{
  "schema_version": 1,
  "problem_number": "AMR-050-0019",
  "title": "Nonconstant Antipedal Area Ratios in Four-Periodic Elliptic Billiards",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a periodic elliptic billiard, let A' be the area of the polygon formed by consecutive tangents to the ellipse, and let A*_M be the antipedal area of the original orbit with respect to a fixed point M. We give an exact counterexample to the claimed constancy of A'/A*_M for periods divisible by four, listed as k402 in Table 5 of Reznik, Garcia and Koiller's 2021 article Fifty New Invariants of N-Periodics in the Elliptic Billiard. The denominator is unprimed and the printed point condition is All; taking M to be the center O is allowed. On a standard connected family of convex primitive four-periodic orbits in a fixed noncircular ellipse with one fixed confocal elliptic caustic, put s=a^2+b^2, D=a^2 b^2 and d(t)=(b^2-a^2) sin(t) cos(t). We derive A'/A*_O=D(D+d(t)^2)/(D^2+s^2 d(t)^2), with sharp range [D s^2/(s^2-2D)^2, 1] on the displayed family. For a=4,b=3, two orbits share the caustic with semiaxes 16/5 and 9/5 and give the ratios 1 and 90000/113569. The analytic continuous-family certificate proves reflection and strictly interior caustic tangency; all witness polygons are finite, convex and positively oriented. The four-periodic family and caustic are known geometry and are explicitly credited. This note refutes the literal k402 assertion and frozen dataset record AMR-050-0019 (raw ID 5100019), not every source invariant. A bounded primary-source search did not identify a correction of this exact row but does not certify novelty or absolute priority. Two standalone exact rational checkers accompany the analytic proof. This AI-assisted manuscript is unrefereed; no independent human review or proof-assistant verification is claimed. Alper Ferudun, Mercury Software GmbH. Contact: alper@mercurycodelab.com. GitHub: https://github.com/AlperTheKing.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "keywords": [
    "elliptic billiards",
    "four-periodic orbit",
    "confocal elliptic caustic",
    "Poncelet polygon",
    "antipedal polygon",
    "signed area",
    "exact counterexample",
    "k402",
    "fixed center"
  ],
  "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-050-0019/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0019/paper.pdf?v=28d48dd82f8c",
  "doi": "10.5281/zenodo.23077039",
  "zenodo_record_url": "https://zenodo.org/records/23077039",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Literal retained source k402 assertion A'/A*_M constant throughout a fixed-ellipse fixed-confocal-caustic family for N divisible by four and every fixed M is false. Two convex primitive N=4 orbits in one explicit continuous family, ellipse semiaxes4,3 and caustic16/5,9/5, M=O, give1 and90000/113569; all derived polygons finite convex positive. General exact formula and sharp range established on displayed family for every a>b>0. Entire exact AMR-050-0019 assertion refuted, not all source-paper conjectures, not other-period classification. The known four-periodic family and outer-rectangle geometry are credited. This concerns the literal unprimed original-orbit antipedal ratio k402 retained in both printed source versions, not all invariants or a new orbit construction. AI-assisted, self-audited, unrefereed preprint; no independent human review, formalization or absolute-priority certification.",
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      "sha256": "28d48dd82f8cd0e42792139d831dddbed48d0c792f2cb9661f6f5e4caf98259e"
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    "source.zip": {
      "sha256": "f46e8639999380bc61e7d8139eb0f2477c3954037f67de5d8286d6cc29d20658"
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    "verification_report.md": {
      "sha256": "a58b1ee570197b42b34e13a4eaaf8f5d1efadaf49f7ae3ba1ab7acd438769a74"
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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."
}