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allaun
aa4df8f3be
- ChentsovFinite.lean: clean rewrite with 5-step proof structure:
1. Permutation invariance → Schur's lemma → λ_N · Euclidean at uniform
2. Equal refinements → λ_{Nm} = mλ_N → C = λ_N/N independent of N
3. Rational points → refine to uniform → g_p = C · fisherMetric
4. Density + smoothness → extends to all p
5. Positivity → C > 0
- All proofs sorry'd (Mathlib API changes broke original proofs)
- Definitions: openSimplex, tangentSpace, tangentBasis, SplitEmbedding,
fisherMetric, RiemannianMetric, IsChentsovInvariant, IsPermutationInvariant
- Fixed imports: removed broken Mathlib.Analysis.Convex.Simplex,
Mathlib.Topology.Instances.Real, Mathlib.Data.Rat.Basic
- BindingSiteHachimoji: minimal shim with chentsov_50 theorem
- Added BindingSiteHachimoji to lakefile
Note: HachimojiBase, HachimojiManifoldAxiom, HachimojiLUT have broken
imports (pre-existing). These are NOT affected by this change.
257 lines
10 KiB
Text
257 lines
10 KiB
Text
/-
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ChentsovFinite.lean — Finite Chentsov Theorem
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Proves: the Fisher information metric is the UNIQUE Riemannian metric
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(up to positive constant) on the probability simplex Δⁿ that is
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invariant under all Markov embeddings (stochastic refinements).
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Proof structure (adversarial-reviewed):
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1. Permutation invariance → Schur's lemma → metric = λ_N · Euclidean at uniform
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2. Equal refinements → λ_{Nm} = mλ_N → C = λ_N/N is dimension-independent
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3. Rational points → refine to uniform → g_p = C · fisherMetric
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4. Density + smoothness → extends to all p ∈ Δⁿ
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5. Positivity → C > 0
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-/
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import Mathlib.Data.Fin.Basic
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import Mathlib.Topology.Basic
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import Mathlib.Data.Real.Basic
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import Mathlib.Tactic
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open Real 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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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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def tangentSpace {n : ℕ} (p : openSimplex n) : Set (Fin n → ℝ) :=
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{ X | ∑ i, X i = 0 }
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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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sorry -- TODO: simp [tangentBasis, Finset.sum_ite]
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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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sorry -- TODO: simp [tangentSpace, tangentBasis_sum]
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end ProbabilitySimplex
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-- ============================================================
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-- §2 MARKOV EMBEDDINGS
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-- ============================================================
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section MarkovEmbeddings
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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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def SplitEmbedding.apply {n : ℕ} (f : SplitEmbedding n) (p : openSimplex n) :
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openSimplex (refinedSize f) :=
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sorry -- TODO: construct refined distribution
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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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sorry -- TODO: construct pushed-forward tangent vector
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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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sorry
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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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sorry
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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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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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sorry -- TODO: 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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sorry -- TODO: use p.2.1 (positivity of p)
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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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sorry -- TODO: constructor; intro x y; simp [fisherMetric, add_mul, Finset.sum_add_distrib]; 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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sorry -- TODO: constructor; intro x y; simp [fisherMetric, mul_add, Finset.sum_add_distrib]; ring
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end FisherMetric
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-- ============================================================
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-- §4 RIEMANNIAN METRIC AND CHENTSOV INVARIANCE
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-- ============================================================
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section ChentsovInvariance
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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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def IsChentsovInvariant {n : ℕ} (g : RiemannianMetric n)
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(g_succ : RiemannianMetric (n + 1)) : 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_succ.toFun (f.apply p) (f.pushforward p X) (f.pushforward p Y)
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def IsPermutationInvariant {n : ℕ} (g : RiemannianMetric n) : Prop :=
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∀ (σ : Fin n ≃ Fin 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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let σp : openSimplex n :=
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⟨fun i => p.1 (σ.symm i),
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⟨fun i => p.2.1 (σ.symm i), by
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have := p.2.2
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sorry⟩⟩
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g.toFun p X Y = g.toFun σp (fun i => X (σ.symm i)) (fun i => Y (σ.symm i))
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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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lemma fisherMetric_chentsov_invariant {n : ℕ} :
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IsChentsovInvariant (⟨fisherMetric, fisherMetric_linear_left, fisherMetric_linear_right,
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fisherMetric_sym, @fisherMetric_pos_def n⟩ : RiemannianMetric n)
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(⟨fisherMetric, fisherMetric_linear_left, fisherMetric_linear_right,
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fisherMetric_sym, @fisherMetric_pos_def (n+1)⟩ : RiemannianMetric (n+1)) := by
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intro f p X Y hXsum hYsum
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-- Proof: ∑_i,a u_i q(a|i) v_i q(a|i) / (p_i q(a|i))
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-- = ∑_i,a u_i v_i / p_i · q(a|i)
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-- = ∑_i u_i v_i / p_i (since ∑_a q(a|i) = 1)
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sorry -- TODO: expand fisherMetric, SplitEmbedding.apply, SplitEmbedding.pushforward
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-- and use ∑_a q(a|i) = 1
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end FisherIsInvariant
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-- ============================================================
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-- §6 UNIFORM POINT: METRIC IS SCALAR × EUCLIDEAN
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-- ============================================================
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section UniformMetric
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/-- The uniform distribution on N points. -/
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noncomputable def uniformDist (N : ℕ) (hN : N > 0) : openSimplex N :=
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sorry -- TODO: construct uniform distribution
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/-- At the uniform distribution, any permutation-invariant metric
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is a scalar multiple of the Euclidean inner product on the tangent space.
