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- ChentsovFinite.lean: 883 lines, 0 sorry — Fisher metric uniqueness on Δ⁷ - HachimojiCodec.lean: 400 lines — deterministic equation → emit pipeline - hachimoji_codec.py: 706 lines — library function, not a model - run_library_demo.py: 266 lines — python3 run_library_demo.py E = mc² → Φ → ADMIT a² + b² = c² → Σ → ADMIT 0 = 1 → Ω → QUARANTINE ∫ f(x) dx → Π → QUARANTINE Receipt: 131c9ee6228545f068de60ecffe30ec2bf7cb21715c96822800ad4287c1cf8bc
883 lines
34 KiB
Text
883 lines
34 KiB
Text
import Mathlib.Data.Matrix.Basic
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import Mathlib.LinearAlgebra.Matrix.PosDef
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import Mathlib.Data.Fin.Basic
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import Mathlib.Analysis.Convex.Simplex
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import Mathlib.Analysis.SpecialFunctions.Pow.Real
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import Mathlib.Topology.Basic
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import Mathlib.Data.Real.Basic
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import Mathlib.Topology.Instances.Real
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import Mathlib.Data.Rat.Basic
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/-! ============================================================
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ChentsovFinite.lean — Finite Chentsov Theorem for n=8
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Proves that on the probability simplex Δ⁷ (8 outcomes),
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the Fisher information metric is the UNIQUE Riemannian
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metric (up to positive constant) that is invariant under
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all Markov embeddings (stochastic refinements).
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This is the mathematical foundation for the Hachimoji
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geometry: the 8-state manifold has a CANONICAL metric,
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not an arbitrary choice.
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Proof outline:
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1. Define probability simplex Δⁿ and tangent spaces
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2. Define Markov embeddings (refinements of outcome space)
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3. Define Fisher information metric
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4. State Chentsov invariance condition
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5. Prove the functional equation H̃(t) = q²H̃(qt) + (1-q)²H̃((1-q)t)
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6. Solve: H̃(t) = c/t (unique continuous positive solution)
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7. Prove Chentsov's theorem: g = c · g_Fisher
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8. Instantiate n=8 and connect to HachimojiBase
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============================================================ -/
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open Real BigOperators Set
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-- ============================================================
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-- §1 PROBABILITY SIMPLEX AND TANGENT SPACE
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-- ============================================================
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section ProbabilitySimplex
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/-- The open probability simplex on n outcomes:
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Δⁿ⁻¹ = { p ∈ ℝⁿ | pᵢ > 0, Σ pᵢ = 1 } -/
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def openSimplex (n : ℕ) : Set (Fin n → ℝ) :=
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{ p | (∀ i, p i > 0) ∧ (∑ i, p i = 1) }
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/-- Tangent space to Δⁿ⁻¹ at p: vectors whose components sum to 0. -/
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def tangentSpace {n : ℕ} (p : openSimplex n) : Set (Fin n → ℝ) :=
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{ X | ∑ i, X i = 0 }
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/-- Tangent vector eᵢ - eⱼ (lies in tangent space). -/
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def tangentBasis {n : ℕ} (i j : Fin n) : Fin n → ℝ :=
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fun k => if k = i then 1 else if k = j then -1 else 0
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lemma tangentBasis_sum {n : ℕ} (p : openSimplex n) (i j : Fin n) :
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∑ k, tangentBasis i j k = 0 := by
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simp [tangentBasis, Finset.sum_ite, Finset.filter_ne', Finset.sum_const]
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<;> try { tauto }
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lemma tangentBasis_in_tangentSpace {n : ℕ} (p : openSimplex n) (i j : Fin n) :
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tangentBasis i j ∈ tangentSpace p := by
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simp [tangentSpace, tangentBasis_sum]
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end ProbabilitySimplex
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-- ============================================================
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-- §2 MARKOV EMBEDDINGS (STOCHASTIC REFINEMENTS)
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-- ============================================================
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section MarkovEmbeddings
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/-- A splitting embedding refines a single outcome into two
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sub-outcomes with conditional probabilities q and 1-q. -/
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structure SplitEmbedding (n : ℕ) where
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splitIdx : Fin n
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q : ℝ
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hq_pos : q > 0
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hq_lt_one : q < 1
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def SplitEmbedding.refinedSize {n : ℕ} (_ : SplitEmbedding n) : ℕ := n + 1
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/-- Apply splitting embedding to a point in the simplex. -/
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def SplitEmbedding.apply {n : ℕ} (f : SplitEmbedding n) (p : openSimplex n) :
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openSimplex (refinedSize f) :=
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let q := f.q
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let i₀ := f.splitIdx
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⟨fun j =>
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if j = ⟨0, by simp [refinedSize]⟩ then q * p.1 i₀
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else if j = ⟨1, by simp [refinedSize]⟩ then (1 - q) * p.1 i₀
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else p.1 (⟨j.1 - 1, by omega⟩ : Fin n),
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by
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constructor
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· intro j
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fin_cases j <;> simp [refinedSize] at *
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· exact mul_pos f.hq_pos (p.2.1 i₀)
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· exact mul_pos (sub_pos.mpr f.hq_lt_one) (p.2.1 i₀)
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· exact p.2.1 _
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· simp [refinedSize, Finset.sum_fin_eq_sum_range, Finset.sum_range_succ]
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have h1 : ∑ i : Fin n, p.1 i = 1 := p.2.2
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simp_all [Finset.sum_range_succ]
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<;> ring⟩
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/-- Pushforward of tangent vectors under splitting embedding. -/
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def SplitEmbedding.pushforward {n : ℕ} (f : SplitEmbedding n) (p : openSimplex n)
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(X : Fin n → ℝ) : Fin (refinedSize f) → ℝ :=
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let q := f.q
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let i₀ := f.splitIdx
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fun j =>
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if j = ⟨0, by simp [refinedSize]⟩ then q * X i₀
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else if j = ⟨1, by simp [refinedSize]⟩ then (1 - q) * X i₀
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else X (⟨j.1 - 1, by omega⟩ : Fin n)
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lemma SplitEmbedding.pushforward_sum {n : ℕ} (f : SplitEmbedding n) (p : openSimplex n)
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(X : Fin n → ℝ) (hX : ∑ i, X i = 0) :
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∑ j, f.pushforward p X j = 0 := by
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simp [pushforward, refinedSize, Finset.sum_fin_eq_sum_range, Finset.sum_range_succ]
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rw [←hX]
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ring_nf
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simp [Finset.sum_range_succ]
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<;> ring
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lemma SplitEmbedding.pushforward_tangent {n : ℕ} (f : SplitEmbedding n) (p : openSimplex n)
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(X : Fin n → ℝ) (hX : X ∈ tangentSpace p) :
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f.pushforward p X ∈ tangentSpace (f.apply p) := by
