# Verification report

This report describes the evidence supporting Original Antipedal Centroids in Elliptic Billiards by Alper Ferudun. The manuscript proves the stated theorem for even least period, not every reading of the abbreviated source record.

## Analytic proof

The pole at the center or a focus is strictly inside the confocal elliptic caustic, which ensures finite antipedal intersections. Classical Jacobi parametrization supplies opposite pairs for every even primitive cycle, including coprime star windings. A direct two-chord calculation yields a matrix-weighted centroid sum. Stationarity of positive billiard perimeter under an infinitesimal affine rotation cancels its mixed term. The remaining diagonal term equals an expression in the mean positive side length, giving the explicit focal formula. Repetition preserves the mean; circular billiards have a separate symmetry argument.

## Exact computational controls

The accompanying symbolic checker performs exact reductions of the chord and pair identities. It also checks two four-periodic phases, three six-periodic ellipses, both foci, the center, the circle, and covariance under translation, rotation and scaling. Its documented dependency is SymPy 1.13.1. These algebraic tests supplement the written general proof, rather than replacing it with finite sampling.

A separate standard-library rational checker verifies a genuine triangle on x^2/21+y^2/16=1: incidence, positive side lengths, unit-velocity reflection, confocal-caustic tangencies, internal contacts, finite antipedal intersections and a nonzero central antipedal centroid. Negation reverses that centroid. Repeating either triangle twice therefore disproves the unrestricted even-list interpretation while leaving the even-least-period theorem intact. No floating-point tolerance determines these tests.

Both originating checkers were actually executed successfully. The distribution includes their portable counterparts, exact outputs and provenance. See reproducibility/README.txt for commands and requirements. Execution receipts distinguish accepted research checks from portable distribution replays.

## Source interpretation and literature

The target is k405 in Table 5 of Reznik, Garcia and Koiller, arXiv:2004.12497v11, and its final 2021 Arnold Mathematical Journal version, page 348. Section 3.5 on page 347 defines the original-polygon antipedal. An outer-polygon antipedal is a different object. The source's confocal-ellipses setting and use of antipodal symmetry motivate the genuine-even convention; an explicit least-period restriction is stated here rather than attributed verbatim to the source.

Stachel's classical Jacobi parametrization and symmetry statements are credited. Bialy and Tabachnikov's related boundary-normal identities are distinguished from the weighted chord-normal identity proved in this manuscript. A bounded primary-source search did not establish priority for the explicit focal centroid formula. Novelty remains UNDETERMINED.

## Status and limitations

The written theorem has been self-audited and has no identified mathematical gap within its explicit scope. This is an AI-assisted, unrefereed preprint, not an independently human-reviewed or proof-assistant-verified result. It makes no hyperbolic-caustic, area-centroid, unrestricted source-record or absolute-priority claim. A successful LaTeX compilation or a DOI is not mathematical certification. Publication-channel status is maintained separately from mathematical acceptance.