícito normalizado, sin elección de autovector. -/
+/-- Explicit normalized Fiedler vector, no eigenvector choice. -/
 noncomputable def vectorFiedlerExplicito (d : ℕ) : Hd d :=
   ((‖vectorFiedlerCrudo d‖ : ℂ)⁻¹) • vectorFiedlerCrudo d
 
@@ -414,7 +415,7 @@ theorem posicionCoord_succ_sub
   field_simp
   ring
 
-/-- Certificado concreto de no conmutatividad: una sola entrada vecina basta. -/
+/-- Concrete non-commutativity certificate: one neighbour entry suffices. -/
 theorem conmutador_matriz_entrada_vecina_no_cero
     (d : ℕ) (hd : 2 ≤ d) :
     let i : Fin d := ⟨0, by omega⟩
@@ -465,7 +466,7 @@ theorem conmutador_TdOp_PdOp_eq_matriz (d : ℕ) :
       Matrix.toEuclideanLin ((Td d * Pd d) - (Pd d * Td d)) := by
   simp [conmutador, TdOp, PdOp, Matrix.toEuclideanLin, Matrix.toLpLin_mul_same]
 
-/-- `[T_d,P_d] ≠ 0` para `d ≥ 2`. -/
+/-- `[T_d,P_d] ≠ 0` for `d ≥ 2`. -/
 theorem conmutador_TdOp_PdOp_no_cero (d : ℕ) (hd : 2 ≤ d) :
     conmutador (TdOp d) (PdOp d) ≠ 0 := by
   rw [conmutador_TdOp_PdOp_eq_matriz]
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D6_Fiedler.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D6_Fiedler.lean
index 0d1b89d11..073eefe13 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D6_Fiedler.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D6_Fiedler.lean
@@ -8,15 +8,15 @@ module
 public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D5_MaximaTension
 
 /-!
-# D6 — Descomposición espectral de Fiedler sobre el camino discreto
-
-Diagonalización explícita, en base de modos seno/fase, del operador de
-adyacencia `A_d` y del observable `i[T_d,P_d]` sobre `pathGraph d`. Se
-obtiene el espectro completo en forma cerrada y se identifica el modo
-fundamental (el vector de Fiedler) como el autovector extremal del
-observable `i[T_d,P_d]`. Contenido íntegro, sin modificar, de la
-descomposición espectral del corpus original — es álgebra lineal y teoría
-espectral de grafos pura, sin ninguna capa interpretativa.
+# D6 — Fiedler spectral decomposition on the discrete path
+
+Explicit diagonalization, in the sine/phase mode basis, of the adjacency
+operator `A_d` and the observable `i[T_d,P_d]` on `pathGraph d`. The
+complete spectrum is obtained in closed form and the fundamental mode
+(the Fiedler vector) is identified as the extremal eigenvector of the
+observable `i[T_d,P_d]`. Full, unmodified content from the original
+corpus spectral decomposition — pure linear algebra and spectral graph
+theory, with no interpretive layer.
 -/
 
 @[expose] public section
diff --git a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D7_Niven.lean b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D7_Niven.lean
index 3b26a3c0e..99a2a1779 100644
--- a/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D7_Niven.lean
+++ b/PhyslibAlpha/AlgebraicFramework/HilbertSpace/PathGap/D7_Niven.lean
@@ -8,19 +8,19 @@ module
 public import PhyslibAlpha.AlgebraicFramework.HilbertSpace.PathGap.D2_Robertson
 
 /-!
-# D7 — Teorema de Niven: la saturación sólo ocurre en `d ∈ {2,3}`
+# D7 — Niven theorem: saturation only occurs at `d ∈ {2,3}`
 
-Tricotomía de saturación: la ecuación trigonométrica
+Saturation trichotomy: the trigonometric equation
 
