Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/quan_loncar_series_phase25/README.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Quan--Loncar series, Phase 25

Phase 25 differentiates the nonlinear pole equation rather than the numerical root-finding iterations. For left/right null vectors y,x of the converged operator it implements

domega/dp = -(y^H A_p x)/(y^H A_omega x).

implicit_pole.py is backend-neutral: finite differences currently supply the two matrix partial derivatives, while the same function is the custom VJP/JVP boundary for a future automatic-differentiation backend.

The first physical validation parameter is the center filling fraction of the quadratic hole taper. The implicit pole and Q derivatives are compared with two independently re-solved perturbed cavities on the same tracked Ey branch.

After that validation passes, run_phase25_production_gradient.py evaluates the same derivative at the final Phase-24 K=14, A=56 pole without differentiating through or replaying the nonlinear solve.