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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_phase13/DERIVATION_PHASE13.md
Snapshot: Quan–Lončar complete lineage, phases 1–28. The body below is unabridged.

Phase 13 derivation: separated even-x in-plane degree four

Controlled sequence

On the corrected Fryett sector (s_x,s_y,s_z)=(+1,-1,+1), hold every cutoff, contour, quadrature, and thickness setting fixed and compute

  1. hole Zernike degree 3, waveguide y degree 3;
  2. hole Zernike degree 3, waveguide y degree 4;
  3. hole Zernike degree 4, waveguide y degree 4.

This separation is required because the Phase-12 restart changed the physical hole basis and the bare-waveguide Green basis simultaneously.

At thickness degree zero, every real disk mode retains exactly one vector component in the target (s_y,s_z)=(-1,+1) block. The degree-four radial family contains five real modes: three cosine modes enter as E_y, while two sine modes enter as E_x. The reduced dimension grows from 450 to 675.

Result

\[ Q_{(p_h,p_w)=(3,3)}=197.3778, \]
\[ Q_{(3,4)}=192.5795, \qquad Q_{(4,4)}=192.5352. \]

The separated relative changes are

\[ \Delta Q_{\rm wg}/Q=-2.431062\%, \qquad \Delta Q_{\rm hole}/Q=-0.023002\%, \]

with a net degree-three to degree-four change of -2.453504%.

The waveguide-only overlap is 0.993938 and the lifted hole-basis overlap is 0.999107. The final null-vector coefficient fractions are

\[ (w_x,w_y,w_z)=(0.0330812,0.9669188,0). \]

Eigenvalue and smallest-singular-value residuals are below 1.7e-11 for the waveguide row and 2.0e-13 for the joint row. Reciprocity remains below 3.6e-16.

Interpretation

The degree-four physical hole correction is already far below the 2% provisional threshold. The bare-waveguide y correction remains slightly above that threshold, so the complete in-plane axis cannot yet be called converged. A separated degree-five row is mandatory.

This behavior does not trigger the abandonment gate. The pole is stable, its overlap is high, the new physical-basis increment is small, and the remaining waveguide correction is in the 2-10% range for which the protocol requires one more order.