Historical derivation — preserved in full

Source: engines/quan_loncar_v1/lineage/phases_01_28/quan_loncar_complete_lineage_phases_01_28/fryett_q_series_phase7/DERIVATION_PHASE7.md
Snapshot: Quan–Lončar complete lineage, phases 1–28. The body below is unabridged.

Phase 7 derivation: contour search in the physical polarization sectors

1. Target symmetry blocks

For a fundamental scalar profile even in y and z, the polar-vector mirror signs place an E_y-like field in

\[ (s_x,s_y,s_z)=(+1,-1,+1)\quad\text{or}\quad(-1,-1,+1). \]

The x sign distinguishes the two longitudinal standing-wave parities.

2. Beyn moment construction

Let A_s(omega) be one finite symmetry-block pole operator and let Gamma be a positively oriented complex-frequency contour. For a fixed probing matrix V, form

\[ M_0=\frac{1}{2\pi i}\oint_\Gamma A_s(z)^{-1}V\,dz, \qquad M_1=\frac{1}{2\pi i}\oint_\Gamma zA_s(z)^{-1}V\,dz. \]

If M_0=U Sigma W^H has numerical rank r, the enclosed nonlinear eigenfrequencies are the eigenvalues of

\[ B=U_r^H M_1 W_r\Sigma_r^{-1}. \]

Each edge of the rectangle is integrated with Gauss--Legendre quadrature. The moment singular-value gap, quadrature-order stability, direct pole residual, and lower-half-plane sign are all retained as acceptance diagnostics.

3. Meromorphic safeguard and branch-complete scan

The finite bare-waveguide Green representation contains its own poles. A broad contour is therefore not treated as an analytic nonlinear-eigenvalue region unless the moment singular values exhibit an order-stable rank gap.

Independently, the operator is diagonalized on a near-real frequency line. Successive right eigenvectors are paired by the global maximum-overlap assignment. At a local minimum of a tracked eigenvalue magnitude, its complex zero is estimated by

\[ \omega_*\simeq\omega_j-\frac{\lambda_j} {(\lambda_{j+1}-\lambda_{j-1})/(\omega_{j+1}-\omega_{j-1})}. \]

This estimate uses the complex eigenvalue, not merely its magnitude. A pole can therefore be seeded even when its imaginary part and linewidth are much smaller than the real-frequency grid spacing. Every seed must still pass the full complex Newton residual and duplicate clustering tests.

4. Physical-sector pole results

The branch scan and complex refinements resolve ten distinct degree-0 poles in the two target sectors. The broad contour calculation is not used for pole counting because its moment rank saturated every tested probe dimension. This is a deliberate convergence rejection, not an assumption that the approximate contour eigenvalues were physical.

The target is selected by

1. E_y-compatible mirror sector;
2. proximity to the paper normalization omega/omega_ref=1;
3. continuous null-vector continuation as the basis grows.

The selected pole is

\[ \omega_0/\omega_{\rm ref} =1.0019129644-0.0016200099i, \qquad Q_0=309.2305. \]

Its in-plane continuation is

\[ Q_0=309.2305,\quad Q_1=295.0186,\quad Q_2=508.3151,\quad Q_3=508.7904. \]

Thus

\[ \frac{Q_3-Q_2}{Q_2}=9.3505\times10^{-4}. \]

The eigenvalue and smallest-singular-value residuals of the degree-3 pole are 5.69e-12 and 5.28e-12, respectively. Reciprocity remains at roundoff. This establishes an in-plane plateau, not convergence of the complete series: thickness, transverse-momentum tail, contour deformation, and geometry remain independent axes.

5. The off-frequency narrow pole

The even-x sector contains

\[ \omega/\omega_{\rm ref}=0.8488719510-4.50144\times10^{-5}i, \qquad Q=9428.89. \]

Under the 740-nm normalization its wavelength is about 872 nm. It demonstrates that the determinant can produce substantially narrower resonances, but it is not selected as the Fryett branch and has not yet passed local-basis continuation.