{
  "schema_version": 1,
  "problem_number": "AMR-050-0012",
  "title": "Center-Pedal Area Products in Elliptic Billiards",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We prove constancy of the product of the signed area of an elliptic billiard orbit and the signed area of its pedal polygon at the common center, for a fixed nondegenerate confocal elliptic caustic and least period odd or divisible by four. Primitive star trajectories are included. A reflection-coordinate telescope expresses the centered pedal area as a fixed multiple of a Jacobi trace. After complexifying the billiard parameter, the original area and this trace have complementary simple-pole sets; odd pairing and reversed-edge pairing remove every pole of their product. This establishes invariant k203,b under an explicitly stated genuine-period convention. Source record: AMR-050-0012 (raw ID 5100012, k203,b), Hugging Face dataset ulamai/UnsolvedMath. Exact standard-library controls verify the centered-pedal prefactor and actual low-period geometry. Primitive six-periodic controls show why an unrestricted repeated-list interpretation would be false. Hyperbolic and degenerate caustics are excluded. This is a self-audited, AI-assisted, unrefereed preprint; no independent human review, formal proof-assistant verification or absolute priority claim is made. Prior geometry, Jacobi parametrization and the established complex-pole method are credited, and novelty remains undetermined after a bounded primary-literature search.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "keywords": [
    "elliptic billiards",
    "primitive periodic orbit",
    "confocal elliptic caustic",
    "Poncelet polygon",
    "center-pedal polygon",
    "signed area product",
    "meromorphic trace",
    "Jacobi parametrization",
    "k203,b"
  ],
  "manuscript_version_date": "2026-10-01",
  "publication_date": "2026-10-01",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "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-050-0012/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0012/paper.pdf?v=8ba41a5466bc",
  "doi": "10.5281/zenodo.23085904",
  "zenodo_record_url": "https://zenodo.org/records/23085904",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The original signed orbit area times the original-side center-pedal signed area is constant for every primitive elliptic-caustic billiard family of least period odd or divisible by four, including coprime star windings. A reflection-coordinate telescope gives an explicit center-pedal Jacobi trace factor, and complementary pole cancellation proves the general invariant k203,b. The genuine-period convention is explicit because the source does not literally define minimal period; an unrestricted padded-list extension is false by exact doubled-N6 controls. No hyperbolic/degenerate-caustic, whole invariant-list, absolute-priority or independent-review claim. Least period is explicit; a repeated six-periodic walk padded to twelve listed 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.",
  "files": {
    "paper.pdf": {
      "sha256": "8ba41a5466bc289a9f714b57527f596fbe39b3422bc6bcf0a95d39bbde50c40c"
    },
    "source.zip": {
      "sha256": "9511d261c55301ce236efe1c1e52216ef1e6e1433fa74e15f6b62e0a00f77fb4"
    },
    "verification_report.md": {
      "sha256": "039440fda2b035a4d9533e872a5735aa003bad0db94ecf8329245947ab141646"
    }
  },
  "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."
}