{
  "schema_version": 1,
  "problem_number": "AMR-050-0062",
  "title": "Products of the Two Focal Inversion Areas in Elliptic Billiards",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For every odd-periodic real billiard family in an ellipse, we prove that the product of the trajectory-order signed areas of the two polygons obtained by unit inversion of the original vertices about the two original boundary foci is a positive phase-independent constant. The result includes every primitive odd period, admitted coprime star winding, orientation reversal and fixed odd repeated traversal; an r-fold traversal multiplies the product by r^2. Circular boundaries are treated directly. A real-caustic parity lemma excludes an omitted nondegenerate odd hyperbolic or degenerate family. In confocal Jacobi coordinates, each inverse-area trace has at most simple poles. For odd periods the two focal pole sets are disjoint, and reversal forces the opposite trace to vanish at each possible pole. Their product is therefore holomorphic on a compact quotient torus and constant. Real geometry establishes positive areas and finite inversion vertices. The target is invariant k903,a in arXiv:2004.12497v11, corresponding to frozen record AMR-050-0062 / 5100062. Reznik, Garcia and Helman's published N=3 formula is expressly credited, together with the classical Jacobi parametrization and meromorphic method. Exact controls verify the common product 1/324 for two non-equivalent triangular phases in one fixed family, odd-repeat scaling and a separate circle case. Genuine period-four phases instead give products 1 and 25/16, showing that the odd hypothesis cannot be removed. A separately executed 85-decimal diagnostic samples 42 odd-period families, including 27 stars, at five phases each; these finite checks are not the general proof. The quantity is signed area in trajectory order, not the unsigned area of a self-crossed polygon's bounded faces. 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 product",
    "Poncelet porism",
    "odd periodic trajectories",
    "coprime stars",
    "Jacobi elliptic functions",
    "meromorphic pole 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-050-0062/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0062/paper.pdf?v=c7fbcc7e5a7b",
  "doi": "10.5281/zenodo.23096994",
  "zenodo_record_url": "https://zenodo.org/records/23096994",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every odd-periodic real billiard family in a noncircular ellipse with its fixed nondegenerate confocal elliptic caustic, the product of the trajectory-order signed shoelace areas of the two polygons obtained by unit inversion of the original vertices about the two original boundary foci is a positive phase-independent constant. The result includes every primitive odd period n >= 3, every admitted coprime star winding, orientation reversal, and every fixed odd repeated traversal; the circular limit is proved separately. The exact source quantity k903,a and its signed-area convention are resolved. The published N=3 formula is explicitly credited. No novelty, absolute priority, independent review, or formal proof-assistant verification is claimed. AI-assisted, self-audited, unrefereed preprint. No independent human review, proof-assistant formalization or absolute-priority certification is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "c7fbcc7e5a7b7746ad102cb7cf0f5e9eb6aa149009d88c3654fdbcc3484126c0"
    },
    "source.zip": {
      "sha256": "284d2a707989ca6f0d4068c09bda6d56527af52889f8a83186434d0868035e4d"
    },
    "verification_report.md": {
      "sha256": "c4f6c4321669884afa8b72f5192c55d7be11043e1c7743616e54255d43499bcd"
    }
  },
  "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": true,
  "theorem_scope_resolved": true,
  "whole_source_record_resolved": true,
  "retained_public_get_count": 4,
  "retained_metadata_stable_before_after": false
}