{
  "schema_version": 1,
  "problem_number": "AMR-050-0025",
  "title": "Focal Antipedal Centroids of Rectangles Circumscribed about an Ellipse",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "Let a rectangle be circumscribed about a noncircular nondegenerate ellipse with center O and a focus F. We show that its antipedal quadrilateral with respect to F has vertex centroid O and area centroid O+(F-O)/3, independently of the rectangle's orientation. The quadrilateral is always finite and strictly convex. Applied to the known outer tangent rectangles of simple four-periodic elliptic billiards with a confocal elliptic caustic, this calculation proves both constancy assertions in the experimental invariant k406,b of Reznik, Garcia and Koiller and identifies their values. The proof uses an orthonormal support frame and exact polygon first moments; it does not assert the corresponding general-even-period invariant k407. Source record: AMR-050-0025 (raw ID 5100025, k406,b), Hugging Face dataset ulamai/UnsolvedMath. Self-audited, AI-assisted, unrefereed preprint; no independent human review or formalization is claimed. The source and novelty review credits the known four-periodic outer-rectangle lemma and makes no absolute priority claim. The source ZIP includes the English LaTeX manuscript and a portable standard-library exact rational checker.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.MG",
    "math.DS"
  ],
  "keywords": [
    "elliptic billiards",
    "Poncelet polygons",
    "antipedal quadrilateral",
    "focal centroid",
    "circumscribed rectangle",
    "four-periodic orbits",
    "polygon first moments",
    "signed shoelace area"
  ],
  "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-050-0025/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0025/paper.pdf?v=14cdab9b9421",
  "doi": "10.5281/zenodo.23068631",
  "zenodo_record_url": "https://zenodo.org/records/23068631",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "One complete-proof manuscript for source AMR-050-0025 (raw ID 5100025), invariant k406,b: both focal antipedal centroids are constant for the outer tangent rectangles of simple four-periodic elliptic billiards, N=4 only. More generally, every rectangle circumscribed about a nondegenerate ellipse with center O and focus F has a finite strictly convex focal antipedal quadrilateral with vertex centroid O and area centroid O+(F-O)/3. No general-even-period k407 closure is claimed. Known outer-rectangle geometry and related centroid results are credited. AI-assisted, self-audited and unrefereed; no independent human review or proof-assistant formalization is claimed, and no absolute priority is claimed.",
  "files": {
    "paper.pdf": {
      "sha256": "14cdab9b9421df849b98aed4deafdd7d627f27ae102a277b4616682d99743f47"
    },
    "source.zip": {
      "sha256": "05be49eea829f532b308e3e22fc5c98c7b3dcf7e9950eadb96cf80513438446b"
    },
    "verification_report.md": {
      "sha256": "d3b9d74cbe4cbb2d38a81856b90a7e566d2ef549ce2e5d8dac0d19ddab073ce2"
    }
  },
  "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."
}