bundle/v0/SCIENTIFIC_METHOD.md

# Independent scientific method

## Implemented route

The current scientific backend independently implements the main Gaussian-portfolio route from Section 6.1 and Appendix A.2.2:

1. construct the SOCP for the worst-case Gaussian CVaR objective;
2. solve it directly with Clarabel, retaining primal variables, dual variables, slack variables, objective values, and explicit KKT/cone residuals;
3. obtain the derivative of the optimal conic value with respect to the lower-triangular transport factor via the dual-envelope identity;
4. calculate the weighted Gelbrich distance and its analytic derivative by differentiating `tr(sqrt(P P^T))` through the matrix square root;
5. construct the smoothed bootstrap coverage penalty in Eqs. 14-15 and combine its analytic derivative with the conic-value derivative;
6. apply the paper's constant `1e-4` step, gradient clipping, metric eigenvalue clipping, and Cholesky refactorization.

The implementation does not import or execute author code. The pinned author revision remains a reference for term-by-term comparison.

The Gaussian risk coefficient is an explicit estimand parameter. `gaussian_exact` uses the normal-distribution CVaR coefficient printed in the paper's numerical definition; `moment_class` uses the Appendix/official SOCP coefficient. The former is primary and the latter is a required matched sensitivity.

## Verified locally

The clean matched bounded pair uses task `portfolio_gaussian_main/d001/r00/n010`, identical data and bootstrap seeds, twenty bootstrap distributions, the paper's `beta=0.1`, `gamma=0.05`, `lambda_p=10`, `eta_p=100`, linear quantile interpolation, and three outer iterations. Hardened canary `v14` is the primary `gaussian_exact` estimand; `v15` changes only the risk coefficient to the Appendix/official `moment_class` value. Both ran from clean commit `8be5520` and check:

- analytic Gelbrich derivative against central differences;
- Clarabel dual-envelope derivative against re-solved central differences;
- the complete outer derivative at a separately declared point where the coverage penalty is active;
- primal, dual, equality, cone, and duality-gap residuals at every iteration;
- raw JSONL trace persistence and hashes.

The verifier independently regenerates the selected task and dataset and binds the manifest, config, plan, source, published parameters, trace, and terminal solution. Each solve reports primal/dual feasibility, primal/dual cone membership, complementarity, and duality gap.

The optimizer's terminal factor is re-solved before reporting the final decision, objective, and residuals. Clarabel's native statuses are mapped into the canonical artifact vocabulary. At exactly zero Gelbrich distance, the implementation returns a conservative zero subgradient instead of injecting a numerical regularizer.

## Independently implemented Appendix routes

The empirical-reference routes use exact finite-support optimal transport with uniform marginals. For W1, the cost is `||L^T(x-y)||_2`; for W2, the transport program minimizes its square and the reported distance takes the square root. Coupling-envelope derivatives are checked against central differences at nonzero transport values, and marginal residuals are retained.

- **Appendix A.2.1, empirical W1 portfolio.** The full-support linear-loss reformulation reduces to `min -mean(xi)^T w + epsilon*lambda`, subject to simplex/nonnegativity constraints, `Lq=w`, and `||q||<=lambda`. This algebraically removes redundant sample epigraph variables without changing the value.
- **Appendix A.3.1, W1 absolute regression.** The SOCP minimizes the empirical absolute residual epigraph plus `epsilon*t`, with `Lq=[-w;1]` and `||q||<=t`.
- **Appendix A.3.2, W2 squared regression.** The SOCP minimizes `RMS(w)+epsilon*||L^-1[-w;1]||`. Because the expression is nonnegative, the decision is unchanged by squaring; the scientific squared-loss value and its derivative are computed only after solve.

The canonical clean hardened Appendix bundle is `appendix-routes-bounded-v5`. It uses one deterministic task per route, the declared linear bootstrap quantile, 20/20/10 bootstraps, and two metric updates. The validator binds every route to its config, source/input tree, plan, regenerated dataset, estimand, trace, and terminal state. This is scientific canary evidence only.

## Stopping rule audit

The released examples set every stopping tolerance to zero except relative improvement of the nonpenalized objective, which is `1e-6`. The released implementation stops when `(previous - current) / abs(previous) < 1e-6`. This also fires when the nonpenalized objective worsens, even if the penalized objective improves. Canary `v16` reproduces that exact comparison under the independent plain-gradient route, records the signed relative improvement and reason, and uses a 1,000-iteration safety cap.

From clean commit `d912834`, `v16` stopped after two iterations in 0.334499 seconds. Its signed nonpenalized relative improvement was `-2.030462032685939e-6`, while the penalized total objective improved from `-0.30689005784322865` to `-0.3069101215220537`. The final gradient norm remained far from zero in the trace. This receipt establishes the released stopping behavior and a local timing datum. It does not establish mathematical convergence. A stop under this rule is called `paper_stopped`, not converged.

## Still absent

- a representative cross-suite timing sample or any paper-scale seeded matrix;
- Linux clean-rerun and reviewer quorum.

The theorem audit now exists at `THEOREM_AUDIT_C2.md` and remains `HOLD / inconclusive`; it does not upgrade C2.

Therefore this implementation is genuine scientific canary evidence but cannot score C1-C3.