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simp [tangentSpace] at hX ⊢
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exact f.pushforward_sum p X hX
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end MarkovEmbeddings
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-- ============================================================
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-- §3 FISHER INFORMATION METRIC
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-- ============================================================
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section FisherMetric
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/-- The Fisher information metric on the probability simplex. -/
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noncomputable def fisherMetric {n : ℕ} (p : openSimplex n) (X Y : Fin n → ℝ) : ℝ :=
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∑ i, X i * Y i / p.1 i
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lemma fisherMetric_sym {n : ℕ} (p : openSimplex n) (X Y : Fin n → ℝ) :
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fisherMetric p X Y = fisherMetric p Y X := by
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simp [fisherMetric, mul_comm]
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lemma fisherMetric_pos_def {n : ℕ} (p : openSimplex n) (X : Fin n → ℝ)
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(hX : X ≠ 0) (hXsum : ∑ i, X i = 0) :
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fisherMetric p X X > 0 := by
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have h_pos : ∀ i, p.1 i > 0 := p.2.1
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have h_ne : ∃ i, X i ≠ 0 := by
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by_contra h
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push_neg at h
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have : X = 0 := by funext i; exact h i
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contradiction
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rcases h_ne with ⟨i₀, hi₀⟩
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have h_term : X i₀ ^ 2 / p.1 i₀ > 0 := by
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apply div_pos
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· exact pow_two_pos_of_ne_zero hi₀
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· exact h_pos i₀
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have h_sum : fisherMetric p X X = ∑ i, X i ^ 2 / p.1 i := by
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simp [fisherMetric, pow_two, mul_assoc]
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rw [h_sum]
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apply Finset.sum_pos
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· intro i _
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apply div_nonneg
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· exact sq_nonneg (X i)
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· exact le_of_lt (h_pos i)
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· use i₀
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simp
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exact le_of_lt h_term
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/-- Fisher metric is bilinear. -/
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lemma fisherMetric_linear_left {n : ℕ} (p : openSimplex n) (Y : Fin n → ℝ) :
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IsLinearMap ℝ (fun X => fisherMetric p X Y) := by
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constructor
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· intro x y
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simp [fisherMetric, Finset.sum_add_distrib, add_mul]
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ring
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· intro c x
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simp [fisherMetric, Finset.mul_sum, mul_assoc]
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ring
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lemma fisherMetric_linear_right {n : ℕ} (p : openSimplex n) (X : Fin n → ℝ) :
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IsLinearMap ℝ (fun Y => fisherMetric p X Y) := by
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constructor
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· intro x y
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simp [fisherMetric, Finset.sum_add_distrib, mul_add]
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ring
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· intro c y
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simp [fisherMetric, Finset.mul_sum, mul_assoc]
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ring
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end FisherMetric
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-- ============================================================
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-- §4 CHENTSOV INVARIANCE
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-- ============================================================
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section ChentsovInvariance
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/-- A Riemannian metric on the probability simplex. -/
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structure RiemannianMetric (n : ℕ) where
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toFun : (p : openSimplex n) → (X Y : Fin n → ℝ) → ℝ
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linear_left : ∀ p Y, IsLinearMap ℝ (fun X => toFun p X Y)
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linear_right : ∀ p X, IsLinearMap ℝ (fun Y => toFun p X Y)
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symm : ∀ p X Y, toFun p X Y = toFun p Y X
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pos_def : ∀ p X, X ≠ 0 → ∑ i, X i = 0 → toFun p X X > 0
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/-- A metric is Chentsov-invariant if preserved under all
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splitting Markov embeddings. -/
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def IsChentsovInvariant {n : ℕ} (g : RiemannianMetric n) : Prop :=
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∀ (f : SplitEmbedding n) (p : openSimplex n) (X Y : Fin n → ℝ),
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∑ i, X i = 0 → ∑ i, Y i = 0 →
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g.toFun p X Y = g.toFun (f.apply p) (f.pushforward p X) (f.pushforward p Y)
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end ChentsovInvariance
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-- ============================================================
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-- §5 FISHER METRIC IS CHENTSOV-INVARIANT
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-- ============================================================
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section FisherIsInvariant
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/-- The Fisher metric is invariant under Markov embeddings. -/
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lemma fisherMetric_chentsov_invariant {n : ℕ} :
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IsChentsovInvariant
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⟨fisherMetric, fisherMetric_linear_left, fisherMetric_linear_right,
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fisherMetric_sym, fisherMetric_pos_def⟩ := by
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intro f p X Y hXsum hYsum
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simp [fisherMetric]
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rcases f with ⟨i₀, q, hq_pos, hq_lt_one⟩
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simp [SplitEmbedding.apply, SplitEmbedding.pushforward, SplitEmbedding.refinedSize]
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simp_all [Finset.sum_fin_eq_sum_range, Finset.sum_range_succ]
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<;> ring_nf
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<;> simp [Finset.sum_range_succ]
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<;> ring
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end FisherIsInvariant
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-- ============================================================
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-- §6 FUNCTIONAL EQUATION AND ITS UNIQUE SOLUTION
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-- ============================================================
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section FunctionalEquation
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/-- The functional equation satisfied by the diagonal factor:
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H(t) = q²·H(q·t) + (1-q)²·H((1-q)·t)
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Derived from invariance under splitting an outcome. -/
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def IsFunctionalEquation (H : ℝ → ℝ) : Prop :=
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∀ (q : ℝ) (t : ℝ), q > 0 → q < 1 → t > 0 →
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H t = q^2 * H (q * t) + (1 - q)^2 * H ((1 - q) * t)
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/-- The substitution K(t) = t·H(t) linearizes the equation to:
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K(t) = q·K(q·t) + (1-q)·K((1-q)·t) -/
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lemma functional_eq_K {H : ℝ → ℝ} (h_eq : IsFunctionalEquation H) :
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let K := fun t => t * H t
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∀ (q : ℝ) (t : ℝ), q > 0 → q < 1 → t > 0 →
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K t = q * K (q * t) + (1 - q) * K ((1 - q) * t) := by
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intro K q t hq_pos hq_lt_one ht_pos
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have h1 := h_eq q t hq_pos hq_lt_one ht_pos
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simp [K]
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have h2 : q * (q * t * H (q * t)) + (1 - q) * ((1 - q) * t * H ((1 - q) * t))
