Finite-Frequency Rigidity for Maximal Perimeter Sections

We prove a quantitative nonlinear origin-symmetry theorem for convex bodies in three dimensions whose radial functions are close to one in C2 and whose odd parts have bounded spherical harmonic degree. The even part is unrestricted. An explicit bound controls the odd radial part by derivatives at zero of the parallel-section perimeters; for arbitrary odd parts there is an additional C2 high-frequency-tail term. The proof combines the classical equatorial derivative transform with a finite-rank inverse and an explicit nonlinear remainder estimate. All odd parameter jets vanish for sufficiently regular solution families through the ball, and real-analytic families are symmetric. An exactly evaluated ellipsoid example detects a missing term in a printed directional first-variation formula, without affecting the classical infinitesimal theorem at the ball. The neighborhood depends on the harmonic cutoff, and the general maximal-perimeter symmetry conjecture recorded as AIM-FUNCTIONAL_ANALYSIS-0012 is not resolved. Classical inputs are credited. This is an unrefereed preprint with AI assistance disclosed in the paper; no independent review, proof-assistant formalization or absolute priority is claimed.