{
  "schema_version": 1,
  "problem_number": "AMR-050-0026",
  "title": "Stationary Focal Antipedal Centroids for Even Elliptic Billiard Periods",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Fix a noncircular ellipse and a nondegenerate confocal elliptic caustic supporting a family of billiard orbits of effective even period. Form the outer polygon by intersecting consecutive boundary tangents, then take its antipedal with respect to either focus. We prove that its vertex centroid is fixed throughout the family and lies on the major axis. An exact opposite-edge calculation reduces this centroid to a bilinear trace of a Poncelet polygon inscribed in a circle and circumscribed about a concentric ellipse. We supply a compact-curve proof of trace constancy, adapting the pole cancellation method of Akopyan, Schwartz and Tabachnikov. The calculation proves the constancy assertion k407 of Reznik, Garcia and Koiller, with the effective-period convention stated explicitly. The constant need not be the center of the ellipse. We make no signed-area-centroid or hyperbolic-caustic claim. Self-audited, unrefereed preprint prepared with AI assistance. Established trace methods are credited; no independent human review, formal proof-assistant verification or absolute-priority claim is made. The source identity is AMR-050-0026 in the frozen ulamai/UnsolvedMath v1.6.0 dataset. This manuscript treats the single literal invariant k407, not every invariant in the source list. PDF, English LaTeX source and standard-library reproducibility checkers are included. Author: Alper Ferudun, Mercury Software GmbH.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "keywords": [
    "elliptic billiards",
    "Poncelet polygons",
    "antipedal polygon",
    "vertex centroid",
    "confocal caustic",
    "complex curve",
    "bilinear trace",
    "k407"
  ],
  "manuscript_version_date": "2026-09-30",
  "publication_date": "2026-09-30",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-09-30",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/amr-050-0026/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0026/paper.pdf?v=317178430532",
  "doi": "10.5281/zenodo.23071366",
  "zenodo_record_url": "https://zenodo.org/records/23071366",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every prescribed Poncelet billiard family in a noncircular ellipse with a nondegenerate confocal elliptic caustic and effective even period N, the outer polygon's antipedal with respect to either focus has a finite vertex centroid that is constant along the family and lies on the major axis. This proves the exact vertex-centroid constancy assertion k407 under the source's effective-period convention; no general area-centroid, hyperbolic-caustic, padded-odd-cycle or absolute-priority claim. Vertex-only result. Established trace methods are credited. AI-assisted, self-audited, unrefereed preprint; no independent human review, proof-assistant formalization or absolute-priority certification. No absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "317178430532655300d12aff19391751a3dc32ab6913b5062a4df803d56355b6"
    },
    "source.zip": {
      "sha256": "b43108ae779b1f403e7e0f4c7d632b0ef49c2eb61c6be50178e5da3ce86e0183"
    },
    "verification_report.md": {
      "sha256": "1bccef396bdb4a612c6d28017a85c885b1b396ebf758dcbd93aea7301e4426eb"
    }
  },
  "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."
}