# PURE MATH FORMULA: Fisher Distance on Δ₇
## Zero English inside formulas. Each equality justified. Verifiable numerically.

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> **Domain note:** All Fisher metric computations require \(p_i > 0\) for all \(i\). Raw byte frequencies produce \(p_i = 0\) for unseen classes. Apply **Laplace smoothing**: \(F(E)_i = (f_i + 1)/(|E| + 8)\) to ensure strict positivity. This keeps all computations in the open simplex \(\Delta_7\) and regularizes the Fisher metric near boundaries.

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## 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.

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**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.

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**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.

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**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.

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**Step 4 — Inner product on S⁷:**

    ⟨φ(p), φ(q)⟩ = Σᵢ₌₁⁸ √(pᵢ qᵢ)

**Justification:** Definition of φ + Euclidean inner product.

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**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}).

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## 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)) │
│                                                     │
└─────────────────────────────────────────────────────┘
```

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## 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**

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## 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.  ✓

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## 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.