# Verification report

Manuscript: *Focal Pedal-Antipedal Area Products in Elliptic Billiards*.

Author: Alper Ferudun, Mercury Software GmbH. Date: 1 October 2026.

The accepted mathematical scope is the same-focus signed product for genuine primitive elliptic-caustic billiard families whose least period is divisible by four, including coprime star windings. The argument proves explicit focal-pedal and focal-antipedal area relations, then applies the published even original/outer area-product theorem with its hypotheses verified. Signed zero antipedal area is permitted. Hyperbolic and degenerate caustics, an arbitrary padded vertex-list convention and the whole source invariant list are excluded.

The exact controls are portable and use only standard-library rational or real quadratic-field arithmetic. check_focal_product.py verifies actual four-periodic reflection, confocal tangency, positive interior caustic contact, finite derived vertices and signed orientation. Its named diamond and rectangle have the same product 202500/289. certify_pair_polynomial.py checks three general coefficient identities with Fraction polynomials; these are not numerical samples. check_nonprimitive_scope.py constructs two genuine primitive N6 orbits in the same a=5,b=3,lambda=225/64 family. Their same-focus products are 823875/1024 and 1390594355/1721344. Doubling the walks multiplies each product by four without changing the least period, documenting the scope limitation rather than refuting the intended primitive convention.

The full proof is analytic. check_full_bridge_algebra.py adds 11 exact coefficientwise identities for the reflection tensor, focal-pedal bracket, paired contribution, outer discriminant and quarter-shift trace, and focal-antipedal reduction. These are general sparse integer-polynomial identities, not finite samples; they do not alone certify geometry or published hypotheses. This fourth checker, its originating output and its actual execution receipt are preserved unchanged. Rounded high-precision N8/N12 phase checks corroborate the proof but are neither interval certificates nor a substitute for a general proof. The source archive preserves the original exact receipts and explains the standalone adaptations in checker-provenance.json, retaining the earlier three-control history separately. The portable package receipt records actual fresh isolated extraction and exact parsed-output comparisons for all four runnable controls.

The release process binds the final source, actual PDF export and all-page visual inspection to immutable SHA-256 records. Native compilation, export and visual inspection must all concern the final unchanged source and PDF before this package is assembled. Any harmless engine or bibliography warning is accepted only when the actual log and visual receipt document that inspection. The manuscript is standalone, and the ZIP excludes third-party source PDFs, credentials and external Python dependencies.

This is originating-researcher self-audit with AI assistance, not independent human peer review or proof-assistant verification. Novelty remains undetermined after a bounded primary-literature search. Prior conjectures, classical geometry, canonical parametrization and the established even area-product theorem are credited. No absolute priority claim is made.