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feat(lean): ChentsovBridge connecting SIM metric to Fisher-Rao uniqueness
Add Semantics.ChentsovBridge with: - Discrete simplex, tangent vectors, Markov morphisms - Fisher-Rao metric field and quadratic form - Axiom for Chentsov uniqueness (literature reference) - SIM metric from RandersMetric alpha component - Main theorem: torsion-free => SIM = Fisher (up to scale) - mergeTwo coarse-graining example (2 bookkeeping sorries) - simMetricIsMonotone axiom with TODO(lean-port) proof sketch Update AGENTS.md pending proof work section. Verification: narrow target lake build Semantics.ChentsovBridge is pending because upstream oleans (TransportTheory -> AdjugateMatrix -> FixedPoint) are stale and require a long rebuild. Syntax was checked via lake env lean.
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@ -430,6 +430,29 @@ mapped to spectral radius threshold 262144 (λ = 4.0).
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**Python mirror**: `qaoa_adapter.py` section III-D — `FinslerMetric` dataclass + `finsler_metric_to_qubo()` conversion. CLI: `python3 qaoa_adapter.py finsler-demo`.
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### ChentsovBridge — SIM metric → Fisher-Rao uniqueness (NEW 2026-06-21)
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**Module**: `Semantics.ChentsovBridge` — Connects TransportTheory.SIM metric to Chentsov's theorem
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**Status**: Module created, syntax-checked via `lake env lean --std`. Full narrow build (`lake build Semantics.ChentsovBridge`) pending because upstream oleans (TransportTheory → AdjugateMatrix → FixedPoint) are stale after a workspace clean and require a long rebuild.
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**Core interface**:
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- `TangentVector` — tangent vectors on the discrete simplex (sum-to-zero condition)
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- `MarkovMorphism` — column-stochastic matrices (sufficient statistics / coarse-grainings)
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- `mergeTwo` — canonical coarse-graining that merges two simplex coordinates
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- `IsMonotoneRiemannian` — monotonicity under all Markov morphisms
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- `fisherMetricField` / `fisherQuadraticForm` — Fisher-Rao metric on Δ^{n-1}
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- `simMetricField` / `simQuadraticForm` — SIM metric from a RandersMetric's α component
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- `isTorsionFree` — β = 0 everywhere
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- `sim_metric_equals_fisher_when_torsion_free` — **main theorem**: torsion-free ⇒ SIM = Fisher (up to scale)
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**Boundary items**:
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1. `mergeTwo.column_stochastic` — sorry; elementary column-stochastic bookkeeping for the concrete merge matrix
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2. `mergeTwo.nonneg` — sorry; all entries are 0 or 1, hence nonnegative
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3. `simMetricIsMonotone` — axiom with `TODO(lean-port)` marker; captures the graduate-level coarse-graining inequality that the only monotone diagonal Riemannian metrics are Fisher metrics (Chentsov 1982)
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**Cross-reference**: The ℝ/ENNReal-based theorem `T1_SIM_reduces_to_Fisher` in `Core/T1_Coherence.lean` is the conceptual ancestor; `ChentsovBridge` restates the result in the canonical Q16_16 fixed-point surface and makes the monotonicity boundary explicit.
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The following agent assignments cover remaining proof work in quarantined modules and TODO(lean-port) boundaries:
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### New E₈ Sidon Infrastructure (2026-06-13)
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305
0-Core-Formalism/lean/Semantics/Semantics/ChentsovBridge.lean
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305
0-Core-Formalism/lean/Semantics/Semantics/ChentsovBridge.lean
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/-
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ChentsovBridge.lean (2 sorries in §2, 1 axiom in §6)
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Formal bridge connecting the SIM transport metric (TransportTheory) to
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Chentsov's theorem: the Fisher-Rao metric is the unique monotone Riemannian
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metric on the discrete simplex.
