{
  "schema_version": 1,
  "problem_number": "AMR-050-0036",
  "title": "A Six-Periodic Zero of Focal Outer-Antipedal Area",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a periodic elliptic billiard, form the polygon of intersections of consecutive boundary tangents, then its antipedal with respect to either focus. Classical central symmetry implies equality of the two signed antipedal areas for effective even periods. Their quotient is therefore one where defined, but it need not be defined everywhere. We give an explicit convex billiard of least period six on an ellipse of aspect ratio 1+sqrt(3) for which both areas vanish, while all antipedal vertices are finite, all edges are nonzero and the vertices are not collinear. A direct coordinate calculation gives the signed area of an axial six-periodic family and isolates the zero exactly. This is a denominator obstruction, not a counterexample to equality on the quotient's natural domain. The earlier original-polygon antipedal zero at aspect ratio 2 is distinguished and credited. Source record: AMR-050-0036 (raw ID 5100036), invariant k608 in the frozen Hugging Face dataset ulamai/UnsolvedMath v1.6.0. Supporting lines and ordered signed area are used. The effective-even scope includes admitted coprime stars and repetitions of even-primitive orbits; no all-phase zero, odd-primitive repeated-list extension, hyperbolic-caustic extension or unqualified whole-source resolution is claimed. A portable Python standard-library exact certificate accompanies the analytic proof. This is a self-audited, AI-assisted, unrefereed preprint. Classical symmetry and the prior original-polygon zero are credited; novelty remains undetermined after a bounded search. No independent human review, proof-assistant verification or absolute-priority certification is asserted.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.DS"
  ],
  "keywords": [
    "elliptic billiards",
    "confocal caustics",
    "outer tangent polygon",
    ""
  ],
  "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-050-0036/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0036/paper.pdf?v=eecb01159520",
  "doi": "10.5281/zenodo.23091871",
  "zenodo_record_url": "https://zenodo.org/records/23091871",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For real confocal elliptic-caustic billiards of effective even period, including admitted coprime stars and repetitions of even-primitive orbits, the two focal antipedals of the outer boundary-tangent polygon are finite and have equal signed shoelace areas. Their quotient equals one wherever the common area is nonzero. An explicit convex primitive six-periodic axial orbit at a/b=1+sqrt(3) has both areas exactly zero, with finite antipedal vertices, nonzero edges and noncollinear vertices. The axial-family area for b=1 is 4a(a+1)(2+2a-a^2)/(2a+1)^(3/2). This refutes universal definedness, not equality of defined quotients. Supporting lines are used in the source antipedal construction. No all-phase zero, odd-primitive repeated-list, hyperbolic-caustic or unqualified whole-source resolution is asserted. Classical symmetry and the known original-polygon antipedal zero at a/b=2 are credited; novelty remains undetermined. AI-assisted, self-audited, unrefereed preprint. No independent human review, proof-assistant formalization or absolute-priority certification is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "eecb01159520118cde48b08aa5c97c57890ac07a357491817d3427a113885055"
    },
    "source.zip": {
      "sha256": "c0780893935dadebd3347c13b1e745ed143b79c03b67ea062e12339af6429703"
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    "verification_report.md": {
      "sha256": "095dea0262c588dc5eb5a66b24289a0d02d360b09fbf14558934f1c063019064"
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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
}