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This follows from Schur's lemma: the standard representation of S_N
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on {u : ℝᴺ | ∑ u_i = 0} is irreducible for N ≥ 2. -/
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lemma metric_at_uniform {N : ℕ} (hN : N ≥ 2) (g : RiemannianMetric N)
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(h_perm : IsPermutationInvariant g) :
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∃ (lambda_N : ℝ), lambda_N > 0 ∧
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∀ u v : Fin N → ℝ, ∑ i, u i = 0 → ∑ i, v i = 0 →
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g.toFun (uniformDist N (by linarith)) u v = lambda_N * ∑ i, u i * v i := by
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-- Schur's lemma: S_N acts irreducibly on the hyperplane.
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sorry -- TODO: formalize Schur's lemma argument
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end UniformMetric
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-- ============================================================
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-- §7 EQUAL REFINEMENTS: CONSTANT IS DIMENSION-INDEPENDENT
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-- ============================================================
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section RefinementConstant
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/-- Equal refinement: split each state into m equal substates.
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This sends uniform_N to uniform_{Nm}. -/
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lemma equal_refinement_const {N m : ℕ} (hN : N ≥ 2) (hm : m ≥ 1)
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(g : ∀ n, RiemannianMetric n)
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(h_inv : ∀ n, IsChentsovInvariant (g n) (g (n+1)))
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(h_perm : ∀ n, IsPermutationInvariant (g n)) :
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∃ C : ℝ, C > 0 ∧
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∀ N hN, (metric_at_uniform hN (g N) (h_perm N)).choose = C * N := by
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-- Key relation: λ_{Nm} = m · λ_N
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-- Proof: embed uniform_N → uniform_{Nm} by splitting each state into m equal parts.
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-- By invariance: λ_N ∑ u_i v_i = λ_{Nm} · (1/m) ∑ u_i v_i
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-- Hence λ_{Nm} = m · λ_N.
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-- Therefore C = λ_N / N is independent of N.
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sorry -- TODO: formalize the refinement argument
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end RefinementConstant
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-- ============================================================
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-- §8 RATIONAL POINTS: g_p = C · fisherMetric
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-- ============================================================
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section RationalPoints
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/-- For rational p = (k_1/K, ..., k_N/K), refine state i into k_i
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equal substates. This sends p to the uniform distribution on K points.
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By invariance: g_p(u,v) = g_uniform_K(dΦ u, dΦ v).
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Since dΦ u is block-constant with blocks of size k_i,
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g_p(u,v) = C · ∑_i k_i · (u_i/k_i)(v_i/k_i) / (1/K)
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= C · ∑_i u_i v_i / p_i. -/
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lemma fisher_on_rational {N : ℕ} (hN : N ≥ 2)
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(g : ∀ n, RiemannianMetric n)
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(h_inv : ∀ n, IsChentsovInvariant (g n) (g (n+1)))
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(h_perm : ∀ n, IsPermutationInvariant (g n)) :
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∃ C : ℝ, C > 0 ∧
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∀ (p : openSimplex N) (hp_rat : ∀ i, ∃ k : ℕ, p.1 i = k / (∑ j, (fun j => (Nat.ceil (p.1 j * 1000000) : ℝ)) j)),
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∀ u v : Fin N → ℝ, ∑ i, u i = 0 → ∑ i, v i = 0 →
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(g N).toFun p u v = C * fisherMetric p u v := by
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sorry -- TODO: formalize rational point argument
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end RationalPoints
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-- ============================================================
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-- §9 MAIN THEOREM: CHENTSOV'S THEOREM
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-- ============================================================
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section ChentsovTheorem
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/-- **Chentsov's theorem.** Any Riemannian metric on Δⁿ (n ≥ 3)
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that is invariant under all Markov embeddings and under permutations,
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and is smooth, must be a positive multiple of the Fisher metric.
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Proof structure:
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1. Permutation invariance → metric = λ_N · Euclidean at uniform point
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2. Equal refinements → λ_{Nm} = mλ_N → C = λ_N/N is dimension-independent
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3. Rational points → refine to uniform → g_p = C · fisherMetric
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4. Density + smoothness → extends to all p
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5. Positivity → C > 0 -/
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theorem chentsov_theorem (n : ℕ) (hn : n ≥ 3) (g : RiemannianMetric n)
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(h_inv : IsChentsovInvariant g sorry)
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(h_perm : IsPermutationInvariant g)
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(h_smooth : True) :
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∃ (c : ℝ), c > 0 ∧ ∀ (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 = c * fisherMetric p X Y := by
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sorry
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end ChentsovTheorem
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