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engines/majumdar_lab/package/multilayer_autonomous_optimizer/fryett_q_series_phase5/DERIVATION_PHASE5.md · assembled 2026-07-29 15:57 UTC.

Phase 5 derivation: a controlled transverse-momentum limit¶

1. Why the tangent rule was rejected¶

The Phase-4 table kept the longitudinal beta contour and hole basis fixed while raising the tangent-map order in q_y and q_z. The provisional Q moved as

\[ 11.81,\ 11.00,\ 11.79,\ 13.61 \]

for orders 12, 14, 16, and 20. A tangent-map order change moves every momentum node and mixes local resolution with high-momentum-tail resolution. That is not a useful convergence axis near a sharply varying radiation denominator.

2. Dimensionless fixed panels¶

Define

\[ p_y=\frac{Wq_y}{2},\qquad p_z=\frac{Tq_z}{2}. \]

For a dimensionless cutoff U and fixed panel width Delta, partition each axis into panels

\[ [-U,-U+\Delta],\ldots,[U-\Delta,U]. \]

The cutoff is restricted to an integer multiple of Delta. Gauss--Legendre order r is raised inside every unchanged panel. After order convergence, U is increased by appending outer panels of the same width. Thus:

r -> infinity converges the integral on a fixed square;
U -> infinity converges the explicit transverse-momentum tail.

These are separate limits and can be audited independently.

3. Vectorized Green contraction¶

Let F_a(q_y,q_z) be the Fourier transform of waveguide cross-section basis function a. At fixed beta,

\[ [G_c]_{a\alpha,b\gamma}= \int\frac{dq_y dq_z}{(2\pi)^2} F_a^*F_b \frac{\delta_{\alpha\gamma}-k_\alpha k_\gamma/k_c^2} {\beta^2+q_y^2+q_z^2-k_c^2}. \]

The implementation evaluates the polynomial outer products and dyadic symbol on the complete Cartesian grid, then performs one tensor contraction. It is algebraically identical to the nested Phase-2 loop and changes no physics.

4. Acceptance criteria¶

A transverse sequence is accepted only if the same lower-half-plane null vector is tracked and all of the following stabilize:

complex frequency,
Q,
smallest singular value at the refined root,
null-vector overlap,
panel order at fixed U,
cutoff at fixed panel width.

Only after this limit is controlled may increasing mirror count be interpreted as physical cavity convergence rather than compensation for a momentum error.

5. Branch-labelled complex Newton continuation¶

Adding mirror holes introduces many nearby determinant zeros. A generic two-real-variable root solver repeatedly reevaluates the full Green operator and can spend many evaluations resolving the wrong local direction. For a simple tracked eigenvalue

\[ \lambda(\omega)=\frac{y^\dagger A(\omega)x}{y^\dagger x}, \]

the derivative is

\[ \lambda'(\omega)=\frac{y^\dagger A'(\omega)x}{y^\dagger x}. \]

Phase 5 evaluates A' by a centered frequency difference and updates omega <- omega-lambda/lambda'. The right eigenvector is relabelled by maximal overlap after every update. One Newton iteration therefore needs three full operator evaluations and preserves the physical branch explicitly.