{
  "schema_version": 1,
  "problem_number": "AMR-050-0046",
  "title": "Focal Inversion Area Ratios in Elliptic Billiards",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every nondegenerate confocal elliptic-caustic billiard of genuine least period n >= 6 with n congruent to 2 modulo 4, we prove that the original ordered signed area divided by the signed area after unit inversion about either boundary-ellipse focus is a positive phase-independent constant, identical at the two foci. The proof includes all admitted coprime star windings and establishes finite inverse vertices and a nonzero denominator. In confocal Jacobi coordinates, the two area traces have the same two possible simple poles on a reduced quotient torus. Residue cancellation and imaginary antiperiodicity force proportionality; real geometry proves positivity. Repetitions of admitted primitive cycles preserve the ratio, and the circular case is treated directly. The target is k805 in arXiv:2004.12497v11, corresponding to frozen record AMR-050-0046 / 5100046. Garcia and Reznik's 2022 Proposition 4.16 already gives the simple-six-periodic formula, which is expressly credited. Exact controls reproduce the ratio 32/27 at two six-periodic phases. Two primitive triangles in one fixed family, each traversed twice to give listed length six, instead have ratios (10584 + 72 sqrt(105))/25 and (10584 - 72 sqrt(105))/25; this excludes only the stronger unrestricted listed-length interpretation. A separately executed 85-decimal diagnostic covers 27 target families and 18 odd-period controls, five phases each; these finite checks are not the general proof. No hyperbolic or degenerate caustic extension or unqualified whole-source closure is claimed. This English preprint is AI-assisted, originating-researcher self-audited and unrefereed. Novelty remains undetermined; no independent human review, proof-assistant verification, guaranteed indexing or absolute-priority claim is made.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "keywords": [
    "elliptic billiards",
    "focal inversion",
    "signed area ratio",
    "Poncelet porism",
    "Jacobi elliptic functions",
    "meromorphic residue cancellation",
    "genuine least period",
    "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-050-0046/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0046/paper.pdf?v=5771503ea55e",
  "doi": "10.5281/zenodo.23094118",
  "zenodo_record_url": "https://zenodo.org/records/23094118",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every nondegenerate confocal elliptic-caustic billiard of genuine least period n >= 6 with n congruent to 2 modulo 4, including all admitted coprime star windings, the original ordered signed area divided by its unit focal-inverted ordered signed area is a positive phase-independent constant, identical at the two foci. The inverse vertices are finite and the denominator is nonzero. Repeated traversals of those primitive cycles preserve the ratio; the circle is treated separately. An exact primitive triangle repeated twice refutes only the stronger unrestricted listed-length assertion. No hyperbolic or degenerate caustic or unqualified whole-source closure 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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  "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
}