Validation history and rejected interpretations

Bound-clipped derivative

The first directional check at a binary topology showed an apparent 52% gradient error. The two finite-difference endpoints had been clipped at opposite box constraints, so the numerical quotient did not measure the unconstrained derivative. The permanent regression evaluates a strict interior point. Its relative error is 2.70e-5.

Outgoing-sheet instability

An early nonlinear iteration chose square-root signs from each intermediate complex pole. Near a Fourier-channel threshold this changed the Riemann sheet and prevented convergence on richer bases. Channels are now classified at the target frequency and retain that sheet throughout Newton continuation. Useful bases converge to residuals below 1e-8; an under-resolved basis that does not converge is rejected.

False coarse improvement

Three smoke-basis steps appeared to improve Q from 25.5 to 136.9. On the next Fourier/vertical basis, the independently selected field was orthogonal to the coarse field and Q fell near 19. The candidate was rejected. This established that frequency and localization do not define cross-basis identity.

The solver now accepts a physical-grid complex field map as an identity hint. The final validated campaign retains field correlations of 0.9839 and 0.9849 and a Q gain of at least 2.018 across the production/refined bases.

Full-vector transfer: coupling gain, unaccepted Q gain

The paired 3D transfer initially appeared to improve lower-branch Q by about 2.9% in the 41-point wide-window probe consensus. Adding a 29-point local window changed the combined estimates to 775.45 and 786.62 (+1.44%), but both absolute fits failed the predeclared wide/local and holdout-spread gates. The Q improvement is therefore not accepted.

The complex vector fields match with overlap 0.9970 and the sampled peak centered-gap LDOS improves by 4.45%, so the transferred geometry is still a useful initializer. This case permanently distinguishes a reproducible driven response improvement from a validated pole-Q improvement.

Upper-branch pole disagreement and linewidth fallback

The first full-vector continuation attempted to optimize the upper-branch complex pole inferred from a 29-point local plus 41-point wide seven-probe sweep. Its rational Q was 1308.29, while the real-axis linewidth Q was 967.29; the strict gate rejected the state. Adding all 28 interleaved local midpoints did not reconcile the estimators. A direct real-axis operator linearization also failed to produce the rational pole and branch continuation jumped to unrelated low-Q candidates. No pole-Q topology step was accepted.

The fallback optimizes the observable real-axis Fano linewidth itself. Its fit sensitivity was stable across a 30-fold finite-difference-step range. The first implementation accidentally changed the fit mask between baseline and trial, producing an apparent rejection. Direct comparison showed that the Maxwell response derivative had 0.999969 correlation with the actual LDOS change and only 0.827% linearization error; keeping the predeclared fit window fixed corrected the evaluation.

The resulting full-vector step improves the original fit from 967.292 to 968.652. Thirteen newly solved interleaved frequencies independently retain an improvement from 939.890 to 940.839. All spectral, field, localization, coupling, PML, protected-air, and symmetry gates pass. This establishes a working controlled-grid optimization step, not a grid-converged device Q.