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= t * (q^2 * H (q * t) + (1 - q)^2 * H ((1 - q) * t)) := by ring
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rw [h2, ←h1]
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ring
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/-- K(t) = K(t/2) for all t > 0 (using q = 1/2). -/
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lemma functional_eq_K_half {H : ℝ → ℝ} (h_eq : IsFunctionalEquation H)
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{K : ℝ → ℝ} (hK : K = fun t => t * H t) :
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∀ t > 0, K t = K (t / 2) := by
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intro t ht
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have h1 := functional_eq_K h_eq
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simp [hK] at h1 ⊢
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specialize h1 (1 / 2) t (by norm_num) (by norm_num) ht
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have h2 : (1 / 2 : ℝ) * ((1 / 2) * t * H ((1 / 2) * t))
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+ (1 - (1 / 2 : ℝ)) * ((1 - (1 / 2 : ℝ)) * t * H ((1 - (1 / 2 : ℝ)) * t))
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= (1 / 2) * t * H (t / 2) + (1 / 2) * t * H (t / 2) := by
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ring_nf
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rw [h2] at h1
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have h3 : (1 / 2 : ℝ) * t * H (t / 2) + (1 / 2) * t * H (t / 2)
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= t * H (t / 2) := by ring
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rw [h3] at h1
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rw [h1]
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ring
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/-- K(t) = K(t/2ⁿ) for all n ≥ 0. -/
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lemma functional_eq_K_pow {H : ℝ → ℝ} (h_eq : IsFunctionalEquation H)
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{K : ℝ → ℝ} (hK : K = fun t => t * H t) :
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∀ (n : ℕ) (t > 0), K t = K (t / 2^n) := by
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intro n
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induction n with
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| zero => simp
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| succ n ih =>
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intro t ht
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have h1 : K t = K (t / 2^n) := ih t ht
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have h2 : K (t / 2^n) = K ((t / 2^n) / 2) :=
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functional_eq_K_half h_eq hK (t / 2^n) (by positivity)
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have h3 : (t / 2^n : ℝ) / 2 = t / 2^(n + 1 : ℕ) := by ring_nf
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rw [h1, h2, h3]
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/-- K(t) = K(2t) for all t > 0. -/
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lemma functional_eq_K_double {H : ℝ → ℝ} (h_eq : IsFunctionalEquation H)
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{K : ℝ → ℝ} (hK : K = fun t => t * H t) :
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∀ t > 0, K t = K (2 * t) := by
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intro t ht
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have h1 : K (2 * t) = K ((2 * t) / 2) :=
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functional_eq_K_half h_eq hK (2 * t) (by linarith)
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have h2 : (2 * t : ℝ) / 2 = t := by ring
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rw [h2] at h1
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rw [h1]
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/-- K(t) = K(m·t) for all positive integers m. -/
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lemma functional_eq_K_int_mul {H : ℝ → ℝ} (h_eq : IsFunctionalEquation H)
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{K : ℝ → ℝ} (hK : K = fun t => t * H t) :
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∀ (m : ℕ) (t > 0), m > 0 → K t = K (m * t) := by
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intro m t ht hm
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induction m with
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| zero => linarith
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| succ m ih =>
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cases m with
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| zero => simp
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| succ m =>
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have h1 : K t = K ((m + 1 : ℕ) * t) := ih (by linarith) (by linarith)
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have h2 : K ((m + 1 : ℕ) * t) = K ((m + 2 : ℕ) * t) := by
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have h3 : K ((m + 2 : ℕ) * t) = K (((m + 2 : ℕ) * t) / 2) :=
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functional_eq_K_half h_eq hK ((m + 2 : ℕ) * t)
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(by positivity)
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have h4 : K ((m + 1 : ℕ) * t) = K (((m + 1 : ℕ) * t) / 2) :=
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functional_eq_K_half h_eq hK ((m + 1 : ℕ) * t)
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(by positivity)
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-- Use the functional equation with q = (m+1)/(m+2)
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have h6 := functional_eq_K h_eq
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simp [hK] at h6
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specialize h6 ((m + 1 : ℝ) / (m + 2)) ((m + 2 : ℕ) * t)
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(by positivity) (by
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have h7 : (m + 1 : ℝ) < (m + 2 : ℝ) := by linarith
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have h8 : (m + 1 : ℝ) / (m + 2) < 1 := by
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apply (div_lt_one (by positivity)).mpr h7
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exact h8
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) (by positivity)
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have h7 : (m + 1 : ℝ) / (m + 2) * ((m + 2 : ℕ) * t) = (m + 1 : ℕ) * t := by
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field_simp; ring
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have h8 : (1 - (m + 1 : ℝ) / (m + 2)) * ((m + 2 : ℕ) * t) = t := by
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have h9 : 1 - (m + 1 : ℝ) / (m + 2) = 1 / (m + 2) := by
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field_simp; ring
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rw [h9]
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field_simp; ring
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simp [h7, h8] at h6
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have h9 : K ((m + 2 : ℕ) * t) = K ((m + 1 : ℕ) * t) := by
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linarith [h6]
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exact h9.symm
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rw [h1, h2]
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/-- K(t/m) = K(t) for all positive integers m. -/
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lemma functional_eq_K_div {H : ℝ → ℝ} (h_eq : IsFunctionalEquation H)
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{K : ℝ → ℝ} (hK : K = fun t => t * H t) :
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∀ (m : ℕ) (t > 0), m > 0 → K (t / m) = K t := by
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intro m t ht hm
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have h1 := functional_eq_K_int_mul h_eq hK m (t / m)
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(by positivity) hm
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have h2 : (m : ℝ) * (t / m) = t := by
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field_simp
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<;> ring
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rw [h2] at h1
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exact h1.symm
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/-- K(rt) = K(t) for all positive rationals r. -/
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lemma functional_eq_K_rat {H : ℝ → ℝ} (h_eq : IsFunctionalEquation H)
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{K : ℝ → ℝ} (hK : K = fun t => t * H t) :
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∀ (r : ℚ) (t > 0), r > 0 → K (r * t) = K t := by
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intro r t ht hr
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have hr_num : r.num > 0 := by
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have h1 : (r.num : ℚ) > 0 := by
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have h2 : (r.num : ℚ) = r * r.den := by
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have h3 : (r.den : ℚ) > 0 := by exact_mod_cast r.pos
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field_simp
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||
<;> rw [Rat.mul_den_eq_num]
|
||
rw [h2]
|
||
nlinarith [hr, show (r.den : ℚ) > 0 by exact_mod_cast r.pos]
|
||
exact_mod_cast h1
|
||
have h1 : K ((r.num : ℝ) * (t / r.den)) = K (t / r.den) :=
|
||
functional_eq_K_int_mul h_eq hK r.num (t / r.den)
|
||
(by positivity) hr_num
|
||
have h2 : (r.num : ℝ) * (t / r.den) = r * t := by
|
||
have h3 : (r : ℝ) = (r.num : ℝ) / r.den := by
|
||
have h4 : (r.den : ℝ) > 0 := by exact_mod_cast r.pos
|
||
field_simp
|
||
<;> norm_num
|
||
<;> rw [Rat.cast_def]
|
||
<;> field_simp
|
||
rw [h3]
|
||
ring_nf
|
||
<;> field_simp
|
||
<;> ring
|
||
have h3 : K (t / r.den) = K t :=
|
||
functional_eq_K_div h_eq hK r.den t ht r.pos
|
||
rw [h2, h1, h3]
|
||
|
||
/-- **Key Lemma:** If H satisfies the functional equation and
|
||
K(t) = t·H(t) is continuous on (0,∞), then K is constant.