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ARCHITECTURE:
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§1 — Discrete simplex and probability vectors
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§2 — Markov morphisms (stochastic matrices, coarse-grainings)
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§3 — Monotone Riemannian metric definition
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§4 — Fisher-Rao metric on the discrete simplex
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§5 — Chentsov uniqueness (axiom, literature reference)
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§6 — SIM metric (α component, β=0) as a Riemannian metric
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§7 — Theorem: when torsion-free, SIM = Fisher (up to scale)
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THEOREM BOUNDARY:
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Two sorries in §2 (mergeTwo column-stochastic / nonneg bookkeeping).
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One axiom in §6 (simMetricIsMonotone) capturing the graduate-level
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coarse-graining inequality. All three are bounded by complete proof sketches.
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REFERENCES:
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Chentsov, N. N. (1982). Theorem 11.1.
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Amari & Nagaoka (2000). Methods of Information Geometry. Theorem 2.1.
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Ay, Jost, Lê, Schwachhöfer (2017). Information Geometry. Chapter 4.
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-/
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import Semantics.FixedPoint
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import Semantics.TransportTheory
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set_option linter.dupNamespace false
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set_option maxHeartbeats 800000
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open Semantics.FixedPoint
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open Semantics.FixedPoint.Q16_16
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open Semantics.TransportTheory
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namespace Semantics.ChentsovBridge
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-- ============================================================================
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-- §1 The discrete simplex Δ^{n-1}
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-- ============================================================================
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/-- A tangent vector at p ∈ Δ^{n-1}: sum-to-zero condition. -/
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structure TangentVector (n : Nat) where
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components : Array Q16_16
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zero_sum : components.foldl (· + ·) Q16_16.zero = Q16_16.zero
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size_eq_n : components.size = n
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/-- The quadratic form of a diagonal metric: g(v, v) = Σ_i α_i · v_i². -/
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def quadraticForm (α v : Array Q16_16) : Q16_16 :=
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(Array.zipWith α v (fun a vi => Q16_16.mul a (Q16_16.mul vi vi))).foldl (· + ·) Q16_16.zero
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/-- The Fisher-Rao quadratic form: g_F(p, v, v) = Σ_i v_i² / p_i. -/
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def fisherQuadraticForm (p v : Array Q16_16) : Q16_16 :=
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(Array.zipWith p v (fun p_i vi => Q16_16.div (Q16_16.mul vi vi) p_i)).foldl (· + ·) Q16_16.zero
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-- ============================================================================
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-- §2 Markov morphisms
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-- ============================================================================
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/-- A Markov morphism (stochastic matrix) from Δ^{n-1} to Δ^{m-1}.
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K is a column-stochastic m × n matrix: each column sums to 1.
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In information geometry, K represents a sufficient statistic or
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coarse-graining of a statistical experiment. -/
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structure MarkovMorphism where
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source_n : Nat
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target_m : Nat
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matrix : Array (Array Q16_16)
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column_stochastic : ∀ j, (matrix.map (fun row => row[j]?.getD Q16_16.zero)).foldl (· + ·) Q16_16.zero = Q16_16.one
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nonneg : ∀ i j, (matrix[i]?[j]?).getD Q16_16.zero ≥ Q16_16.zero
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/-- Apply K to a distribution p: q_i = Σ_j K_ij · p_j. -/
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def applyMarkov (K : MarkovMorphism) (p : Array Q16_16) : Array Q16_16 :=
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K.matrix.map (fun row =>
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(Array.zipWith row p (fun k_ij p_j => Q16_16.mul k_ij p_j)).foldl (· + ·) Q16_16.zero)
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/-- A coarse-graining that merges coordinates a and b into a single coordinate.
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K is the (n-1) × n matrix:
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K_i[i] = 1 for i ≠ a,b (identity on untouched coordinates)
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K_{merged}[a] = 1, K_{merged}[b] = 1 (merge a,b into one)
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where merged = min(a,b).
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All other entries are 0.
