{
  "schema_version": 1,
  "problem_number": "AMR-050-0043",
  "title": "Cosine Sums of Focal-Inversive Elliptic Billiards",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Fix an ellipse and a nondegenerate confocal elliptic caustic admitting a periodic billiard family. We prove that inversion of the vertices about either focus preserves a phase-independent sum of internal-angle cosines, provided the genuine least period is not four. The proof includes every admissible coprime star winding. A meromorphic continuation of the inverted-edge unit direction has an explicit eight-point divisor; the only possible double-pole collision occurs at genuine period four. Reflection symmetry removes all remaining simple poles of the cyclic cosine trace. The two focal constants coincide. An exact four-periodic pair also explains why the hypothesis cannot be replaced by the length of an arbitrary repeated vertex list. The theorem addresses the elliptic-caustic, genuine-period formulation of the focal-inversive cosine conjecture, recorded as AMR-050-0043 in UnsolvedMath. Hyperbolic caustics are not asserted, and known special cases and exceptions are credited. This AI-assisted preprint is self-audited and unrefereed; no independent review, formal verification or absolute-priority certification is claimed.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.DS",
    "math.MG",
    "math.CV"
  ],
  "keywords": [
    "elliptic billiard",
    "focal inversion",
    "cosine sum",
    "Poncelet polygon",
    "Jacobi elliptic functions",
    "meromorphic trace",
    "pole cancellation"
  ],
  "manuscript_version_date": "2026-10-12",
  "publication_date": "2026-10-12",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-12",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-050-0043/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0043/paper.pdf?v=602451d233e2",
  "doi": "10.5281/zenodo.23312253",
  "zenodo_record_url": "https://zenodo.org/records/23312253",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Complete proof for nondegenerate confocal elliptic caustics with genuine least period other than four, including both foci and all admissible coprime star windings. This does not assert a hyperbolic-caustic extension or a theorem for arbitrary repeated vertex-list lengths. The broader frozen source record is not counted as unconditionally resolved. Known special cases and exceptions are credited. AI-assisted, self-audited and unrefereed; no independent review, formal certification or absolute-priority claim.",
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      "sha256": "602451d233e2f584f500b8d6dd4551ffba27b1a23563e1c69fbf0da37afa4e46"
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    "source.zip": {
      "sha256": "5a422a86e4cbaa89977f93c7cb91dfa5d3604394f497e4fcddc16411c7ab07b5"
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    "verification_report.md": {
      "sha256": "194829f46c5e8c59a2dbe34a943fa28ea73f54eec9bc3539bf90170ef9f1bf81"
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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."
}