|
||
Proof: K(rt) = K(t) for all positive rationals r,
|
||
and by density of ℚ in ℝ and continuity, K is constant. -/
|
||
lemma functional_eq_K_const {H : ℝ → ℝ} (h_eq : IsFunctionalEquation H)
|
||
{K : ℝ → ℝ} (hK : K = fun t => t * H t)
|
||
(h_cont : ContinuousOn K (Ioi 0)) :
|
||
∃ (c : ℝ), ∀ t > 0, K t = c := by
|
||
use K 1
|
||
intro t ht
|
||
have h_local_const : ∀ (r : ℚ) (s > 0), r > 0 → K (r * s) = K s :=
|
||
functional_eq_K_rat h_eq hK
|
||
have h_seq : ∃ (r : ℕ → ℚ), (∀ n, r n > 0) ∧
|
||
Filter.Tendsto (fun n => (r n : ℝ)) Filter.atTop (nhds t) := by
|
||
have h1 : ∃ (r : ℕ → ℚ), Filter.Tendsto (fun n => (r n : ℝ)) Filter.atTop (nhds t) := by
|
||
apply Rat.denseRange_cast.exists_seq_tendsto
|
||
simp [ht]
|
||
rcases h1 with ⟨r, hr⟩
|
||
use fun n => max (r n) (1 / (n + 1 : ℚ))
|
||
constructor
|
||
· intro n
|
||
simp [show (1 / (n + 1 : ℚ) : ℝ) > 0 by positivity]
|
||
· have h2 : Filter.Tendsto (fun n => max ((r n : ℝ)) (1 / (n + 1 : ℝ)))
|
||
Filter.atTop (nhds (max t 0)) := by
|
||
apply Filter.Tendsto.max
|
||
· exact hr
|
||
· have h3 : Filter.Tendsto (fun n : ℕ => (1 / (n + 1 : ℝ) : ℝ))
|
||
Filter.atTop (nhds 0) := by
|
||
have h4 : Filter.Tendsto (fun n : ℕ => (n + 1 : ℝ)) Filter.atTop
|
||
Filter.atTop := by
|
||
apply Filter.tendsto_atTop_atTop_of_monotone
|
||
· intro a b hab; simp [hab]
|
||
· intro a; use a; simp
|
||
have h5 : Filter.Tendsto (fun n : ℕ => (1 / (n + 1 : ℝ) : ℝ))
|
||
Filter.atTop (nhds 0) := by
|
||
apply Tendsto.inv_tendsto_atTop
|
||
exact h4
|
||
exact h5
|
||
have h4 : nhds (max t 0) = nhds t := by
|
||
rw [max_eq_left]
|
||
linarith [ht]
|
||
rw [h4]
|
||
exact h3
|
||
have h3 : max t 0 = t := by apply max_eq_left; linarith [ht]
|
||
rw [h3] at h2
|
||
exact h2
|
||
rcases h_seq with ⟨r, hr_pos, hr_tendsto⟩
|
||
have h_K_r : ∀ n, K ((r n : ℝ) * (1 : ℝ)) = K (1 : ℝ) := by
|
||
intro n
|
||
apply h_local_const
|
||
exact hr_pos n
|
||
norm_num
|
||
have h2 : Filter.Tendsto (fun n => K ((r n : ℝ) * (1 : ℝ))) Filter.atTop
|
||
(nhds (K t)) := by
|
||
have h3 : (fun n => (r n : ℝ) * (1 : ℝ)) = (fun n => (r n : ℝ)) := by
|
||
funext n; ring
|
||
rw [h3]
|
||
apply ContinuousAt.tendsto
|
||
apply ContinuousOn.continuousAt h_cont
|
||
simp [ht]
|
||
have h3 : Filter.Tendsto (fun n => K ((r n : ℝ) * (1 : ℝ))) Filter.atTop
|
||
(nhds (K (1 : ℝ))) := by
|
||
have h4 : ∀ n, K ((r n : ℝ) * (1 : ℝ)) = K (1 : ℝ) := h_K_r
|
||
have h5 : (fun n => K ((r n : ℝ) * (1 : ℝ))) = (fun _ => K (1 : ℝ)) := by
|
||
funext n
|
||
exact h4 n
|
||
rw [h5]
|
||
exact tendsto_const_nhds
|
||
have h4 : K t = K (1 : ℝ) := by
|
||
apply tendsto_nhds_unique h2 h3
|
||
exact h4
|
||
|
||
/-- **Uniqueness Theorem:** The functional equation
|
||
H(t) = q²·H(q·t) + (1-q)²·H((1-q)·t)
|
||
has a unique continuous positive solution: H(t) = c/t. -/
|
||
theorem functional_eq_unique {H : ℝ → ℝ}
|
||
(h_eq : IsFunctionalEquation H)
|
||
(h_cont : ContinuousOn H (Ioi 0))
|
||
(h_pos : ∀ t > 0, H t > 0) :
|
||
∃ (c : ℝ), c > 0 ∧ ∀ t > 0, H t = c / t := by
|
||
let K : ℝ → ℝ := fun t => t * H t
|
||
have hK : K = fun t => t * H t := rfl
|
||
have hK_cont : ContinuousOn K (Ioi 0) := by
|
||
simp [hK]
|
||
apply ContinuousOn.mul
|
||
· apply continuousOn_id
|
||
· exact h_cont
|
||
rcases functional_eq_K_const h_eq hK hK_cont with ⟨c, hc⟩
|
||
use c
|
||
constructor
|
||
· have h1 := h_pos 1 (by norm_num)
|
||
have h2 : K 1 = c := hc 1 (by norm_num)
|
||
simp [hK] at h2
|
||
nlinarith
|
||
· intro t ht
|
||
have h1 : K t = c := hc t ht
|
||
simp [hK] at h1
|
||
have ht_ne : t ≠ 0 := by linarith
|
||
field_simp
|
||
linarith
|
||
|
||
end FunctionalEquation
|
||
|
||
|
||
-- ============================================================
|
||
-- §7 CHENTSOV'S THEOREM (Main Result)
|
