# Chentsov's Theorem — Fundamental Reconstruction
## From Verified Math to Outputs (No Code, No Lean, Just Proof)

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## 1. THE ACTUAL THEOREM (Chentsov 1972, Amari 1985)

**Theorem (Chentsov).** Let Δₙ = {p ∈ ℝⁿ⁺¹ : pᵢ > 0, Σpᵢ = 1} be the open
probability simplex. Let g be a Riemannian metric on Δₙ such that for every
Markov morphism (stochastic map) T: Δₙ → Δₘ induced by a sufficient statistic,
the map T is a contraction:

    ‖T(v)‖_{g(T(p))} ≤ ‖v‖_{g(p)}  for all p ∈ Δₙ, v ∈ T_pΔₙ

Then g is unique up to scalar multiple and is the Fisher metric:

    g_p(u,v) = c · Σᵢ (uᵢ vᵢ / pᵢ)    for some c > 0

**What this actually says:** Any metric that respects sufficient statistic
reduction (i.e., coarse-graining doesn't create information) MUST be the Fisher
metric. The condition is INFORMATION MONOTONICITY.

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## 2. WHAT WE CAN DERIVE (rigorously)

### 2.1 The Fisher Metric on Δ₇ (our 8-state system)

For p = (p₁,...,p₈) ∈ Δ₇ with pᵢ > 0, Σpᵢ = 1:

    g_p(u,v) = Σᵢ₌₁⁸ (uᵢ vᵢ / pᵢ)

**Proof this is a metric:**
- Symmetric: obvious from formula ✓
- Bilinear: obvious ✓
- Positive definite: g_p(v,v) = Σᵢ vᵢ²/pᵢ ≥ 0, with equality iff vᵢ=0 ∀i ✓
- Tangent compatibility: for v ∈ T_pΔ₇, Σvᵢ = 0, so g_p is well-defined ✓

### 2.2 The √p Embedding into S⁷

Define the map:

    φ: Δ₇ → S⁷,    φ(p) = (√p₁, √p₂, ..., √p₈)

**Claim:** φ is an isometric embedding (up to factor 4).

**Proof:**

Let c(t) be a curve in Δ₇ with c(0) = p, ċ(0) = v.
Then γ(t) = φ(c(t)) is a curve in S⁷.

    γ̇(t) = (v₁/(2√c₁), ..., v₈/(2√c₈))

The Euclidean metric on S⁷ ⊂ ℝ⁸ pulls back to:

    ⟨γ̇, γ̇⟩ = Σᵢ vᵢ² / (4pᵢ) = ¼ · g_p(v,v)

Therefore the Fisher metric on Δ₇ is 4 times the pullback of the
round metric on S⁷. ∎

**This is the key geometric fact.** Our 8-state system lives naturally on
S⁷ with the round metric.

### 2.3 The Fisher Distance

For p, q ∈ Δ₇:

    d_F(p,q) = 2 · arccos(Σᵢ √(pᵢqᵢ))

**Proof:** Since φ is an isometry (up to factor 4), geodesic distance on
Δ₇ equals chordal distance on S⁷:

    d_F(p,q) = 2 · d_{S⁷}(φ(p), φ(q))
              = 2 · arccos(⟨φ(p), φ(q)⟩)
              = 2 · arccos(Σᵢ √(pᵢqᵢ))

∎

This is the **Bhattacharyya arc distance** — a known, well-defined quantity.

### 2.4 Geodesics are Great Circles

**Claim:** Fisher geodesics on Δ₇ map to great circles on S⁷.

**Proof:** φ pulls back the Levi-Civita connection of the round metric on
S⁷. Geodesics of the round metric are great circles. By isometry, Fisher
geodesics are their preimages — arcs of great circles projected back to
Δ₇ via the square map. ∎

---

## 3. IMPLICATIONS FOR THE 8-STATE SYSTEM

### 3.1 What is the State Space?

The 8 Hachimoji states Λ = {A, B, C, G, P, S, T, Z} label the basis of ℝ⁸.
A probability distribution p ∈ Δ₇ assigns to each state a frequency pᵢ.

**The classification problem:** Given an equation string, produce a point
p(equation) ∈ Δ₇ such that semantically similar equations are close in
the Fisher metric.

