{
  "schema_version": 1,
  "problem_number": "AMR-050-0018",
  "title": "Antipedal Areas of N-Periodics in the Elliptic Billiard: Row k401 Fails at the Centre and Holds at the Foci",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Row k401 of Reznik, Garcia and Koiller's list of experimentally found invariants of N-periodic billiard trajectories in an ellipse (record AMR-050-0018 of the UnsolvedMath collection) states that for N ≡ 2 (mod 4) and every point M the product A'A*_M is constant along the family; here A' is the signed area of the outer polygon, bounded by the tangents to the ellipse at the vertices, and A*_M that of the antipedal polygon of the trajectory with respect to M. The row is false as printed: for the ellipse with semi-axes 8/5 and 1, two members of the convex 6-periodic family, certified in exact arithmetic, give A'A*_O = 801792/21875 and 3241/100 at the centre O, and for every ellipse with semi-axes a > b the two 6-periodic members symmetric in both axes give values that differ by 4(a−b)³(a+b)⁵/(ab(2a+b)(a+2b)). At the foci the row holds. We prove that for every N ≥ 3 and every member of the convex N-periodic family A*_f1 + A*_f2 = 2κA, where A is the signed area of the trajectory and κ an explicit rational function of a, b and the caustic parameter; the proof combines a side-by-side formula for antipedal areas, Stachel's parametrisation by Jacobi elliptic functions and a residue computation. For even N the two focal areas are equal by central symmetry, so A*_fj = κA, a factor that an earlier note of the author obtained by another argument, and with Chavez-Caliz's theorem that A'A is constant for even N the product A'A*_fj is constant; it equals 16a²b² for N = 4 and 4a²(4b² − a²) for N = 6. The vanishing of κ accounts for the families with zero focal antipedal area found earlier, among them N = 6 with a = 2b, observed by Garcia and Reznik. Non-constancy at points other than the centre and the foci is only tested numerically. No printed statement, proof or refutation of k401 was found in the sources searched. This is an unrefereed note.",
  "result_type": "COMPLETE_SCOPED_PROOF",
  "categories": [
    "math.MG",
    "math.DS",
    "math.CV"
  ],
  "keywords": [
    "elliptic billiard",
    "Poncelet porism",
    "N-periodic trajectories",
    "antipedal polygon",
    "outer polygon",
    "signed area",
    "counterexample",
    "Jacobi elliptic functions",
    "residues",
    "Reznik–Garcia–Koiller invariants",
    "UnsolvedMath",
    "AMR-050-0018",
    "math.MG",
    "math.DS",
    "math.CV",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-15",
  "publication_date": "2026-10-15",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-15",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-050-0018/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0018/paper.pdf?v=0e2189ef1511",
  "doi": "10.5281/zenodo.23403151",
  "zenodo_record_url": "https://zenodo.org/records/23403151",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Row k401 of the Reznik-Garcia-Koiller list of invariants of N-periodic billiard trajectories in an ellipse (record AMR-050-0018) is false as printed: an exact 6-periodic counterexample at the centre of the ellipse. At the foci the row holds: the two focal antipedal areas add up to 2 kappa A for every N, and the outer area times a focal antipedal area is constant for every even N.",
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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."
}