Physical Gradient Recovery and Stopping Guarantees for a Folded Stokes Parametrization

Hiroyasu Usami, Ayato Tsuboi, Keisuke Hara

Submitted by Hiroyasu Usami

Published October 29, 2026 · Version v1, October 29, 2026 · DOI 10.66977/xsci.2610.003b

Optics, Applied Mathematics

Abstract

Small derivatives in a constraint-enforcing parametrization can coexist with a substantial physical objective gap. For the rational Stokes map, first derivatives lose one physical-gradient component at full polarization. We recover it from one signed radial curvature for any twice continuously differentiable objective, including nonstationary points with an unknown physical Hessian. Near the fold, mixed-scalar and vector-Hessian inverses give explicit derivative- and curvature-input error gains. For strongly convex physical objectives on the inner sheet with bounded intensity, a normalized head-gradient tolerance $\varepsilon$ and a radial lower-curvature tolerance $\nu$ yield an explicit objective-gap bound with sharp orders $\varepsilon^{4/3}+\nu^2$. Valid derivative-error budgets give sufficient stopping thresholds. An outward interval implementation for the nominal ideal quadratic certifies a gap at most $10^{-6}$ at 8,861 of 9,284 stored warm-start endpoints, using 52 scalar curvature evaluations. Controlled two-frame and noisy Fresnel fits exhibit fold-related inaccurate solver acceptance. Released observations reveal broader failures that the local fold gate misses. Coordinate repair corrects 249 of 295 inaccurate warm-start acceptances in a paired audit; standard physical fallback corrects the remaining 46. Direct physical solvers are cheaper in the tested implementation. Twenty shared-network fits contain no near-fold negative-curvature candidates at the saved checkpoints.

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