Mirror-Constrained Optical Hall Rectification in Asymmetric Gold Gratings: Edelstein Conversion and a Phase-Seeded Kerr Four-Wave Mixing Instability

Transverse optical rectification has been reported in a one-dimensional asymmetric Au grating even though an ideal profile h(x) retains the xz mirror:y to-y. Because J y is odd under, whereas the p-polarized bilinears E x squared, E z squared, and Re(E xE z*) are even, the transverse dc current requires a distinct mirror-odd factor. We formulate a symmetry-complete effective model combining the origin-independent cell invariant = Im(h 1 2h 2*), Drude--Edelstein spin polarization, and spin--orbit conversion. We then propose a dynamical realization of for a finite-width device. A driven pump-like plasmonic mode at k y 0 generates a mirror pair of transverse surface-plasmon-polariton quasinormal modes (K x, plus or minus q y) through the same third-order susceptibility that produces self- and cross-Kerr shifts. Four-wave mixing creates the pair and locks its phase sum but, by itself, cannot change the pair-population difference. Weak coherent backscattering makes the relative drive phase observable; a small phase difference from incident k y, spot displacement, edges, or disorder then unfolds a Kerr directional instability and deterministically selects its sign. The resulting population imbalance supplies the dynamical mirror-odd factor =λ ηη in the structure-dependent j xs y t side-jump-like channel, which vanishes for the centered symmetric grating because =0. We give the complete three-mode equations, the pump-only and saturated-state Bogoliubov problems, the imperfect-pitchfork reduction, and a validated normalized numerical realization. The normalized equations exhibit a robust supercritical pitchfork with an approximately one-half critical exponent and phase-reversed branch selection.