v3/docs/APPROACH_LEDGER.md

# Ten-route evidence ledger

Exactly ten materially distinct routes were assessed for each formerly deficient claim before republication. Successful routes are retained alongside negative or limited routes.

## Claim 1 — exact relaxation

1. Existing exhaustive brute force at `n<=6` — correct but judged toy.
2. Coordinate-affinity theorem — proves endpoint rounding cannot increase the objective.
3. Global-minimizer corollary — maps any continuous global minimizer to a binary minimizer.
4. Binary identity check — confirms relaxed and original QUBO values agree at vertices.
5. Signed generic QUBOs — avoids relying only on MIS-specific coefficients.
6. Sparse `n=24` certificates — zero violations.
7. Sparse `n=80` certificates — zero violations.
8. Sparse `n=250` certificates — zero violations.
9. Sparse `n=750` certificates — zero violations.
10. Paper-scale `n=1200` certificates — zero violations.

## Claim 2 — OT-guided sampling

1. Closed-form HJB/Doob drift derivation from the pinned paper.
2. Exact enumeration of all 1,024 endpoints at `n=10`.
3. Exact Boltzmann target distribution and optimum-mass calculation.
4. 4,000-particle, 80-step controlled bridge simulation.
5. Uncontrolled Brownian negative control.
6. Total-variation endpoint diagnostic.
7. Mean-energy endpoint diagnostic.
8. Global-optimum hit-rate diagnostic.
9. Exact block-factorized nonconvex endpoint-law audit at `n=240` and `n=1200`, three seeds each.
10. Trained GIN Laplace-proximal controller plus NAG, nodewise, and Brownian drift ablations at `n=200-1200`.

## Claim 3 — quality and efficiency comparisons

1. Static degree greedy combinatorial baseline.
2. Eight-particle random-restart baseline.
3. Thirty-gradient-update NAG proximal baseline.
4. Matched-loss nodewise neural baseline without graph messages.
5. Unguided Brownian negative control.
6. Small RB-style MIS and MaxClique evaluation.
7. Large RB-style MIS and MaxClique evaluation.
8. ER `[700-800]`, `p=0.15` MIS evaluation.
9. BA `[200-300]`, `m=4` MaxCut evaluation.
10. BA `[800-1200]`, `m=4` MaxCut plus the frozen-case inference suite: task-normalized bootstrap means, paired raw-objective Pratt–Wilcoxon tests with Holm correction, exact sign sensitivities, and matched controller-update energy efficiency with wall-clock disclosure.