Verification of the odd-period focal inversion area product

Abstract

Attributed verification/research note giving a self-contained flag-curve proof of the historical odd-period focal inversion signed-area product: k903,a in Reznik--Garcia--Koiller arXiv:2004.12497v11 and Conjecture 1 of Reznik--Garcia--Helman arXiv:2012.03020v2. Alper Ferudun, Products of the Two Focal Inversion Areas in Elliptic Billiards, version 1.0, DOI 10.5281/zenodo.23096994, proves the same theorem using substantially the same meromorphic divisor mechanism; this exact overlap and the established general method are explicitly credited. The strict theorem concerns proper confocal nested ellipses, original ordered vertices, original foci, unit inversion, signed straight-edge areas, every admitted coprime winding of a primitive odd-period orbit, reversal and indexed repetitions. A portable standard-library exact verifier accompanies four specified convex winding-one controls, with actual normal/optimized receipts and deliberate negative controls; finite checks do not prove the all-period theorem. The source ZIP includes authored source, code and notes, without third-party full texts or dependencies. The bounded priority comparison leaves documented access gaps and does not establish novelty, first proof, certified independent discovery, literal simultaneity, absolute priority or current openness. Specified PR/commit and Zenodo record dates are reported with these limitations. AI tools were used extensively in research, proof development, drafting, auditing and computation. Separate AI adversarial checks are not independent human peer review. This is an unrefereed research note without conventional human refereeing or proof-assistant certification.