{
  "schema_version": 1,
  "problem_number": "AMR-050-0044",
  "title": "Focal Inversion Area Products in Elliptic Billiards",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For an elliptic billiard with a fixed nondegenerate confocal elliptic caustic, we prove that the original signed shoelace area times the signed area of its unit focal inversion is a positive phase-independent constant when the genuine least period is divisible by four. Both foci give the same constant. The theorem includes all admitted coprime star windings and repetitions of those primitive cycles; concentric circles are treated directly. The general proof derives the focal denominator identity, enumerates every possible complex singularity and cancels all poles on a common Jacobi torus. Isotropic tangency and cyclic reversal reduce the apparent higher-order inverse-area poles to simple ones, and the two areas have the required complementary zeros. Classical parametrization and Jacobi methods are credited, as is the already published simple four-periodic value 4. An unrestricted divisible-by-four listed length is a stronger, false assertion: two exact primitive six-periodic phases in one fixed ellipse and caustic have unequal products after being traversed twice into lists of length twelve. A separate repeated-triangle control confirms the same convention boundary. This is a complete proof of an explicit genuine-period theorem and a counterexample to its unrestricted repeated-list extension, not an unqualified closure of every interpretation of source invariant k804,a or frozen record AMR-050-0044 (5100044) in ulamai/UnsolvedMath v1.6.0. Hyperbolic and degenerate caustics are not included. Three standard-library exact controls and a separately identified 33-family numerical diagnostic support, but do not replace, 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",
    "focal inversion",
    "signed area product",
    "Poncelet porism",
    "Jacobi elliptic functions",
    "isotropic tangency",
    "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-050-0044/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0044/paper.pdf?v=8a529273654b",
  "doi": "10.5281/zenodo.23093706",
  "zenodo_record_url": "https://zenodo.org/records/23093706",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every nondegenerate confocal elliptic-caustic billiard of genuine least period n divisible by four, including admitted coprime star windings and repeated traversals of those primitive cycles, the original signed shoelace area times its unit focal-inverted signed area is a positive phase-independent constant, identical at the two foci. The inverse vertices are finite. Repetition multiplies the product by the square of the repetition count. The circle is treated separately. Exact primitive triangles repeated four times and primitive hexagons repeated twice refute the stronger unrestricted listed-length-divisible-by-four assertion. No hyperbolic or degenerate caustic, unqualified whole-source closure, or absolute novelty claim is made. AI-assisted, self-audited, unrefereed preprint. No independent human review, proof-assistant formalization or absolute-priority certification is claimed.",
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      "sha256": "8a529273654b924f160b937ac893011e9b10dc38368e0533ef5884b3fde997ca"
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    "source.zip": {
      "sha256": "8fc732805ba20293f479b59ac04f31ed8f37f332d5ca33cbf820b2709943c8cf"
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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
}