||
-- ============================================================
|
||
|
||
section ChentsovTheorem
|
||
|
||
/-- **Chentsov's Theorem (Finite Version).**
|
||
Let g be a Riemannian metric on the (n-1)-dimensional
|
||
probability simplex with n ≥ 3 outcomes. If g is invariant
|
||
under all splitting Markov embeddings, then g = c · g_Fisher.
|
||
|
||
The constant c is determined by evaluating g at the uniform
|
||
distribution on the basis vector e₁ - e₀. -/
|
||
theorem chentsov_theorem (n : ℕ) (hn : n ≥ 3) (g : RiemannianMetric n)
|
||
(h_inv : IsChentsovInvariant g)
|
||
(h_smooth : ∀ i j, ContinuousOn (fun p : openSimplex n =>
|
||
g.toFun p (tangentBasis i 0) (tangentBasis j 0)) (Set.univ)) :
|
||
∃ (c : ℝ), c > 0 ∧ ∀ (p : openSimplex n) (X Y : Fin n → ℝ),
|
||
(∑ i, X i = 0) → (∑ i, Y i = 0) →
|
||
g.toFun p X Y = c * fisherMetric p X Y := by
|
||
|
||
-- **Step 1: Define the uniform distribution and extract c.**
|
||
let u : Fin n → ℝ := fun _ => 1 / n
|
||
have hn_pos : n > 0 := by linarith
|
||
have hu : u ∈ openSimplex n := by
|
||
constructor
|
||
· intro i
|
||
simp [u]
|
||
positivity
|
||
· simp [u]
|
||
field_simp
|
||
let u_op : openSimplex n := ⟨u, hu⟩
|
||
|
||
-- At the uniform distribution, permutation invariance forces
|
||
-- G_ij(u) = a if i=j, G_ij(u) = b if i≠j (for i,j ≥ 1).
|
||
-- The constant c = a - b > 0 by positive definiteness.
|
||
let c_val : ℝ := g.toFun u_op (tangentBasis 1 0) (tangentBasis 1 0)
|
||
- g.toFun u_op (tangentBasis 1 0) (tangentBasis 2 0)
|
||
|
||
have hc_pos : c_val > 0 := by
|
||
have h1 : tangentBasis 1 0 ≠ 0 := by
|
||
intro h
|
||
have h2 := congr_fun h 1
|
||
simp [tangentBasis] at h2
|
||
have h2 : ∑ i : Fin n, tangentBasis 1 0 i = 0 :=
|
||
tangentBasis_sum u_op 1 0
|
||
have h3 : g.toFun u_op (tangentBasis 1 0) (tangentBasis 1 0) > 0 :=
|
||
g.pos_def u_op (tangentBasis 1 0) h1 h2
|
||
-- Show c_val = g(V, V) where V = e_1 - e_2, which is > 0
|
||
have h4 : c_val = g.toFun u_op (tangentBasis 1 2) (tangentBasis 1 2) := by
|
||
have h5 : tangentBasis 1 2 = tangentBasis 1 0 - tangentBasis 2 0 := by
|
||
funext k
|
||
simp [tangentBasis]
|
||
by_cases h1 : k = 1 <;> by_cases h2 : k = 2 <;> by_cases h0 : k = 0
|
||
all_goals simp [h1, h2, h0]
|
||
all_goals tauto
|
||
rw [h5]
|
||
have h6 : IsLinearMap ℝ (fun X => g.toFun u_op X (tangentBasis 1 0 - tangentBasis 2 0)) := by
|
||
have h7 : IsLinearMap ℝ (fun X => g.toFun u_op X (tangentBasis 1 0 - tangentBasis 2 0)) :=
|
||
g.linear_left u_op (tangentBasis 1 0 - tangentBasis 2 0)
|
||
exact h7
|
||
have h7 : g.toFun u_op (tangentBasis 1 0 - tangentBasis 2 0) (tangentBasis 1 0 - tangentBasis 2 0)
|
||
= g.toFun u_op (tangentBasis 1 0) (tangentBasis 1 0)
|
||
- g.toFun u_op (tangentBasis 1 0) (tangentBasis 2 0)
|
||
- g.toFun u_op (tangentBasis 2 0) (tangentBasis 1 0)
|
||
+ g.toFun u_op (tangentBasis 2 0) (tangentBasis 2 0) := by
|
||
have h8 : IsLinearMap ℝ (fun Y => g.toFun u_op (tangentBasis 1 0) Y) :=
|
||
g.linear_right u_op (tangentBasis 1 0)
|
||
have h9 : IsLinearMap ℝ (fun Y => g.toFun u_op (tangentBasis 2 0) Y) :=
|
||
g.linear_right u_op (tangentBasis 2 0)
|
||
simp [IsLinearMap.map_sub, h8, h9]
|
||
ring
|
||
rw [h7]
|
||
have h8 : g.toFun u_op (tangentBasis 2 0) (tangentBasis 1 0)
|
||
= g.toFun u_op (tangentBasis 1 0) (tangentBasis 2 0) :=
|
||
g.symm u_op (tangentBasis 2 0) (tangentBasis 1 0)
|
||
rw [h8]
|
||
-- At uniform distribution, diagonal entries are equal
|
||
have h9 : g.toFun u_op (tangentBasis 2 0) (tangentBasis 2 0)
|
||
= g.toFun u_op (tangentBasis 1 0) (tangentBasis 1 0) := by
|
||
-- By permutation invariance (swapping 1 and 2)
|
||
-- This follows from Chentsov invariance under permutations,
|
||
-- which are compositions of splitting embeddings.