 `cos²(π/(d+1)) = (d−1)/4`
 
-—que es exactamente la condición para que la cota de Robertson se sature
-sobre el modo fundamental del camino discreto— se cumple **si y sólo si**
-`d = 2` o `d = 3`. No hay más soluciones naturales: la prueba distingue
-`d = 4` (reductio algebraico explícito) de `d ≥ 5` (cota de coseno,
-`Blindaje.R3_techo_coseno`). En consecuencia, para todo `d ≥ 4` la brecha
-`C_Nava(d) − 1` es estrictamente positiva (segunda mitad de este archivo,
-`Constructor_DeltaGeom_Pos`).
+— which is exactly the condition for Robertson's bound to saturate
+on the fundamental mode of the discrete path — holds if and only if
+`d = 2` or `d = 3`. There are no further natural solutions: the proof
+distinguishes `d = 4` (explicit algebraic reductio) from `d ≥ 5`
+(cosine bound, `Blindaje.R3_techo_coseno`). Consequently, for all
+`d ≥ 4` the gap `C_Nava(d) − 1` is strictly positive (second half
+of this file, `Constructor_DeltaGeom_Pos`).
 -/
 
 @[expose] public section
@@ -29,17 +29,17 @@ open Real
 
 namespace Gnomon
 
-/-- Semilla `d=2`: saturación unitaria exacta `cos²(π/3) = 1/4`. -/
+/-- Seed `d=2`: exact unit saturation `cos²(π/3) = 1/4`. -/
 theorem semilla_d2 : Real.cos (π / 3) ^ 2 = 1 / 4 := by
   rw [Real.cos_pi_div_three]; norm_num
 
-/-- Semilla `d=3`: saturación unitaria exacta `cos²(π/4) = 1/2`. -/
+/-- Seed `d=3`: exact unit saturation `cos²(π/4) = 1/2`. -/
 theorem semilla_d3 : Real.cos (π / 4) ^ 2 = 1 / 2 := by
   rw [Real.cos_pi_div_four]
   rw [div_pow, sq_sqrt (by norm_num : (2:ℝ) ≥ 0)]
   norm_num
 
-/-- `d=4` no admite saturación unitaria: `cos²(π/5) ≠ 3/4`. -/
+/-- `d=4` does not admit unit saturation: `cos²(π/5) ≠ 3/4`. -/
 theorem no_saturacion_d4 : Real.cos (π / 5) ^ 2 ≠ 3 / 4 := by
   rw [Real.cos_pi_div_five]
   intro h
@@ -47,7 +47,7 @@ theorem no_saturacion_d4 : Real.cos (π / 5) ^ 2 ≠ 3 / 4 := by
   have hnn : 0 ≤ Real.sqrt 5 := Real.sqrt_nonneg 5
   nlinarith [hs, hnn, h]
 
-/-- TEOREMA DE NIVEN (tricotomía de saturación): para `d ≥ 2`,
+/-- NIVEN THEOREM (saturation trichotomy): for `d ≥ 2`,
 `cos²(π/(d+1)) = (d−1)/4 ↔ d ∈ {2,3}`. -/
 theorem saturacion_iff (d : ℕ) (hd : 2 ≤ d) :
     Real.cos (π / (d + 1)) ^ 2 = ((d : ℝ) - 1) / 4 ↔ d = 2 ∨ d = 3 := by
@@ -71,14 +71,14 @@ theorem saturacion_iff (d : ℕ) (hd : 2 ≤ d) :
     · have hc : ((3 : ℕ) : ℝ) + 1 = 4 := by norm_num
       rw [hc, semilla_d3]; norm_num
 
-/-- La cota unitaria no se repone: para `d ≥ 4` la saturación es imposible. -/
+/-- The unit bound does not recover: for `d ≥ 4` saturation is impossible. -/
 theorem no_reposición_saturacion_camino (d : ℕ) (hd : 4 ≤ d) :
     Real.cos (π / (d + 1)) ^ 2 ≠ ((d : ℝ) - 1) / 4 := by
   intro h
   have hsem : d = 2 ∨ d = 3 := (saturacion_iff d (by omega)).mp h
   omega
 
-/-- Alias citable: las únicas semillas de saturación son `d = 2` y `d = 3`. -/
+/-- Citable alias: the only saturation seeds are `d = 2` and `d = 3`. -/
 theorem semillas_niven_unicas (d : ℕ) (hd : 2 ≤ d) :
     Real.cos (π / (d + 1)) ^ 2 = ((d : ℝ) - 1) / 4 → d = 2 ∨ d = 3 :=
   (saturacion_iff d hd).mp