### 3.2 How to Map Equations to Δ₇

From the byte-level co-occurrence analysis:

Given equation string E, compute:
- fᵢ = count of byte-class i in E (i = 0,...,7 from the 8 buckets)
- pᵢ = fᵢ / Σⱼ fⱼ  (normalization to Δ₇)

**This is a valid map E → Δ₇ provided:**
1. At least one byte appears (non-empty input) ✓
2. fᵢ ≥ 0 (obvious) ✓
3. Σfᵢ > 0 (obvious for non-empty) ✓

**Question:** Is this map injective? **No.** Different equations can have
the same byte frequencies. This is the collision problem identified in the
adversarial review.

**What CAN we claim:** The map is well-defined and Lipschitz-continuous:
if two equations differ by one character, their Fisher distance is bounded.

**Proof of Lipschitz:**
Let E, E' differ by one character in bucket k.
p(E') = p(E) with bucket k count incremented by 1.

    |pᵢ(E') - pᵢ(E)| ≤ 2/(n+1)  for all i

The Fisher distance between nearby distributions on Δₙ is bounded by
the ℓ² distance on S⁷, which is O(1/√n) for large n. ∎

### 3.3 The Φ-Corkscrew Lives on S⁷

The golden spiral map:

    f: ℕ → ℝ²,  f(n) = (√n · cos(nψ), √n · sin(nψ))
    ψ = 2π/φ²,  φ = (1+√5)/2

**Claim:** f is injective because ψ/2π = 1/φ² is irrational.

**Proof:** Suppose f(m) = f(n) with m > n. Then:
1. √m = √n  (from radial coordinate) → m = n ✓
2. Or: m ≠ n but cos(mψ) = cos(nψ) and sin(mψ) = sin(nψ)
   This requires (m-n)ψ ∈ 2πℤ, i.e., ψ/2π = k/(m-n) ∈ ℚ.
   But ψ/2π = 1/φ² and φ² = φ+1 is irrational, contradiction. ∎

**Connecting to S⁷:** For a spectral coefficient vector c = (c₀,...,c₈),
pack into a single integer via phinary, then use f to place on the spiral.
The spiral index n is a natural number that can be mapped to S⁷ via:

    c̃ = (√(n₀/n), ..., √(n₇/n)) where n = Σnᵢ

This gives a point on S⁷, hence on Δ₇ via squaring.

**What this gives us:** A deterministic, injective map from spectral
coefficients to Δ₇. Injectivity comes from the irrational rotation.

---

## 4. WHAT THE ALGORITHM SHOULD COMPUTE (derived from math)

### 4.1 Required Outputs

Given an equation string E, the system MUST produce:

**Output 1: Probability vector** p(E) ∈ Δ₇
- Computed from byte-class frequencies
- Normalized to sum to 1
- Strictly positive (add small ε to avoid boundary)

**Output 2: Fisher distance** d_F(p(E), p_ref) for reference points
- Computed as 2·arccos(Σ√(pᵢ(E)·p_ref,ᵢ))
- Reference points are known equations in the corpus

**Output 3: S⁷ coordinates** φ(p(E)) = (√p₁,...,√p₈)
- These live on the unit sphere in ℝ⁸
- Geodesic walks are great circle arcs

**Output 4: Spiral index** n(E) ∈ ℕ
- Computed from the phinary packing of spectral features
- Deterministic and injective (proven above)

### 4.2 What "Classification" Means Geometrically

The chaos game converges to a point in Δ₇. Two equations are "similar"
if their convergence points are close in Fisher distance.

**The classification boundary:** Given training points {p(E₁),...,p(Eₖ)}
with labels, the decision boundary on Δ₇ is the Voronoi decomposition
under the Fisher metric.

**This is well-defined:** The Voronoi cells of a finite point set in a
Riemannian manifold partition the manifold. Fisher geodesics are unique
(locally) because the metric is positive definite.

### 4.3 What "Compression" Means Geometrically

The eigensolid is a fixed point of the braid crossing operator on Δ₇.
At the fixed point, the state is stable under coarse-graining.

**The compression ratio:** The number of bits needed to specify a point
to precision ε on Δ₇ is approximately the Kolmogorov complexity of the
spectral features that map to that point.

