math(formula): pure Fisher distance derivation — zero English in formulas, full chain from Chentsov to closed form

2026-07-31 01:25:21 +00:00 · 2026-06-23 05:41:10 -05:00 · 2026-06-23 05:41:10 -05:00 · 7c6e9bfc89
commit 7c6e9bfc89
parent ffbaf3676c
1 changed files with 170 additions and 0 deletions
--- a/docs/fundamental_math/PURE_FORMULA_DERIVATION.md
+++ b/docs/fundamental_math/PURE_FORMULA_DERIVATION.md
@ -0,0 +1,170 @@
+# PURE MATH FORMULA: Fisher Distance on Δ₇
+## Zero English inside formulas. Each equality justified. Verifiable numerically.
+
+---
+
+## THE CHAIN
+
+**Given:** p, q ∈ Δ₇ (probability simplex, 8 dimensions)
+
+**Step 0 — Chentsov's metric:**
+
+    g_p(u,v) = Σᵢ₌₁⁸ (uᵢ vᵢ / pᵢ)
+
+**Justification:** Chentsov 1972, Amari 1985. Unique metric respecting sufficient statistics.
+
+---
+
+**Step 1 — The √p embedding:**
+
+    φ : Δ₇ → S⁷,    φ(p) = (√p₁, √p₂, ..., √p₈)
+
+**Lemma:** ‖φ(p)‖₂ = 1
+
+    ‖φ(p)‖₂² = Σᵢ₌₁⁸ (√pᵢ)² = Σᵢ₌₁⁸ pᵢ = 1
+
+**Justification:** p ∈ Δ₇ ⇒ Σpᵢ = 1 by definition.
+
+---
+
+**Step 2 — Pullback of round metric:**
+
+    (φ* g_{S⁷})_p(u,v) = ¼ · g_p(u,v)
+
+**Proof sketch:**
+
+Let c(t) be a curve in Δ₇, c(0) = p, ċ(0) = v.
+
+    γ(t) = φ(c(t)) = (√c₁(t), ..., √c₈(t))
+
+    γ̇ᵢ(0) = vᵢ / (2√pᵢ)
+
+Round metric on S⁷:
+
+    ⟨γ̇, γ̇⟩_{S⁷} = Σᵢ γ̇ᵢ² = Σᵢ vᵢ² / (4pᵢ) = ¼ · Σᵢ vᵢ²/pᵢ = ¼ · g_p(v,v)
+
+**Justification:** Chain rule + direct computation.
+
+---
+
+**Step 3 — Geodesics are great circles:**
+
+S⁷ has round metric ⇒ geodesics are great circles.
+
+Great-circle distance:
+
+    d_{S⁷}(a,b) = arccos(⟨a,b⟩)
+
+**Justification:** Standard Riemannian geometry of the sphere.
+
+---
+
+**Step 4 — Inner product on S⁷:**
+
+    ⟨φ(p), φ(q)⟩ = Σᵢ₌₁⁸ √(pᵢ qᵢ)
+
+**Justification:** Definition of φ + Euclidean inner product.
+
+---
+
+**Step 5 — Fisher distance (THE FORMULA):**
+
+    d_F(p,q) = 2 · d_{S⁷}(φ(p), φ(q))
+             = 2 · arccos(⟨φ(p), φ(q)⟩)
+             = 2 · arccos( Σᵢ₌₁⁸ √(pᵢ qᵢ) )
+
+**Justification:** Steps 2+3+4 combined. The factor 2 comes from Step 2 (g = 4·φ*g_{S⁷}).
+
+---
+
+## THE CLOSED-FORM RESULT
+
+```
+┌─────────────────────────────────────────────────────┐
+│                                                     │
+│   d_F(p,q) = 2 · arccos( Σᵢ₌₁ⁿ √(pᵢ qᵢ) )         │
+│                                                     │
+│   Domain: p, q ∈ Δₙ (any dimension n ≥ 2)          │
+│   Range: [0, π]                                     │
+│   Equality: d_F(p,q) = 0  ⟺  p = q                │
+│   Max: d_F(p,q) = π  when p, q are antipodal       │
+│        (e.g., p = (1,0,...,0), q = (0,1,0,...,0)) │
+│                                                     │
+└─────────────────────────────────────────────────────┘
+```
+
+---
+
+## VERIFICATION INSTANCE (n=8)
+
+**Inputs:**
+
+    p = (0.3, 0.1, 0.15, 0.05, 0.2, 0.08, 0.07, 0.05)
+    q = (0.2, 0.2, 0.1, 0.1, 0.15, 0.1, 0.1, 0.05)
+
+**Step A — Compute √(pᵢqᵢ):**
+
+    √(0.3×0.2) = 0.24494897
+    √(0.1×0.2) = 0.14142136
+    √(0.15×0.1) = 0.12247449
+    √(0.05×0.1) = 0.07071068
+    √(0.2×0.15) = 0.17320508
+    √(0.08×0.1) = 0.08944272
+    √(0.07×0.1) = 0.08366600
+    √(0.05×0.05) = 0.05000000
+
+**Step B — Sum:**
+
+    S = 0.97586930
+
+**Step C — Arccos:**
+
+    arccos(0.97586930) = 0.22012896
+
+**Step D — Multiply by 2:**
+
+    d_F(p,q) = 2 × 0.22012896 = 0.44025792
+
+**OUTPUT: d_F(p,q) ≈ 0.440258**
+
+---
+
+## PROPERTIES (all verifiable)
+
+**Symmetry:**
+
+    d_F(p,q) = 2·arccos(Σ√(pᵢqᵢ)) = 2·arccos(Σ√(qᵢpᵢ)) = d_F(q,p)  ✓
+
+**Identity:**
+
+    d_F(p,p) = 2·arccos(Σ√(pᵢpᵢ)) = 2·arccos(Σpᵢ) = 2·arccos(1) = 0  ✓
+
+**Triangle inequality:**
+
+    d_F(p,q) ≤ d_F(p,r) + d_F(r,q)  for all p,q,r ∈ Δ₇
+
+    Proof: Great-circle distance on S⁷ satisfies triangle inequality.
+           Pullback by isometry preserves triangle inequality.  ✓
+
+**Bound:**
+
+    0 ≤ d_F(p,q) ≤ π
+
+    Proof: arccos: [-1,1] → [0,π]. The argument Σ√(pᵢqᵢ) ∈ [0,1]
+           by Cauchy-Schwarz: (Σ√(pᵢqᵢ))² ≤ (Σpᵢ)(Σqᵢ) = 1.  ✓
+
+---
+
+## WHY THIS IS THE RIGHT FORMULA
+
+1. **Chentsov's theorem** says: any metric respecting sufficient statistics
+   MUST be the Fisher metric (up to constant).
+
+2. **The √p embedding** maps Δ₇ → S⁷ isometrically (up to factor 4).
+
+3. **Geodesics on S⁷** are great circles with known distance formula.
+
+4. **Pulling back** gives the Fisher distance formula above.
+
+There is no choice in this formula. It is forced by the geometry of the
+probability simplex combined with Chentsov's uniqueness result.