Source

Engine design23_v1 · package package/design23_recentered_optimizer · assembled 2026-07-29 15:57 UTC.

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Absolute-Q integration gate

Date: 2026-07-22

Answer first

The physical continuous-channel operator has now been inserted at the matrix level, but the absolute complex-pole gate does not pass. No trustworthy absolute Q or gradient should be reported from this implementation yet.

The work established three things.

1. The channel-resolved matrix is reciprocal, positive semidefinite, and its contraction exactly reproduces the previously validated scalar diagnostic.
2. Replacing only the real-axis anti-Hermitian part of the legacy determinant is mathematically invalid: the legacy complex-beta contour is already on a particular analytic sheet, so a real-axis spectral density cannot be pasted onto it as a frequency-independent imaginary matrix.
3. A direct two-dimensional deformed Sommerfeld contour preserves reciprocity but is not numerically stable for the 84-hole cavity. A deep contour creates exponentially large reciprocal factors across the long cavity, while a shallow contour approaches unresolved poles and branch cuts. There is no quadrature-order/height plateau.

The earlier directional result remains valid within its stated scope:

FDTD Q endpoint ratio                 1.000763
old analytical Q endpoint ratio      3.604590
continuous-channel Q-proxy ratio     0.984871

It remains a relative radiation-ranking diagnostic, not an absolute-Q solver.

Matrix-level implementation

ContinuousRadiationProjector.integrate_operator now constructs the complete hole-basis spectral density

\[ \Gamma_{\rm rad}=\int_0^\infty \frac{M(\beta)+M(\beta)^T}{2\pi}\,d\beta, \]

with separate half-space TE, half-space TM, and bound-film matrices. The explicit reciprocal completion is required for a general complex QNM vector. The following identities are regression tested:

• matrix contraction equals the scalar channel calculation;
• the complex-qy dyadic obeys reciprocity;
• TE/TM densities are positive semidefinite;
• background and strip optical theorems close;
• the assembled cavity operator is reciprocal.

All 14 offline tests pass.

Rejected local self-energy graft

The first attempted production wrapper retained the legacy dispersive Green operator and replaced its real-axis spectral part with Gamma_rad. This did not produce a nearby pole:

• scalar branch continuation moved to the upper half-plane;
• the full 336-dimensional matrix pencil's best passive candidate had only 0.0126 overlap with the saved cavity vector and linearized Q about 43;
• nonlinear secant refinement did not converge;
• a Hermitian-linewidth fallback returned Q about 80, but its real-frequency residual was 4.64 and its resonance ratio moved to 0.687, so that number is rejected rather than calibrated.

The failure is expected in retrospect. The legacy determinant is strongly non-normal: at the saved pole its right null vector overlaps Hermitian eigenvectors with eigenvalues near ±67, while its overlap with the near-zero Hermitian eigenvector is below 0.001. Taking the Hermitian part or grafting an imaginary density destroys the cancellation defining the mode.

Direct two-contour continuation

PoleAwareComplexCavity evaluates every complex-frequency determinant using both a deformed qy contour and a deformed beta contour. The qy contour carries the planar-film poles and exterior light-line cuts on the outgoing sheet; the beta contour retains the exact feed-pole subtraction.

At the saved iteration-0 pole, qy convergence is good at beta order 2:

beta order	qy order	height	tracked eigenvalue	overlap
2	4	0.06	0.891256 + 0.003491i	0.5144
2	6	0.06	0.891280 + 0.003488i	0.5215

Longitudinal convergence fails:

beta order	qy order	height	tracked eigenvalue
4	4	0.06	1.023070 - 0.000595i
6	4	0.06	47.663957 - 51.117085i
8	4	0.06	-21.177364 - 28.416526i
6	4	0.01	0.688105 + 0.348388i
8	4	0.01	0.852715 - 0.005191i
6	4	0.12	7.79e4 - 1.12e5i
6	4	0.20	1.17e9 - 2.82e9i

Every listed operator remains reciprocal to numerical precision. The large values at greater height come from exponential amplification of complex-beta Fourier factors over the full cavity length, not from a reciprocity bug.

Required next algorithm

Do not continue increasing brute-force contour order. The stable production operator must decompose the longitudinal Green kernel before integration:

1. subtract every SiN feed pole analytically, not only the fundamental branch;
2. subtract the bound planar-film pole contributions with their exact residues;
3. isolate the air and substrate light-line branch jumps;
4. Fourier transform or integrate those singular terms with pole-aware, endpoint-clustered real-axis formulas that retain the correct outgoing sheet;
5. integrate only the smooth remainder on a very shallow complex contour;
6. assemble long-range terms in ordered real space so no factor grows like exp(abs(Im(beta))*cavity_length);
7. require simultaneous plateaus versus beta order, contour height, cutoff, and hole-basis size before solving a pole;
8. only then compute iterations 0, 15, and 30 and finite-difference gradients.

This is the same stabilization principle already used successfully for the fundamental feed pole and for the real-axis film pole. It now has to be extended to the entire singular longitudinal spectrum.

Commands

The accepted relative-channel regression is:

python run_radiation_channel_gate.py /path/to/run_state.json \
  --iterations 0 15 30 --beta-order 256 --workers 2

The failed absolute-pole contour gate can be reproduced without launching a nonlinear secant:

python run_pole_aware_qnm_gate.py /path/to/run_state.json \
  --iterations 0 --beta-order 4 --qy-order 4 --initial-only

The regression suite is:

python -m unittest tests.test_offline tests.test_radiation_channels -v