Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/fryett_q_series_phase4/README.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Fryett Q series: Phase 4

This checkpoint replaces the constant mode in each elliptical hole by a complete real-Zernike basis in the ellipse and normalized Legendre functions through the beam thickness.

For every retained scalar mode and Cartesian component it assembles

\[ A_N(\omega;g)=I-k_0^2\Delta\epsilon \int_{\mathcal C_+}\frac{d\beta}{2\pi} C_N(-\beta;g)^T G_{\rm wg}(\beta,\omega)C_N(\beta;g). \]

The unit-disk radial integral uses Gauss--Legendre quadrature and the periodic angular integral uses the trapezoidal rule. Both converge spectrally for the entire Fourier integrand. The thickness integral is exact because the hole and waveguide use the same normalized Legendre basis.

test_phase4.py verifies Zernike orthonormality, the exact reduction to the Phase-3 constant mode, the thickness selection rule, and reciprocity of an assembled higher-order operator.

run_phase4_convergence.py performs an initial real-frequency scan and then tracks one simple complex pole while independently raising the hole, transverse-momentum, contour, and physical-hole truncations. Branch identity is carried by null-vector overlap and exact coefficient embedding. Its Q values are diagnostics of branch tracking, not yet the reported Fryett result.

The first table is in CONVERGENCE_PHASE4.json. It shows initial stabilization of the longitudinal contour and first-order hole basis, but it rejects the transverse-momentum and mirror-count axes as unconverged.