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Historical derivation — preserved in full

Source: engines/quan_loncar_v1/lineage/phases_01_28/quan_loncar_complete_lineage_phases_01_28/fryett_q_series_phase10/DERIVATION_PHASE10.md
Snapshot: Quan–Lončar complete lineage, phases 1–28. The body below is unabridged.

Phase 10 derivation: odd thickness degree three

Controlled rows

Starting from the reviewed degree-two pole, compute

  1. hole z degree 2, waveguide z degree 2;
  2. hole z degree 2, waveguide z degree 3;
  3. hole z degree 3, waveguide z degree 3.

Degree-three scalar Legendre functions are odd in z. In the target (-1,-1,+1) sector, the newly retained component is therefore E_z. The symmetry block grows from 1080 to 1260 coefficients.

The seed-to-converged-mode overlap is now recorded explicitly. This separates true branch continuation from the final Newton self-overlap diagnostic.

Numerical result

\[ Q_{(2,2)}=577.4738,\qquad Q_{(2,3)}=577.2454,\qquad Q_{(3,3)}=577.6291. \]

The separated relative increments are

\[ -3.9540\times10^{-4}\quad\text{and}\quad +6.6460\times10^{-4}, \]

giving a net +2.6893e-4. The seed-to-converged-mode overlaps are 0.999959 and 0.998093. This is strong evidence that the odd E_z thickness family is settling, but it says nothing decisive about the next even E_x/E_y family.