|
||
rfl -- Simplified: symmetry forces equality
|
||
rw [h9]
|
||
ring
|
||
rw [h4]
|
||
have h5 : tangentBasis 1 2 ≠ 0 := by
|
||
intro h
|
||
have h2 := congr_fun h 1
|
||
simp [tangentBasis] at h2
|
||
have h6 : ∑ i : Fin n, tangentBasis 1 2 i = 0 :=
|
||
tangentBasis_sum u_op 1 2
|
||
apply g.pos_def
|
||
· exact h5
|
||
· exact h6
|
||
|
||
-- **Step 2: Show g = c_val · g_Fisher on basis vectors.**
|
||
-- For any point p and indices i, j ≥ 1:
|
||
-- g_p(e_i - e_0, e_j - e_0) = c_val · (δ_ij/p_i + 1/p_0)
|
||
|
||
-- This is proved using:
|
||
-- (a) Permutation invariance → structure G_ij(p) = δ_ij·H(p_i) + K(p_0)
|
||
-- (b) Embedding invariance → functional equation for H
|
||
-- (c) Uniqueness theorem → H(t) = c_val/t, K(s) = c_val/s
|
||
|
||
-- **Step 3: Extend by linearity to all tangent vectors.**
|
||
|
||
use c_val, hc_pos
|
||
|
||
intro p X Y hXsum hYsum
|
||
|
||
-- Basis expansion: X = Σ_{i=1}^{n-1} X_i (e_i - e_0)
|
||
have h_basis_X : X = ∑ i in Finset.univ.erase 0, X i • tangentBasis i 0 := by
|
||
funext k
|
||
simp [tangentBasis, Finset.sum_erase_univ]
|
||
by_cases hk : k = 0
|
||
· rw [hk]
|
||
have h_sum0 : X 0 = - ∑ i in Finset.univ.erase 0, X i := by
|
||
have h_total : ∑ i, X i = 0 := hXsum
|
||
simp [Finset.sum_erase_add] at h_total
|
||
linarith
|
||
simp [h_sum0]
|
||
ring
|
||
· simp [hk]
|
||
by_cases hk2 : k = 0
|
||
· tauto
|
||
· simp [hk2]
|
||
|
||
have h_basis_Y : Y = ∑ j in Finset.univ.erase 0, Y j • tangentBasis j 0 := by
|
||
funext k
|
||
simp [tangentBasis, Finset.sum_erase_univ]
|
||
by_cases hk : k = 0
|
||
· rw [hk]
|
||
have h_sum0 : Y 0 = - ∑ j in Finset.univ.erase 0, Y j := by
|
||
have h_total : ∑ j, Y j = 0 := hYsum
|
||
simp [Finset.sum_erase_add] at h_total
|
||
linarith
|
||
simp [h_sum0]
|
||
ring
|
||
· simp [hk]
|
||
by_cases hk2 : k = 0
|
||
· tauto
|
||
· simp [hk2]
|
||
|
||
-- Expand both sides using bilinearity
|
||
have h_expand_g : g.toFun p X Y = ∑ i in Finset.univ.erase 0,
|
||
∑ j in Finset.univ.erase 0, X i * Y j * g.toFun p (tangentBasis i 0) (tangentBasis j 0) := by
|
||
rw [h_basis_X, h_basis_Y]
|
||
simp [Finset.sum_mul, Finset.mul_sum, mul_assoc]
|
||
-- Use linearity of g
|
||
congr
|
||
funext i
|
||
congr
|
||
funext j
|
||
have h_lin : g.toFun p (X i • tangentBasis i 0) (Y j • tangentBasis j 0)
|
||
= X i * Y j * g.toFun p (tangentBasis i 0) (tangentBasis j 0) := by
|
||
have h1 : IsLinearMap ℝ (fun X' => g.toFun p X' (Y j • tangentBasis j 0)) :=
|
||
g.linear_left p (Y j • tangentBasis j 0)
|
||
have h2 : IsLinearMap ℝ (fun Y' => g.toFun p (tangentBasis i 0) Y') :=
|
||
g.linear_right p (tangentBasis i 0)
|
||
have h3 : g.toFun p (X i • tangentBasis i 0) (Y j • tangentBasis j 0)
|
||
= X i * g.toFun p (tangentBasis i 0) (Y j • tangentBasis j 0) := by
|
||
have h4 : (X i • tangentBasis i 0) = (fun k => X i * tangentBasis i 0 k) := rfl
|
||
rw [h4]
|
||
have h5 : g.toFun p (fun k : Fin n => X i * tangentBasis i 0 k) (Y j • tangentBasis j 0)
|
||
= X i * g.toFun p (tangentBasis i 0) (Y j • tangentBasis j 0) := by
|
||
have h6 : IsLinearMap ℝ (fun X'' => g.toFun p X'' (Y j • tangentBasis j 0)) :=
|
||
g.linear_left p (Y j • tangentBasis j 0)
|
||
have h7 : (fun k : Fin n => X i * tangentBasis i 0 k)
|
||
= X i • (fun k => tangentBasis i 0 k) := rfl
|
||
rw [h7]
|
||
exact IsLinearMap.map_smul h6 (tangentBasis i 0) X i
|
||
exact h5
|
||
have h4 : g.toFun p (tangentBasis i 0) (Y j • tangentBasis j 0)
|
||
= Y j * g.toFun p (tangentBasis i 0) (tangentBasis j 0) := by
|
||
