{ "schema_version": 1, "problem_number": "AMR-050-0014", "title": "Outer-Pedal Area Products in Elliptic Billiards", "author": { "name": "Alper Ferudun", "affiliation": "Mercury Software GmbH", "url": "https://github.com/AlperTheKing" }, "abstract": "For a fixed confocal elliptic-caustic billiard family with least period congruent to two modulo four, we prove that the signed area of the outer tangent polygon times the signed area of its pedal polygon with respect to any fixed point is constant. Coprime star trajectories are included. The tangent-intersection polygon is an explicit affine image of a half-step shift of the billiard orbit. After complexification, its area and the outer-pedal area have disjoint possible simple-pole sets. Central pairing separates the pedal map into two area traces; two reversal symmetries provide the complementary zeros that make their product entire and elliptic. This proves invariant k303,a under an explicit least-period convention. For the center pedal, the same argument also covers every odd period, proving k303,b for all traversal lengths not divisible by four, including repeated traversals. We evaluate the six-periodic constant and give exact counterexamples to the unrestricted arbitrary-point repeated-list interpretation and to adding period four in the centered case. No absolute priority claim is made. Source records: AMR-050-0014 (raw ID 5100014, k303,a) and AMR-050-0015 (raw ID 5100015, k303,b), ulamai/UnsolvedMath v1.6.0. Version 1.1 extends the existing manuscript, whose prior version DOI is 10.5281/zenodo.23087221 and concept DOI is 10.5281/zenodo.23087220. No separate manuscript is added. The k303,a arbitrary-fixed-point theorem retains its explicit genuine least-period condition N congruent to two modulo four. The additional k303,b center-pedal theorem covers every closed traversal length not divisible by four, including repeated traversals and all admitted signed star windings. Both use a fixed noncircular ellipse and fixed nondegenerate confocal elliptic caustic, boundary-tangent feet and signed shoelace areas. The centered N=4 products 8*a^2*b^2 and 2*(a^2+b^2)^2 differ by 2*(a^2-b^2)^2; this refutes an all-period extension, not k303,b. No hyperbolic, degenerate or whole invariant-list extension is asserted. This is a self-audited, AI-assisted, unrefereed preprint. No independent human review, formal proof-assistant verification, first-proof or absolute priority certification is claimed. Novelty remains undetermined after bounded primary literature searches. Credited confocal geometry, Jacobi parametrization and the established meromorphic pole-cancellation method are prior mathematics.", "result_type": "COMPLETE_PROOF", "categories": [ "math.DS", "math.MG" ], "keywords": [ "elliptic billiards", "primitive periodic orbit", "confocal elliptic caustic", "Poncelet polygon", "outer tangent polygon", "tangent pedal polygon", "arbitrary fixed point", "signed area product", "meromorphic trace", "Jacobi parametrization", "k303,a", "center pedal", "repeated traversal", "k303,b" ], "manuscript_version_date": "2026-10-01", "publication_date": "2026-10-01", "publication_date_kind": "first public online release", "version": "1.1", "date_modified": "2026-10-01", "presentation_revision_only": false, "doi_archived_file_version": "1.0", "status": "unrefereed preprint", "canonical_url": "https://eulersolve.org/papers/amr-050-0014/", "pdf_url": "https://eulersolve.org/papers/amr-050-0014/paper.pdf?v=5ff816c279e3", "doi": "10.5281/zenodo.23088516", "zenodo_record_url": "https://zenodo.org/records/23088516", "license": "https://creativecommons.org/licenses/by/4.0/", "scope_caveat": "Retained k303,a: For every fixed noncircular ellipse a>b>0, fixed nondegenerate confocal elliptic caustic 0=6), the signed area A' of the consecutive boundary-tangent outer polygon times the signed area A'_M of its orthogonal pedal polygon is phase-independent for every fixed finite real point M. All coprime windings, including signed star trajectories, are included. This closes exact k303,a under the EXPLICIT_GENUINE_PERIOD_CONVENTION; the source does not literally define least period. Arbitrary padded vertex lists are not covered, and that unrestricted extension is exactly refuted. No hyperbolic or degenerate caustic, neighboring invariant, entire invariant list, novelty, independent human review, or formal verification claim. Added k303,b: For every fixed noncircular ellipse a>b>0 and fixed nondegenerate confocal elliptic caustic 0b>0, fixed nondegenerate confocal elliptic caustic 0=6), the signed area A' of the consecutive boundary-tangent outer polygon times the signed area A'_M of its orthogonal pedal polygon is phase-independent for every fixed finite real point M. All coprime windings, including signed star trajectories, are included. This closes exact k303,a under the EXPLICIT_GENUINE_PERIOD_CONVENTION; the source does not literally define least period. Arbitrary padded vertex lists are not covered, and that unrestricted extension is exactly refuted. No hyperbolic or degenerate caustic, neighboring invariant, entire invariant list, novelty, independent human review, or formal verification claim. Least period congruent to two modulo four is explicit; a primitive triangle traversed twice and listed as six vertices does not qualify. The experimental statement of Reznik, Garcia and Koiller, Stachel's canonical parametrization, and the related meromorphic methods of Chavez-Caliz and Roitman, Garcia and Reznik are credited. AI-assisted, self-audited, unrefereed preprint; no independent human review, formalization or absolute-priority certification.", "archived_record_url": "https://zenodo.org/records/23087221", "original_public_archive_preserved": true } ] }