Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/fryett_q_series_phase5/README.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Fryett Q series: Phase 5

Phase 5 replaces the unconverged tangent-mapped transverse momentum integral with fixed panels in

\[ p_y=q_yW/2,\qquad p_z=q_zT/2. \]

Increasing Gauss order leaves all panels unchanged. Increasing the cutoff by an integer number of panel widths adds only outer panels, preserving every previous interior node. This separates local quadrature error from the omitted large-momentum tail.

The homogeneous Green contraction is vectorized over the two-dimensional momentum grid and is algebraically identical to the Phase-2 loop. The faster implementation is used transparently by the Phase-4 outgoing contour, hole projection, and branch-labelled pole tracker.

The convergence runner first varies panel order at fixed cutoff, then grows the cutoff at fixed panel width and order. Only if that pole stabilizes does it add the remaining physical mirror holes.

CONVERGENCE_PHASE5.json records the first complete 90-hole spectrum. The lowest hole basis produces a best resolved branch with Q of a few hundred, but the first Zernike enlargement moves the same branch to Q near 99. The full geometry is therefore assembled successfully, while its hole basis is decisively unconverged.