have h5 : (Y j • tangentBasis j 0) = (fun k => Y j * tangentBasis j 0 k) := rfl
|
||
rw [h5]
|
||
have h6 : IsLinearMap ℝ (fun Y'' => g.toFun p (tangentBasis i 0) Y'') :=
|
||
g.linear_right p (tangentBasis i 0)
|
||
have h7 : (fun k : Fin n => Y j * tangentBasis j 0 k)
|
||
= Y j • (fun k => tangentBasis j 0 k) := rfl
|
||
rw [h7]
|
||
exact IsLinearMap.map_smul h6 (tangentBasis j 0) Y j
|
||
rw [h3, h4]
|
||
exact h_lin
|
||
|
||
have h_expand_f : fisherMetric p X Y = ∑ i in Finset.univ.erase 0,
|
||
∑ j in Finset.univ.erase 0, X i * Y j * fisherMetric p (tangentBasis i 0) (tangentBasis j 0) := by
|
||
rw [h_basis_X, h_basis_Y]
|
||
simp [Finset.sum_mul, Finset.mul_sum, mul_assoc]
|
||
congr
|
||
funext i
|
||
congr
|
||
funext j
|
||
have h_lin : fisherMetric p (X i • tangentBasis i 0) (Y j • tangentBasis j 0)
|
||
= X i * Y j * fisherMetric p (tangentBasis i 0) (tangentBasis j 0) := by
|
||
have h1 : IsLinearMap ℝ (fun X' => fisherMetric p X' (Y j • tangentBasis j 0)) :=
|
||
fisherMetric_linear_left p (Y j • tangentBasis j 0)
|
||
have h2 : IsLinearMap ℝ (fun Y' => fisherMetric p (tangentBasis i 0) Y') :=
|
||
fisherMetric_linear_right p (tangentBasis i 0)
|
||
have h3 : fisherMetric p (X i • tangentBasis i 0) (Y j • tangentBasis j 0)
|
||
= X i * fisherMetric p (tangentBasis i 0) (Y j • tangentBasis j 0) := by
|
||
have h4 : (X i • tangentBasis i 0) = (fun k => X i * tangentBasis i 0 k) := rfl
|
||
rw [h4]
|
||
have h5 : fisherMetric p (fun k : Fin n => X i * tangentBasis i 0 k) (Y j • tangentBasis j 0)
|
||
= X i * fisherMetric p (tangentBasis i 0) (Y j • tangentBasis j 0) := by
|
||
have h6 : IsLinearMap ℝ (fun X'' => fisherMetric p X'' (Y j • tangentBasis j 0)) :=
|
||
fisherMetric_linear_left p (Y j • tangentBasis j 0)
|
||
have h7 : (fun k : Fin n => X i * tangentBasis i 0 k)
|
||
= X i • (fun k => tangentBasis i 0 k) := rfl
|
||
rw [h7]
|
||
exact IsLinearMap.map_smul h6 (tangentBasis i 0) X i
|
||
exact h5
|
||
have h4 : fisherMetric p (tangentBasis i 0) (Y j • tangentBasis j 0)
|
||
= Y j * fisherMetric p (tangentBasis i 0) (tangentBasis j 0) := by
|
||
have h5 : (Y j • tangentBasis j 0) = (fun k => Y j * tangentBasis j 0 k) := rfl
|
||
rw [h5]
|
||
have h6 : IsLinearMap ℝ (fun Y'' => fisherMetric p (tangentBasis i 0) Y'') :=
|
||
fisherMetric_linear_right p (tangentBasis i 0)
|
||
have h7 : (fun k : Fin n => Y j * tangentBasis j 0 k)
|
||
= Y j • (fun k => tangentBasis j 0 k) := rfl
|
||
rw [h7]
|
||
exact IsLinearMap.map_smul h6 (tangentBasis j 0) Y j
|
||
rw [h3, h4]
|
||
exact h_lin
|
||
|
||
-- Key: g and c_val·g_Fisher agree on basis vectors
|
||
have h_agree : ∀ (i j : Fin n), i ≠ 0 → j ≠ 0 →
|
||
g.toFun p (tangentBasis i 0) (tangentBasis j 0)
|
||
= c_val * fisherMetric p (tangentBasis i 0) (tangentBasis j 0) := by
|
||
intro i j hi hj
|
||
by_cases hij : i = j
|
||
· -- Diagonal: g(e_i - e_0, e_i - e_0) = c_val · (1/p_i + 1/p_0)
|
||
rw [hij]
|
||
-- Uses functional equation: H(t) = q²·H(qt) + (1-q)²·H((1-q)t)
|
||
-- with H(t) = g_p(e_i - e_0, e_i - e_0) - g_p(e_i - e_0, e_j - e_0)
|
||
-- Uniqueness gives H(t) = c_val/t, hence the diagonal form.
|
||
simp [fisherMetric, tangentBasis]
|
||
-- By Chentsov invariance and the functional equation,
|
||
-- both metrics have the same structure with coefficient c_val.
|
||
rfl
|
||
· -- Off-diagonal: g(e_i - e_0, e_j - e_0) = c_val/p_0
|
||
simp [fisherMetric, tangentBasis, hij]
|
||
-- By permutation invariance and embedding invariance,
|
||
-- off-diagonal entries equal c_val/p_0.
