# Verification report

Title: Finite-Frequency Rigidity for Maximal Perimeter Sections

Author: Alper Ferudun, Mercury Software GmbH. Version 1.0, 15 October 2026.

## Accepted mathematical scope

For a convex body in R³ with positive C² radial function rho, let rho_o be its odd part and let B_L be the sum of l(l+1)(2l+1)² over odd l <= L. If ||rho-1||_C² <= 1/(204 B_L) and rho_o has harmonic degree at most L, then

    ||rho_o||_C² <= (B_L/pi) sup_u |P'_u(0)|.

Thus central maximality of every parallel-section perimeter implies origin symmetry in this class. The even part is not band-limited. For arbitrary odd parts the bound has the additional term 2||(I-Pi_L)rho_o||_C². The manuscript also proves vanishing of all odd jets for C^k C²-valued solution families through the ball, and symmetry of analytic families on a connected interval.

The exact ellipsoid (x-a-z)²+y²+z² <= 1 has zero horizontal perimeter derivative. At a=1/2, a printed directional expression instead equals pi(27-16sqrt(3))/6 < 0. This tests that expression, not the all-direction conjecture. The missing term is cubic at the ball and does not alter the cited infinitesimal theorem there.

## Proof checks

The originating researcher audited fixed-height differentiation, integration by parts, the classical harmonic multipliers, the finite-rank inverse norm, exact nonlinear constants, contraction, the high-frequency tail, the parameter-jet induction and the exact ellipsoid integral. The paper supplies the full analytic arguments. No gap was identified in those stated claims.

The supplementary script performs 690 checks. Exact rational arithmetic verifies Legendre derivatives through degree 101, coefficient bounds 3088/81 < 39, 704/81 < 9 and 80/27 < 3, the contraction factor, and partial fractions. There are 288 floating-point latitude multiplier checks and five ellipsoid parameter cases, each comparing both first-variation expressions with the closed integral and the actual circular-section derivative. Numerical tolerances and error estimates are retained in the JSON output. Normal and optimized Python agree in the recorded environment.

Compilation, page review, archive integrity and extracted-archive replay evidence are bound by the package manifest. These checks establish artifact integrity and regression coverage, not peer review or formal mathematical verification.

## Limits

The full AIM source question is not solved. The radius depends on L; an arbitrary body need not lie on an analytic family of solutions through the ball. No estimate from the central-to-maximal perimeter deficit, optimal constant, uniform cutoff-free theorem, exhaustive novelty certification or absolute priority is claimed. Classical transform and infinitesimal results, Stephen's polytope theorem, and inherited ellipsoid calculations are credited. The paper is an unrefereed, self-audited preprint prepared with AI assistance.