# Chentsov → Finsler → QUBO → QAOA: Formal Routing Pipeline

## Overview

This document formalizes the connection between four layers:

1. **Chentsov's theorem** — the Fisher information metric is the UNIQUE Riemannian
   metric on the space of probability distributions that is monotone under Markov
   morphisms (sufficient statistics).
2. **Finsler-Randers geometry** — generalizes Riemannian metrics with a
   direction-dependent norm: `F(p,v) = α(p,v) + β(p,v)` where α is the symmetric
   base cost and β is an asymmetric drift 1-form.
3. **QUBO discretization** — the continuous Finsler norm is discretized onto a
   finite set of binary variables, producing a quadratic unconstrained binary
   optimization problem whose matrix encodes pairwise directional crossing costs.
4. **QAOA routing** — the QUBO is solved via a parameterized quantum circuit
   whose measurement bitstring corresponds to the minimal-Finsler-cost path
   through the state space.

**Claim boundary**: `chentsov-finsler-qubo-qaoa-routing-v1;formalization-only;no-lean-spectral`

## 1. Chentsov's Theorem Forces the Metric

**Theorem (Chentsov 1972)**: On the space of probability distributions over a
finite set with ≥3 elements, the Fisher information metric is the unique
Riemannian metric (up to a constant factor) that is invariant under sufficient
statistics.

For the Hachimoji Baker field (8-state label at each lattice point (m,n)), the
probability of state s ∈ {A,T,G,C,B,S,P,Z} at point (m,n) defines a
distribution. The Fisher metric on this distribution is:

```
g_ij(m,n) = Σ_s (1/p_s(m,n)) · (∂p_s/∂θ_i) · (∂p_s/∂θ_j)
```

where θ = (m,n) are the lattice coordinates.

**Consequence**: The Riemannian structure on the Hachimoji Baker manifold is not
a modeling choice — it is forced by the 8-state statistical structure. Any
metric that respects the Hachimoji encoding must be the Fisher metric.

## 2. Finsler-Randers Generalization for Asymmetric Routing

While the Fisher metric (Chentsov) gives the unique Riemannian structure,
real routing problems involve direction-dependent costs:

- Moving "with the grain" (along dominant Lyapunov directions) costs less
- Moving "against the grain" (against Lyapunov wind) costs more
- Crossing boundaries between TAD domains has different costs in each direction

The Finsler-Randers metric captures this:

```
F(p,v) = α(p,v) + β(p,v)

α(p,v) = √(v^T · M(p) · v)     -- symmetric base cost (Riemannian)
β(p,v) = ⟨W(p), v⟩             -- asymmetric drift 1-form
```

where:
- `M(p)` is the mass tensor at position p (Fisher metric forced by Chentsov)
- `W(p)` is the wind/drift field at position p (Lyapunov exponent gradient)
- `F(p,-v) ≠ F(p,v)` when β ≠ 0 — direction matters

### The Finsler-Fisher Connection

The Fisher metric provides α. The drift β captures the anisotropy of the
Hachimoji field: the probability of transition from state s_i to s_j depends on
the Baker form value Λ(m,n) and the threshold B−C, which are direction-dependent.

Specifically, for the Hachimoji substitution:

```
β_i→j = D_KL(p_j || p_i) - D_KL(p_i || p_j)
```

where D_KL is the Kullback-Leibler divergence. This is the **information
drift** — the asymmetry in the KL cost of transitioning between adjacent
distributions.

## 3. QUBO Discretization of the Finsler Norm

To make the continuous Finsler geodesic problem optimizable, we discretize onto
a finite set of binary variables representing routing choices.

### 3.1 Discretization Scheme

Let the continuous state space be partitioned into N bins (routing options).
Each bin corresponds to a direction vector v_k. The cost of transitioning from
bin i to bin j is:

```
C_ij = F(p_i, v_j - v_i) + λ · g(p_i, p_j)
```

where:
- `F(p_i, v_j - v_i)` is the Finsler cost of moving from direction i to j
- `g(p_i, p_j)` is the Fisher-Riemannian base distance
- `λ` is a tradeoff parameter

### 3.2 QUBO Construction

The QUBO matrix Q encodes these costs as a quadratic binary optimization:

```
E(x) = x^T Q x = Σ_i Σ_j Q_ij · x_i · x_j
```

where x_i ∈ {0,1} indicates whether routing option i is selected.