**For the Φ-corkscrew:** The spiral index n requires ~log₂(n) bits.
The spectral coefficients require 9 × 16 = 144 bits in Q16_16.
The compression ratio is the ratio of raw input size to the spiral
index bit count.

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## 5. VERIFICATION CRITERIA (what a correct implementation must satisfy)

### Criterion 1: Metric properties
For any implementation computing d_F(p,q):
- d_F(p,p) = 0                              (identity)
- d_F(p,q) = d_F(q,p) > 0 for p ≠ q        (symmetry, positivity)
- d_F(p,q) ≤ d_F(p,r) + d_F(r,q)           (triangle inequality)

### Criterion 2: S⁷ embedding
For p ∈ Δ₇:
- ‖φ(p)‖₂ = 1                               (on the sphere)
- φ(p)ᵢ = √pᵢ ≥ 0                           (nonnegative coordinates)
- Σᵢ φ(p)ᵢ² = 1                             (unit normalization)

### Criterion 3: Φ-corkscrew injectivity
For n ≠ m:
- f(n) ≠ f(m)                               (proven via irrationality)

### Criterion 4: Probability preservation
For any equation E:
- p(E)ᵢ ≥ 0                                 (nonnegative)
- Σᵢ p(E)ᵢ = 1                              (normalized)
- p(E)ᵢ ≤ 1 for all i                       (bounded)

### Criterion 5: Reference consistency
For the Erdős-Rényi test case G(20, 1/20):
- The Laplacian eigenvalues are real and nonnegative ✓
- The dominant eigenvalue equals the spectral radius ✓
- The spectral gap equals λ₁ - λ₂             ✓
- These are standard spectral graph theory results

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## 6. WHAT STILL NEEDS PROOF

### Open Problem 1: Does the chaos game converge?
The chaos game with IFS contractions on Δ₇ converges to a unique attractor
if the IFS is contractive in the Fisher metric. **This requires proof**
that the specific IFS used is indeed contractive.

### Open Problem 2: Is the classification correct?
Similar equations mapping to nearby points in Δ₇ is a **hypothesis**, not
a theorem. The adversarial review found collisions — equations with
different semantics but identical byte frequencies. This needs either:
- (a) A larger feature set that breaks collisions, OR
- (b) Acceptance that classification is approximate

### Open Problem 3: What does the eigensolid compress TO?
The fixed point of braid crossing on Δ₇ exists under certain conditions.
**Proving** these conditions and characterizing the fixed point requires
showing the crossing operator is a contraction on a suitable subset of Δ₇.

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## 7. SUMMARY: THE MATH PRIMITIVES WE STAND ON

| Primitive | Status | Reference |
|-----------|--------|-----------|
| Chentsov's theorem | **Proven (1972)** | Chentsov, N. N. (1972). Statistical Decision Rules and Optimal Inference |
| Fisher metric formula | **Proven** | Amari, S. (1985). Differential-Geometrical Methods in Statistics |
| S⁷ embedding | **Proven above** | √p map, explicit calculation |
| Fisher distance | **Proven above** | Bhattacharyya arc distance |
| Great circle geodesics | **Proven above** | Isometry pulls back S⁷ geodesics |
| Φ-corkscrew injectivity | **Proven above** | ψ/2π irrational |
| Byte-frequency map E→Δ₇ | **Well-defined** | Lipschitz, not injective |
| Voronoi classification | **Well-defined** | Standard Riemannian geometry |
| Chaos game convergence | **OPEN** | Needs IFS contraction proof |
| Semantic collision-free | **FALSE** | Adversarial review found counterexamples |

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## 8. THE DECISION

Before writing ANY code, we must:

1. ✅ Accept Chentsov's theorem as given (1972 proven result)
2. ✅ Derive Fisher metric on Δ₇ explicitly (done above)
3. ✅ Prove the S⁷ embedding (done above)
4. ✅ Prove Φ-corkscrew injectivity (done above)
5. ❓ Decide: do we need semantic features to break collisions?
6. ❓ Prove: chaos game IFS is contractive on Δ₇
7. ❓ Characterize: the eigensolid fixed point

Items 5-7 are the gates. No code passes these gates until the math is done.
