{
  "schema_version": 1,
  "problem_number": "AMR-050-0048",
  "title": "A Focal Inversion Area Ratio for Four-Periodic Elliptic Billiards",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "The experimental invariant list of Reznik, Garcia and Koiller prints the value 2 for the focal-inverse area of the outer tangent polygon divided by that of a four-periodic elliptic billiard orbit. We give a rational counterexample to this value and a self-contained area calculation showing that the correct constant is 1/2. The result concerns vertexwise inversion of both polygons about the same focus and signed shoelace area. It applies to primitive four-periodic orbits with a confocal elliptic caustic. This is a reciprocal correction to a tabulated constant, not a refutation of the underlying constancy or a claim about the corresponding general-period conjecture.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "keywords": [
    "elliptic billiards",
    "Poncelet polygons",
    "focal inversion",
    "confocal ellipse",
    "four-periodic orbits",
    "signed shoelace area",
    "counterexample",
    "reciprocal correction"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-050-0048/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0048/paper.pdf?v=4fc41546f94b",
  "doi": "10.5281/zenodo.23065068",
  "zenodo_record_url": "https://zenodo.org/records/23065068",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "One manuscript for source record AMR-050-0048 (raw ID 5100048), invariant k806,b. The literal printed value 2 for outer-over-original focal-inverse area is refuted; the correct ratio 1/2 is proved for primitive four-periodic physical orbits with a nondegenerate confocal elliptic caustic. Both polygons undergo vertexwise inversion about the same focus, using signed shoelace area. This is a reciprocal numerical correction, not a refutation of constancy and not a solution of the general-period conjecture. Known four-periodic product invariants are credited. AI-assisted, self-audited and unrefereed; no independent human review or proof-assistant formalization is claimed, and no absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "4fc41546f94bf787fb1dccf2b9db39f75885bc922abcabfe8caae7776ddc4c00"
    },
    "source.zip": {
      "sha256": "505a7b84a1b607f27d190e2efa7517a337a0df592c3060820554baf8bc57e3f3"
    },
    "verification_report.md": {
      "sha256": "0e7bc1bcbaaca2853af985d26574c02bdd09748194b073a04eb581cdb6c1eccf"
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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."
}