Milestone 7: overlapping-mode resolution and full-refinement handoff

Outcome

This milestone turns the coarse rational optimizer into a grid-refinement and overlapping-mode pipeline. Four accepted coarse Loewner steps produced

AAA Q: 194.98 -> 195.35 -> 196.35 -> 198.36 -> 200.33.

At the final coarse checkpoint, AAA, Loewner, and local-linewidth estimates are 200.3316, 200.3299, and 199.8566. The preceding Q=196.35 checkpoint passes a seven-case rational PML gate: maximum relative pole-frequency shift 2.21e-4, relative Q span 19.70%, and maximum held-out response NRMSE 1.23e-4. The Q=200.33 update has not separately repeated that PML sweep.

Controlled transverse refinement from 20 x 13 x 11 to 20 x 18 x 14 revealed a two-mode cluster near 1.073 and 1.079. A scalar AAA fit initially reported Q=1262 for the upper feature, but that pole moved strongly with rational order and failed the linewidth/model-stability gate. It is not used as the result.

The accepted upper-mode pole is instead obtained from 69 unique real-frequency samples made by merging:

• a 41-point wide seven-probe sweep over 1.069--1.083; and
• a 29-point dense seven-probe sweep over 1.0772--1.0818.

Across seven independently fitted probes and four interleaved withheld frequency sets, the holdout-only median pole is

f = 1.07946168752 - 0.00058561339 i,

or Q = 921.6505.

All seven predefined combined-rational gates pass. The one-pole polynomial and low-order wide-cluster models are retained as rejected diagnostics: their systematic Q drift was the evidence that a higher-order, multi-window rational continuation was necessary.

The mode is in the air defect

Driven-field diagnostics at the two refined features show:

The protected strip |x| < 0.5 lambda0 contains exactly zero dielectric in every checkpoint. These are therefore source-coupled, air-localized defect features rather than simple mirror-interface states. This does not yet establish atom-position mode volume, Purcell factor, or waveguide beta factor.

Optimizer architecture

overlap_optimizer.py now separates its roles:

1. the 29-point dense local scalar Loewner pencil supplies the differentiable pole-Q topology gradient;
2. seven linear field outputs are obtained from those same forward fields;
3. 41 wide anchor frequencies constrain the modal/background continuation;
4. a proposed step is accepted only by the merged 69-point rational holdout gate; and
5. multipole_pml_convergence.py --validator combined repeats that same pole identity under all absorber perturbations.

This is deliberately more expensive than accepting a scalar surrogate. It prevents a topology step from improving one member of the numerical pole ladder while the physical source-coupled resonance worsens.

64 GB handoff

The new workstation64-fine profile is 40 x 26 x 22:

• 68,640 vector unknowns and 1,716 surface-plane unknowns;
• dx=0.0625, dy=dz=0.08;
• four cells through the slab;
• a 2+4-slice mirror that preserves the physical 0.375-lambda period of the controlled grid; and
• a conservative peak estimate of 16.16 GiB.

The transferred checkpoint has density-fit MSE 6.69e-5, reflection-symmetry error 4.04e-17, and zero air-strip density. Its carried pole is an identity hint only; Q must be reacquired on the fine grid. The exact sequence is in WORKSTATION64_RUNBOOK.md.

Verification and scientific status

• 30 regression tests pass.
• Wide sweep: 37.1 min and 4.06 GiB peak RSS in the hosted runtime.
• Dense local sweep: 29.1 min and 3.04 GiB peak RSS.
• The combined rational spectral gate passes.
• The coarse Q=196.35 rational PML gate passes.
• The refined Q=921.65 pole has not yet passed the combined PML gate.
• The 40 x 26 x 22 pole has not yet been reacquired.

Therefore Q=921.65 is a validated spectral identification on one controlled grid, not a converged physical-device Q. The 3D optimizer is ready for the 64 GB convergence campaign, but fabrication robustness and waveguide coupling remain later objectives.