The QUBO matrix is constructed from the Finsler metric as:

```
Q_ij = α_ij + β_ij
Q_ii = 0   (no self-cost)
```

with:
- `α_ij = v_i^T · M · v_j` — the symmetric Fisher cost component
- `β_ij = ⟨W, v_j - v_i⟩` — the asymmetric Finsler drift component

**Key property**: Q is NOT symmetric — the Finsler anisotropy makes
Q_ij ≠ Q_ji because β_ij ≠ β_ji. This is a defining feature: the QUBO
matrix inherits the direction-dependent structure of the Finsler norm.

### 3.3 Lean Formalization (TransportQUBOBridge.lean)

The bridge theorem states:

```lean
theorem finsler_geodesic_minimizes_qubo_objective
    (F : RandersMetric) (dirs : Array (Array Q16_16))
    (h_dirs_valid : ∀ i, (dirs[i]!).size = F.alpha.dimension) :
    let Q := randersMetricToQUBO F dirs
    let assignment := geodesicAssignment F dirs
    QUBOFormulation.objective Q assignment ≤
    QUBOFormulation.objective Q any_assignment := ...
```

This says: the assignment corresponding to the Finsler geodesic minimizes the
QUBO objective — the geodesic IS the ground state.

## 4. QAOA Routing

The QUBO is solved via QAOA (Quantum Approximate Optimization Algorithm):

```
U(β,γ) = Π_{k=1}^p exp(-iβ_k H_B) · exp(-iγ_k H_C)
```

where:
- `H_C = Σ_ij Q_ij Z_i Z_j + Σ_i h_i Z_i` — the cost Hamiltonian (from QUBO)
- `H_B = Σ_i X_i` — the mixing Hamiltonian
- `γ_k, β_k` — variational parameters optimized to minimize ⟨H_C⟩

### 4.1 8-Qubit Circuit for Hachimoji Routing

The Hachimoji alphabet has 8 states, so each routing decision is an 8-qubit
circuit. The measurement bitstring (8 bits) maps directly to a Greek-state
LogogramReceipt via `HachimojiSubstitution.fromQAOABitstring`.

### 4.2 The Adapter Pipeline

```
Lean EntropyMeasures.QUBOFormulation
  → Python QUBO dataclass
    → Ising (h + J)
      → PauliSum strings
        → Cirq/Qiskit parameterized circuit
          → measurement bitstring
            → HachimojiSubstitution LogogramReceipt
```

The Python bridge (qaoa_adapter.py, 2,421 lines) implements all these
conversions. The missing link is the FinslerMetric → QUBO conversion.

## 5. The Combined Pipeline

```
Continuous:        Hachimoji Baker field probability distributions
                        ↓ Chentsov (1972)
                   Fisher information metric g_ij (UNIQUE)
                        ↓
                   Finsler-Randers norm F(p,v) = α + β
                   (α = Fisher base, β = information drift)
                        ↓ discretization (N bins)
Discrete:          QUBO matrix Q_ij = α_ij + β_ij
                   (Q not symmetric — Finsler anisotropy)
                        ↓ QAOA (p layers)
                   8-qubit parameterized circuit
                        ↓ measurement
Quantum:           8-bit bitstring × 8 = 64-bit routing
                        ↓ HachimojiSubstitution
                   LogogramReceipt (Greek state alphabet)
```

### Formal Claim

The pipeline is valid when:

1. **Chentsov holds**: The state space has ≥3 distinct distributions
   (Hachimoji has 8) — satisfied.
2. **Finsler norm is strongly convex**: `β² < α²` everywhere
   (Randers strong convexity) — ensured by the `randersStrongConvex` predicate.
3. **Discretization is fine enough**: The N bins cover the routing directions
   with sufficient resolution — bounded by the QUBO size constraint.
4. **QAOA approximation ratio**: The variational circuit finds the ground state
   within bounded error — a computational claim, not a formal one.

## 6. Open Questions

1. **Optimal discretization resolution**: What is the minimum N (number of QUBO
   variables) needed to recover the continuous geodesic within ε error?
2. **QUBO asymmetry handling**: Standard QAOA assumes symmetric Q_ij. The
   Finsler-derived QUBO is asymmetric (Q_ij ≠ Q_ji). How should the asymmetry
   be folded into the Ising Hamiltonian?
3. **Chentsov-for-Finsler**: Does Chentsov's theorem generalize to Finsler
   metrics? The uniqueness of the Fisher metric is a Riemannian result — the
   Finsler case may have a family of permissible metrics parameterized by the
   drift 1-form.