{
  "schema_version": 1,
  "problem_number": "AMR-050-0001",
  "title": "Four-Periodic Counterexamples to Two Elliptic Billiard Invariants",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We give two exact counterexamples using one connected family of convex, primitive four-periodic billiards in a noncircular ellipse with a fixed confocal elliptic caustic. Write A for the orbit's ordered signed area, A' for the area of its consecutive tangent-intersection polygon, A'' for the area of its caustic-contact polygon, and theta_i for the internal angles of the original orbit. The quantity (A'/A) product_i sin(theta_i/2), listed as k107 for N congruent to 0 modulo 4, varies on this family. So does the product A'A'', listed as old k111 for even N in arXiv:2004.12497v11. The first assertion survives in the final 2021 journal table; the old second row does not. The renumbered final-journal k111 is a different, odd-period identity and is not contradicted. With s=a^2+b^2, D=a^2 b^2 and d(t)=(b^2-a^2) sin(t) cos(t), the displayed family has A'A''=16D(D+d(t)^2)/s^2 and (A'/A) product_i sin(theta_i/2)=2(D+d(t)^2)^2/(D s^2). Both vary whenever a>b>0. For a=4, b=3, an axis diamond and an axis-aligned rectangle share the caustic with semiaxes 16/5 and 9/5. Their area products are 331776/625 and 576, and their angle-area quantities are 288/625 and 625/1152. Boundary membership, reflection, interior segment tangency, primitive period and positive signed areas are established explicitly. The contradiction does not rely on self-intersection, hyperbolic caustics or numerical inference. The standard four-periodic geometry and Garcia and Reznik's Proposition 4.9 area identities are credited. The old product row's omission is documented without speculating about its reason. This is one version-qualified corrective note covering frozen dataset records AMR-050-0001 and AMR-050-0005 (raw IDs 5100001 and 5100005), not two deposits or a solution of the full source list. Bounded literature checks do not establish absolute novelty or priority. The manuscript is AI-assisted and unrefereed; no independent human review or formal proof-assistant verification is claimed. Author affiliation and contact: Mercury Software GmbH; alper@mercurycodelab.com; https://github.com/AlperTheKing.",
  "result_type": "COMPLETE_COUNTEREXAMPLE",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "keywords": [
    "elliptic billiards",
    "four-periodic orbit",
    "confocal elliptic caustic",
    "Poncelet polygon",
    "tangent polygon",
    "contact polygon",
    "signed area",
    "counterexample",
    "k107",
    "version-qualified old k111"
  ],
  "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-050-0001/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0001/paper.pdf?v=df4f6b6ad1af",
  "doi": "10.5281/zenodo.23075934",
  "zenodo_record_url": "https://zenodo.org/records/23075934",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "The literal retained k107 assertion (A'/A)*product_i sin(theta_i/2), for N congruent to 0 modulo 4, is false: two genuine primitive convex N=4 billiards in one explicit connected family, with outer semiaxes 4,3 and the same confocal caustic semiaxes 16/5,9/5, have values 288/625 and 625/1152. This completely refutes the exact AMR-050-0001 universal constancy claim. One joint corrective manuscript also includes the already accepted linked AMR-050-0005 counterexample to the older even A'A'' assertion, with values 331776/625 and 576; that removed row is not final journal k111. No other-period, neighboring-record, novelty, formal-verification or independent-review claim is made. Published four-periodic area identities are credited. The k107 multiplication survives in the final 2021 table; the linked even k111 product is a removed arXiv v11/dataset assertion, not the renumbered odd-period journal k111. AI-assisted, self-audited, unrefereed preprint; no independent human review, formalization or absolute-priority certification. No neighboring assertion or whole-source-list closure is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "df4f6b6ad1afd46ae274db52eb75bee63a722322642c72e6f829b3785f708d46"
    },
    "source.zip": {
      "sha256": "dd418508a4c390c0868f20712a70fee5977915c129369e9f5a6ea9457feeb280"
    },
    "verification_report.md": {
      "sha256": "01efa472c7c476eb7354ff238f4229c2a1be65e726c58218289b7fc467d3697c"
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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."
}