{
  "schema_version": 1,
  "problem_number": "AMR-050-0023",
  "title": "Original Antipedal Centroids in Elliptic Billiards",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "We establish explicit, phase-independent vertex-centroid formulas for antipedals of the original polygon in elliptic billiards with a nondegenerate confocal elliptic caustic and even least period. The pole is the ellipse center or either focus, and all consecutive antipedal intersections are finite. The result covers admissible coprime star trajectories and repetitions of even primitive orbits; the circle is treated separately. A paired-chord identity and perimeter stationarity yield the focal coefficient in terms of semiaxes, caustic parameter and mean side length. An even number of listed vertices is not sufficient: exact twice-repeated primitive triangles show that the unrestricted even-list extension fails. This note gives a precisely scoped interpretation of invariant k405 in Reznik-Garcia-Koiller's Table 5, rather than declaring every interpretation of the abbreviated dataset record AMR-050-0023 resolved. Source record 5100023 belongs to the frozen Hugging Face dataset ulamai/UnsolvedMath v1.6.0. The construction is the original-polygon antipedal and its unweighted vertex mean, not an outer-polygon antipedal or an area centroid. No hyperbolic or degenerate-caustic extension is claimed. Portable exact symbolic and rational verification files accompany the analytic proof. This is an AI-assisted, self-audited, unrefereed preprint. Classical ingredients are credited; novelty remains undetermined, and 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",
    "antipedal polygon",
    "vertex centroid",
    "Poncelet porism",
    "perimeter variation",
    "exact computation"
  ],
  "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-050-0023/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0023/paper.pdf?v=efab7f38f6a5",
  "doi": "10.5281/zenodo.23092466",
  "zenodo_record_url": "https://zenodo.org/records/23092466",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every genuine even-period billiard in a nondegenerate confocal elliptic caustic, including admitted coprime stars and repetitions of an even primitive orbit, the original-polygon antipedal has finite intersections, center-pole vertex mean O and focal vertex mean O+gamma(F-O), with explicit gamma=-1+(a^2*b^2-lambda*c^2)/((b^2-lambda)*c^2)*(1-b^2*L/(2*N*a*b*sqrt(lambda))). The circle is treated separately. Exact twice-repeated primitive triangles refute the stronger unrestricted even-list interpretation. No hyperbolic or degenerate caustic, area-centroid, unqualified frozen-record closure, whole-source-list, or novelty claim is made. AI-assisted, self-audited, unrefereed preprint. No independent human review, proof-assistant formalization or absolute-priority certification is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "efab7f38f6a5041af5c5af5d5311aefcb949b30d601fca790fafa583079d677d"
    },
    "source.zip": {
      "sha256": "1a8d84e45b79898583ed93574d51024c1f358f73a40ce6d4180d56d6c8e5c193"
    },
    "verification_report.md": {
      "sha256": "501206eb611853e0973d8a860844f6eddbe1c91dc54f403d59037ee4a89f366e"
    }
  },
  "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
}