Historical derivation — preserved in full

Source: design23_autonomous_optimizer_v1.0.0/design23_autonomous_optimizer/multilayer_nanobeam_phase14/README.md
Snapshot: design23-v1.0.0. The body below is unabridged.

Multilayer nanobeam — Phase 14

This phase supplies the custom reverse-mode rule for the complete nonlinear multilayer resonance. All 75 gap/radius partials are contracted with the QNM left/right vectors in one complex-beta pass. The pole gradient follows from

ds/dp = -(y^H A_p x) / (y^H A_s x),

so neither the eigensolver nor the complex secant iterations appear on the autodiff tape. The result exposes both ds/dgeometry and dQ/dgeometry in the exact packed geometry ordering used by the optimizer.