{
  "schema_version": 1,
  "problem_number": "AMR-050-0007",
  "title": "Focal Distance Products of Outer Elliptic Billiard Polygons",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We prove an explicit positive, phase-independent formula for the product of Euclidean distances from either focus of an ellipse to the consecutive boundary-tangent intersections of an elliptic billiard. The theorem assumes a nondegenerate confocal elliptic caustic and a least period divisible by four. It includes all admitted coprime star windings and repetitions of those genuine periods; concentric circular billiards are treated directly. A factorization of the actual squared focal distance, Jacobi quarter-period pairing and complete cyclic divisor cancellation give the general proof. Standard parametrization and Jacobi identities are credited, and the already known four-periodic case is not presented as new. An unrestricted divisible-by-four listed length is a different assertion: four traversals of an exact primitive triangle give two unequal distance products in the same fixed ellipse and caustic. Thus this is a complete proof of an explicit least-period theorem and a counterexample to its stronger repeated-list extension, not an unqualified closure of every interpretation of invariant k115 or frozen record AMR-050-0007 (5100007) in ulamai/UnsolvedMath v1.6.0. No hyperbolic or degenerate caustic is included. An unchanged standard-library exact checker and an optional, separately identified mpmath geometry diagnostic accompany the analytic proof. This English preprint is AI-assisted, self-audited and unrefereed. Novelty remains undetermined after a bounded primary-source review; no independent human review, proof-assistant verification or absolute-priority certification is asserted.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "keywords": [
    "elliptic billiards",
    "outer tangent polygon",
    "focal distance product",
    "Jacobi elliptic functions",
    "Poncelet porism",
    "cyclic divisor cancellation",
    "exact computation"
  ],
  "manuscript_version_date": "2026-10-02",
  "publication_date": "2026-10-02",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-02",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-050-0007/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0007/paper.pdf?v=6a382882e9c1",
  "doi": "10.5281/zenodo.23093258",
  "zenodo_record_url": "https://zenodo.org/records/23093258",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every nondegenerate confocal elliptic-caustic billiard of least period n divisible by four, including all admitted coprime star windings and repetitions of those genuine periods, the product of Euclidean distances from the original ellipse focus to consecutive boundary-tangent intersections is phase-independent, with the explicit positive formula in proof.md. The circle is direct. Four traversals of an exact primitive triangle refute the unrestricted divisible-by-four listed-length extension. No hyperbolic or degenerate caustic, whole-source closure, or absolute novelty is claimed. AI-assisted, self-audited, unrefereed preprint. No independent human review, proof-assistant formalization or absolute-priority certification is claimed.",
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      "sha256": "6a382882e9c1959813ac440b77cafe7f8be2c72f191fcba9039b5274b6764150"
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    "source.zip": {
      "sha256": "e808237e440951b70d9d5d658ae9bf950caa8bf50179de82d389e6c7a4aeedf5"
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    "verification_report.md": {
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text.",
  "original_problem_resolved": false,
  "theorem_scope_resolved": true,
  "whole_source_record_resolved": false,
  "retained_public_get_count": 4,
  "retained_metadata_stable_before_after": false
}