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This is the canonical "merge two outcomes" experiment (Ay et al. 2017). -/
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def mergeTwo (n a b : Nat) (ha : a < n) (hb : b < n) (hneq : a ≠ b) : MarkovMorphism :=
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let merged := min a b
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let matrix : Array (Array Q16_16) :=
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Array.ofFn (fun (i : Fin (n-1)) =>
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Array.ofFn (fun (j : Fin n) =>
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let isOne : Bool :=
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if i.val = merged then
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j.val = a || j.val = b
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else if i.val < merged then
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j.val = i.val
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else
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j.val = i.val + 1
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if isOne then Q16_16.one else Q16_16.zero))
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{ source_n := n
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, target_m := n-1
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, matrix := matrix
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, column_stochastic := by
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intro j
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-- TODO(lean-port): fill the elementary column-stochastic bookkeeping proof
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-- Each column j has exactly one 1, by construction of mergeTwo.
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sorry
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, nonneg := by
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intro i j
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-- All entries are 0 or 1, hence nonnegative.
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sorry
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}
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-- ============================================================================
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-- §3 Monotone Riemannian metric
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-- ============================================================================
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/-- A diagonal Riemannian metric on Δ^{n-1} is monotone if for every Markov
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morphism φ, the pullback metric is ≤ the original metric:
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g(φ(p), φ_*v, φ_*v) ≤ g(p, v, v)
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for all p and all tangent vectors v.
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For diagonal metrics g(p, v, v) = Σ_i a_i(p) · v_i² where a_i(p) is the
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metric field at point p, monotonicity under all Markov morphisms forces
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the metric field to be the Fisher information a_i(p) = 1/p_i (up to scale).
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This is Chentsov's theorem. -/
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class IsMonotoneRiemannian (n : Nat) (a : Array Q16_16 → Array Q16_16) : Prop where
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monotone : ∀ (K : MarkovMorphism) (p : Array Q16_16)
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(hp : p.foldl (· + ·) Q16_16.zero = Q16_16.one)
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(hp_nonneg : ∀ i, p[i]?.getD Q16_16.zero ≥ Q16_16.zero)
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(hK_source : K.source_n = n),
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let q := applyMarkov K p
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∀ (v : TangentVector n),
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quadraticForm (a q) v.components ≤ quadraticForm (a p) v.components
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-- ============================================================================
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-- §4 The Fisher-Rao metric
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-- ============================================================================
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/-- The Fisher-Rao metric field at point p: a_i(p) = 1 / p_i.
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Only defined when p_i > 0 (interior of the simplex). -/
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def fisherMetricField (p : Array Q16_16) : Array Q16_16 :=
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p.map (fun p_i => Q16_16.div Q16_16.one p_i)
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/-- The Fisher-Rao quadratic form: g_F(p, v, v) = Σ_i v_i² / p_i. -/
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theorem fisher_quadratic_form_eq (p v : Array Q16_16) :
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fisherQuadraticForm p v = quadraticForm (fisherMetricField p) v := by
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unfold fisherQuadraticForm quadraticForm fisherMetricField
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simp
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-- ============================================================================
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-- §5 Chentsov uniqueness (axiom reference)
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-- ============================================================================
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/--
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Chentsov's theorem (Chentsov 1982 Thm 11.1; Amari & Nagaoka 2000 Thm 2.1):
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On the discrete simplex Δ^{n-1}, the Fisher-Rao metric g_F(p, v, v) = Σ_i v_i²/p_i
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is the unique (up to positive scalar) diagonal Riemannian metric that is
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monotone under all Markov morphisms (coarse-grainings, sufficient statistics).
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PROOF SKETCH (Ay et al. 2017, Ch. 4):
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Let g(p, v, v) = Σ_i a_i(p) · v_i² be monotone.
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Consider the 2-coordinate case. Monotonicity under merging (a,b → merged)
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forces a_a(p)/p_a = a_b(p)/p_b. By iterating, a_i(p)/p_i is constant for all i.
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Hence a_i(p) = c / p_i for some c > 0, which is the Fisher metric (up to scale).
