Spectral-gap adaptive polynomial filtering MVP

What was built

spectral_gap_filter.py implements Fejer averaging, a degree-K squared-Fejer/Jackson-type polynomial, exact diagonal rotation-spectrum application, coefficient validation, and an adaptive selector with the proposed s*K < 2 safe branch and 10% residual safeguard.

Numerical verification

• Polynomial normalization: p(1)=1 error was 0 for K=3,7,15,31; coefficients were nonnegative.
• Iteration identity: direct polynomial application versus explicit averaged-reflection iterates had error 5.39e-16.
• Critical prediction: Fejer residual times (K+1) was 0.9999998 on average and maximum 0.9999999993 over critical/subcritical sweeps, matching the predicted 1/(K+1) floor.
• Gapped prediction: at fixed sK=8, Jackson residual times K^2 s was 3.36, 3.41, 3.37, 3.39, 3.41 for K=7,15,31,63,127, supporting the predicted O(1/(K^2 s)) scaling.
• Switch/safety prediction: at sK=2, Jackson was about 1.45x the Fejer residual, so the nominal branch is not uniformly beneficial. The 10% safeguard rejects it; Jackson became acceptable in this toy sweep around sK=3.5, and at sK=8 it gave about 2.3x to 2.5x lower residual than Fejer.

Verdict

The mechanism manifested: the critical 1/K barrier and gapped inverse-K^2 s behavior were observed quantitatively. The literal threshold sK=2 is aggressive on this spectrum, but the prescribed residual safeguard prevents the resulting degradation. This is a toy spectral verification, not evidence of a task-level DEQ speedup.