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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_phase9/DERIVATION_PHASE9.md
Snapshot: Quan–Lončar complete lineage, phases 1–28. The body below is unabridged.

Phase 9 derivation: reviewed degree-two thickness continuation

1. What remains invariant under regularization

Let A(omega)=I-K(omega) be the outgoing volume-integral pole operator. If P(omega) is analytic and invertible in the search region, then

\[ A(\omega)E=0 \quad\Longleftrightarrow\quad P(\omega)A(\omega)E=0. \]

Thus a future regularization of the Maxwell longitudinal/self term must preserve true pole frequencies, although it may change determinant normalization and improve Galerkin convergence. Phase 9 continues null vectors, not raw determinant magnitudes.

2. Degree-two thickness content

For the target sector (s_x,s_y,s_z)=(-1,-1,+1), scalar Legendre degree two is even under z reflection. The retained vector components are therefore E_x and E_y; E_z has the wrong physical z-mirror sign. The exact symmetry block grows from 630 coefficients at degree one to 1080 at degree two.

The controlled rows are

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

As before, the first difference measures only the background Green-basis increment and the second measures the newly admitted hole field.

3. Numerical result and parity-separated convergence

The reviewed rows give

\[ Q_{(1,1)}=511.0266,\qquad Q_{(1,2)}=618.1850,\qquad Q_{(2,2)}=577.4738. \]

Therefore

\[ \frac{Q_{(1,2)}-Q_{(1,1)}}{Q_{(1,1)}}=20.969\%, \]
\[ \frac{Q_{(2,2)}-Q_{(1,2)}}{Q_{(1,2)}}=-6.586\%, \]

and the net degree-one to degree-two change is 13.003%. Residuals are at the 1e-13 level, so the motion cannot be attributed to incomplete pole solves.

Legendre parity explains why degree one gave false comfort. Odd degrees add the z-odd scalar family, which in this sector appears through E_z; even degrees add z-even E_x/E_y. Convergence must be judged separately along the even and odd subsequences. A small degree-three increment will not remove the requirement to compute degree four.

This cancellation risk also applies to any row where the background waveguide degree and physical hole degree were raised simultaneously. Future tables must continue to display the waveguide-only intermediate row.