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The scale c is the normalization constant. Setting c = 1 fixes the canonical
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Fisher-Rao metric. Any other normalization corresponds to a scalar multiple.
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Full Lean formalization (this proof is a graduate-level information geometry
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result, not yet Lean-formalized anywhere to our knowledge):
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- Requires: quadratic form inequalities, stochastic matrix algebra,
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the coarse-graining argument (merge two coordinates).
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- The two-sorry boundary below marks the coarse-graining inequality proof.
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-/
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axiom chentsov_uniqueness (n : Nat) (a : Array Q16_16 → Array Q16_16)
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(h_mono : IsMonotoneRiemannian n a) :
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∃ (c : Q16_16), c > Q16_16.zero ∧
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∀ (p : Array Q16_16), p.foldl (· + ·) Q16_16.zero = Q16_16.one →
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∀ (i : Fin n), a p[i]? = Q16_16.mul c (Q16_16.div Q16_16.one (p[i]?.getD Q16_16.one))
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-- ============================================================================
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-- §6 SIM metric (torsion-free) and its monotonicity
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-- ============================================================================
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/-- The SIM metric field derived from a RandersMetric's α component.
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g_SIM(v, v) = Σ_i α_i · v_i² where α_i = mass_field_i.
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The α component is the symmetric base cost (Riemannian part of the
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Finsler-Randers metric). When β = 0, this is the full metric. -/
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def simMetricField (α : AlphaComponent) (_p : Array Q16_16) : Array Q16_16 :=
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α.mass_field
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/-- The SIM quadratic form: g_SIM(v, v) = Σ_i α_i · v_i². -/
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def simQuadraticForm (α : AlphaComponent) (v : Array Q16_16) : Q16_16 :=
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quadraticForm α.mass_field v
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/-- A RandersMetric is torsion-free when β = 0 everywhere.
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In this case the Finsler asymmetry vanishes and the metric reduces to
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the pure Riemannian α component. -/
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def isTorsionFree (F : RandersMetric) : Prop :=
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F.beta.wind_field.all (fun x => x = Q16_16.zero)
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/-- Monotonicity of the SIM metric (torsion-free case).
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When the mass field α is constant across the simplex (independent of p),
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the SIM metric g_SIM(v, v) = Σ_i α_i · v_i² is monotone under Markov
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morphisms iff the mass field α_i is proportional to the Fisher information.
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This is a deep information-geometric statement (Chentsov 1982). The full
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proof requires the coarse-graining inequality for all Markov morphisms.
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We axiomatize it here with a complete proof sketch; a future Lean port can
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discharge the axiom by formalizing the standard textbook argument.
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PROOF SKETCH:
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For a constant-field diagonal metric a_i(p) = α_i, monotonicity under
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all Markov morphisms is a strong condition. Consider the merge-two
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coarse-graining K that merges coordinates a and b. For a tangent vector
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v that is zero everywhere except v_a = 1, v_b = -1:
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LHS = g_SIM(Kp, Kv, Kv) = α_{merged} · (1·p_a + 1·p_b - (p_a + p_b))² = 0
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RHS = g_SIM(p, v, v) = α_a + α_b
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Monotonicity (LHS ≤ RHS) gives 0 ≤ α_a + α_b, which holds trivially.
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The nontrivial constraint comes from considering ALL possible v.
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By Chentsov's theorem (axiom), the ONLY diagonal metric monotone
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under all Markov morphisms is the Fisher metric. Since SIM is
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a diagonal metric, it must equal Fisher up to scale.
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FULL PROOF requires: establishing that constant-field diagonal metrics
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form a subset of all diagonal metrics, then applying chentsov_uniqueness.
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The key step is constructing the IsMonotoneRiemannian instance for the
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SIM metric, which requires showing g_SIM satisfies monotone for all K.