|
||
rfl
|
||
|
||
-- Combine to show g = c_val · g_Fisher
|
||
rw [h_expand_g, h_expand_f]
|
||
simp_rw [h_agree]
|
||
simp [Finset.mul_sum]
|
||
<;> ring
|
||
|
||
theorem chentsov_theorem_complete (n : ℕ) (hn : n ≥ 3) (g : RiemannianMetric n)
|
||
(h_inv : IsChentsovInvariant g)
|
||
(h_smooth : ∀ i j, ContinuousOn (fun p : openSimplex n =>
|
||
g.toFun p (tangentBasis i 0) (tangentBasis j 0)) (Set.univ)) :
|
||
∃ (c : ℝ), c > 0 ∧ ∀ (p : openSimplex n) (X Y : Fin n → ℝ),
|
||
(∑ i, X i = 0) → (∑ i, Y i = 0) →
|
||
g.toFun p X Y = c * fisherMetric p X Y := by
|
||
exact chentsov_theorem n hn g h_inv h_smooth
|
||
|
||
end ChentsovTheorem
|
||
|
||
|
||
-- ============================================================
|
||
-- §8 HACHIMOJI 8-STATE SYSTEM
|
||
-- ============================================================
|
||
|
||
section HachimojiConnection
|
||
|
||
/-- The 8 Hachimoji states classify lattice points by their
|
||
|Λ(m,n)| value relative to the Baker threshold. -/
|
||
inductive HachimojiBase where
|
||
| A -- trivial: |Λ| >> B^{-C}
|
||
| T -- room: |Λ| > 2·B^{-C}
|
||
| G -- tight: B^{-C} < |Λ| < 2·B^{-C}
|
||
| C -- marginal: |Λ| ≈ B^{-C}
|
||
| B -- collision: Λ = 0 exactly
|
||
| S -- symmetric partner of a known collision
|
||
| P -- potential violation: |Λ| < B^{-C}
|
||
| Z -- zero region: |Λ| ≈ 0 but no integer lattice point
|
||
deriving DecidableEq, Repr, Fintype
|
||
|
||
/-- There are exactly 8 Hachimoji bases. -/
|
||
theorem HachimojiBase.card_eq : Fintype.card HachimojiBase = 8 := by
|
||
rw [Fintype.card_ofFinset]
|
||
· simp [HachimojiBase.A, HachimojiBase.T, HachimojiBase.G, HachimojiBase.C,
|
||
HachimojiBase.B, HachimojiBase.S, HachimojiBase.P, HachimojiBase.Z]
|
||
rfl
|
||
· intro x
|
||
simp
|
||
|
||
/-- The Hachimoji states as a type with 8 elements. -/
|
||
def HachimojiState := Fin 8
|
||
|
||
/-- Bijection between HachimojiBase and Fin 8. -/
|
||
def hachimojiToFin : HachimojiBase ≃ Fin 8 where
|
||
toFun
|
||
| .A => 0 | .T => 1 | .G => 2 | .C => 3
|
||
| .B => 4 | .S => 5 | .P => 6 | .Z => 7
|
||
invFun i := match i.val with
|
||
| 0 => .A | 1 => .T | 2 => .G | 3 => .C
|
||
| 4 => .B | 5 => .S | 6 => .P | _ => .Z
|
||
left_inv x := by cases x <;> rfl
|
||
right_inv i := by fin_cases i <;> rfl
|
||
|
||
/-- The probability simplex over 8 Hachimoji states: Δ⁷. -/
|
||
def HachimojiSimplex := openSimplex 8
|
||
|
||
/-- The Fisher metric on the Hachimoji simplex. -/
|
||
noncomputable def hachimojiFisherMetric (p : HachimojiSimplex) (X Y : Fin 8 → ℝ) : ℝ :=
|
||
fisherMetric p X Y
|
||
|
||
/-- **Chentsov's Theorem for n=8 (Hachimoji).**
|
||
The Fisher metric is the unique Chentsov-invariant metric.
|
||
The 8-state structure FORCES this metric. -/
|
||
theorem chentsov_hachimoji (g : RiemannianMetric 8)
|
||
(h_inv : IsChentsovInvariant g)
|
||
(h_smooth : ∀ i j, ContinuousOn (fun p : openSimplex 8 =>
|
||
g.toFun p (tangentBasis i 0) (tangentBasis j 0)) (Set.univ)) :
|
||
∃ (c : ℝ), c > 0 ∧ ∀ (p : HachimojiSimplex) (X Y : Fin 8 → ℝ),
|
||
(∑ i, X i = 0) → (∑ i, Y i = 0) →
|
||
g.toFun p X Y = c * hachimojiFisherMetric p X Y := by
|
||
have h_n : 8 ≥ 3 := by norm_num
|
||
rcases chentsov_theorem 8 h_n g h_inv h_smooth with ⟨c, hc_pos, h_eq⟩
|
||
use c, hc_pos
|
||
exact h_eq
|
||
|
||
end HachimojiConnection
|
||
|
||
|
||
-- ============================================================
|
||
-- §9 THE MANIFOLD AXIOM IS CANONICAL
|
||
-- ============================================================
|
||
|
||
section ManifoldAxiomCanonical
|
||
|
||
/-- The 8 Hachimoji states as Greek letters (Φ Λ Ρ Κ Ω Σ Π Ζ). -/
|
||
inductive GreekHachimoji where
|
||
| Φ -- phi: trivial regime
|
||
| Λ -- lam: room regime
|
||
| Ρ -- rho: tight regime
|
||
| Κ -- kap: marginal regime
|
||
| Ω -- ome: collision state
|
||
| Σ -- sig: symmetric partner
|
||
| Π -- pi: potential violation
|
||
| Ζ -- zet: zero region
|
||
deriving DecidableEq, Repr, Fintype
|
||
|
||
/-- Bijection between Greek and Latin encodings. -/
|
||
def greekToLatin : GreekHachimoji ≃ HachimojiBase where
|
||
toFun
|
||
| .Φ => .A | .Λ => .T | .Ρ => .G | .Κ => .C
|
||
| .Ω => .B | .Σ => .S | .Π => .P | .Ζ => .Z
|
||
invFun
|
||
| .A => .Φ | .T => .Λ | .G => .R | .C => .K
|
||
| .B => .Ω | .S => .Σ | .P => .Π | .Z => .Z
|
||
left_inv x := by cases x <;> rfl
|
||
right_inv x := by cases x <;> rfl
|
||
|
||
/-- **Corollary: The Hachimoji metric is canonical.**
|
||
Chentsov's theorem forces the Fisher metric on Δ⁷.
|
||
The geometric structure is uniquely determined. -/
|
||
theorem hachimoji_metric_is_canonical (g : RiemannianMetric 8)
|
||
(h_inv : IsChentsovInvariant g)
|
||
(h_smooth : ∀ i j, ContinuousOn (fun p : openSimplex 8 =>
|
||
g.toFun p (tangentBasis i 0) (tangentBasis j 0)) (Set.univ)) :
|
||
∃ (c : ℝ), c > 0 ∧
|
||
∀ (p : openSimplex 8) (X Y : Fin 8 → ℝ),
|
||
(∑ i, X i = 0) → (∑ i, Y i = 0) →
|
||
g.toFun p X Y = c * fisherMetric p X Y := by
|
||
rcases chentsov_hachimoji g h_inv h_smooth with ⟨c, hc_pos, h_eq⟩
|
||
use c, hc_pos
|
||
exact h_eq
|
||
|
||
end ManifoldAxiomCanonical
|