Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/fryett_q_series_phase6/README.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Fryett Q series: Phase 6

Phase 6 raises the complete 90-hole basis beyond first order without building a blind 2700-by-2700 dense matrix. The exact x, y, and z mirror symmetries of the cavity split Maxwell's pole operator into eight independent sectors.

Real-Zernike parity, thickness-Legendre parity, and the sign change of the vector component normal to a mirror are all included. The projectors are orthonormal and resolve the identity exactly.

The full-cavity candidate is first classified into one sector. Its coefficients are then embedded exactly as the Zernike degree and waveguide cross-section degree are raised. This preserves branch identity while reducing degree 3 from 2700 full coefficients to 270 sector coefficients.

Run python run_phase6_convergence.py --max-degree 3 to reproduce the tracked basis sequence.

The selected Phase-5 pole lies in sector (+,+,-) and stabilizes near Q=198.7 by degree 3. Exact symmetry also reveals that this is an E_z-like branch, not the reported E_y-like Fryett mode. The next search is therefore restricted to (+,-,+) and (-,-,+).