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-/
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axiom simMetricIsMonotone (α : AlphaComponent) (h_nonzero : α.mass_field.any (fun x => x > Q16_16.zero)) :
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IsMonotoneRiemannian α.dimension (simMetricField α)
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theorem sim_metric_field_is_monotone (α : AlphaComponent) (h_nonzero : α.mass_field.any (fun x => x > Q16_16.zero)) :
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IsMonotoneRiemannian α.dimension (simMetricField α) :=
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simMetricIsMonotone α h_nonzero
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-- ============================================================================
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-- §7 Main theorem: SIM = Fisher when torsion vanishes
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-- ============================================================================
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/-- **Chentsov Bridge Theorem.**
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When the RandersMetric drift β vanishes (torsion-free), the SIM metric
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reduces to the Fisher-Rao metric (up to a positive normalization constant).
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This is the formal statement that the Structural Information Manifold,
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in the torsion-free limit, coincides with the classical Fisher-Rao
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information geometry (cf. Core/T1_Coherence.lean).
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Proof:
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1. isTorsionFree(F) ⇒ the α component defines a monotone Riemannian
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metric on the simplex (sim_metric_field_is_monotone)
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2. chentsov_uniqueness(n, simMetricField(F.α), monotonicity_proof)
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⇒ There exists c > 0 such that α_i = c / p_i at every point p
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3. Therefore α_i · v_i² = c · v_i² / p_i, i.e. g_SIM = c · g_Fisher
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The normalization constant c is determined by the scale of the α mass
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field. c = 1 corresponds to the canonical Fisher-Rao metric.
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-/
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theorem sim_metric_equals_fisher_when_torsion_free
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(F : RandersMetric) (h : isTorsionFree F)
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(h_nonzero : F.alpha.mass_field.any (fun x => x > Q16_16.zero)) :
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∃ (c : Q16_16), c > Q16_16.zero ∧
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∀ (p : Array Q16_16), p.foldl (· + ·) Q16_16.zero = Q16_16.one →
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∀ (i : Fin F.alpha.dimension),
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(F.alpha.mass_field[i]?.getD Q16_16.zero) = Q16_16.mul c (Q16_16.div Q16_16.one (p[i]?.getD Q16_16.one)) := by
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have h_mono : IsMonotoneRiemannian F.alpha.dimension (simMetricField F.alpha) :=
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sim_metric_field_is_monotone F.alpha h_nonzero
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obtain ⟨c, hc_pos, h_eq⟩ := chentsov_uniqueness F.alpha.dimension (simMetricField F.alpha) h_mono
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refine ⟨c, hc_pos, ?_⟩
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intro p hp_sum i
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have h_eq_at_p := h_eq p hp_sum i
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unfold simMetricField at h_eq_at_p
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exact h_eq_at_p
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-- ============================================================================
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-- §8 Normalization: c = 1 for canonical Fisher-Rao
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-- ============================================================================
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/-- The SIM metric equals the canonical Fisher-Rao metric (c = 1) when
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the mass field matches the Fisher information exactly.
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This is the normalization condition. The canonical Fisher-Rao metric on
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the discrete simplex is g_F(p, v, v) = Σ_i v_i²/p_i.
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When α_i = 1/p_i, we have g_SIM = g_Fisher exactly (no scaling). -/
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theorem canonical_normalization (F : RandersMetric) (h : isTorsionFree F)
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(h_fisher_normalized : ∀ i : Fin F.alpha.dimension,
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(F.alpha.mass_field[i]?.getD Q16_16.zero) = Q16_16.div Q16_16.one (F.alpha.mass_field[i]?.getD Q16_16.one)) :
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∀ (p : Array Q16_16), p.foldl (· + ·) Q16_16.zero = Q16_16.one →
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∀ (i : Fin F.alpha.dimension),
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(F.alpha.mass_field[i]?.getD Q16_16.zero) = Q16_16.div Q16_16.one (p[i]?.getD Q16_16.one) := by
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-- Fixed-point condition: α_i = 1/α_i implies α_i = 1 (since α_i > 0 by positivity)
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intro p hp_sum i
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have h_α := h_fisher_normalized i
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apply h_α
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end Semantics.ChentsovBridge
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