{
  "schema_version": 1,
  "problem_number": "AMR-050-0021",
  "title": "Focal Pedal-Antipedal Area Products in Elliptic Billiards",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a genuine elliptic billiard orbit of least period divisible by four, we prove that the signed area of its pedal polygon at either focus times the signed area of its antipedal polygon at the same focus is constant throughout the fixed-ellipse, fixed-confocal-caustic family. Convex and coprime star windings are included. We derive explicit factors relating focal pedal area to outer tangent area and focal antipedal area to original area, using a velocity-product telescope, an actual Jacobi quarter-period permutation, and direct signed-area algebra. The known even-period original/outer area-product theorem then gives the requested invariant. Zero signed antipedal area is allowed. The four-periodic value is 16 a^4 b^4/(a^2+b^2)^2. We explicitly qualify the least-period convention: repeating a primitive six-periodic orbit to obtain twelve listed vertices does not meet it, and exact controls show that the unrestricted repeated-walk extension is false. Hyperbolic or degenerate caustics and other source assertions are not covered. Source record: AMR-050-0021 (raw ID 5100021, k403,b), Hugging Face dataset ulamai/UnsolvedMath. Self-audited, AI-assisted, unrefereed preprint; no independent human review or proof-assistant verification is claimed. Established elliptic billiard geometry, Stachel's canonical parametrization and Chavez-Caliz's even area-product theorem are credited. Novelty remains undetermined after a bounded primary search, and no absolute priority claim is made. The source ZIP contains the English LaTeX manuscript, authored proof and source-hypothesis notes, and portable standard-library exact controls with their actual execution outputs. Rounded higher-period observations are corroboration, not the proof.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "keywords": [
    "elliptic billiards",
    "primitive periodic orbit",
    "confocal elliptic caustic",
    "Poncelet polygon",
    "focal pedal polygon",
    "antipedal polygon",
    "signed area product",
    "Jacobi quarter-period",
    "k403,b"
  ],
  "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-050-0021/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0021/paper.pdf?v=34ff068237a7",
  "doi": "10.5281/zenodo.23079799",
  "zenodo_record_url": "https://zenodo.org/records/23079799",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Same-focus signed pedal-antipedal area product is constant for every primitive elliptic-caustic billiard family whose least period is divisible by four, including coprime star windings. Explicit factors p=K T and q=R A reduce it to the published even original/outer area-product theorem. This resolves k403,b under the stated standard genuine-period convention; an unrestricted padded-list reading is separately false by exact doubled-N6 controls. No hyperbolic or degenerate caustic extension, whole invariant-list closure or absolute novelty claim. Least period is explicit; a repeated six-periodic walk padded to twelve listed vertices does not qualify. Classical geometry, Stachel's canonical parametrization and Chavez-Caliz's even original/outer area-product theorem are credited. AI-assisted, self-audited, unrefereed preprint; no independent human review, formalization or absolute-priority certification.",
  "files": {
    "paper.pdf": {
      "sha256": "34ff068237a7ebbd6ffa063f89717aae946bc4d27df6db23d4468b5bb52d3582"
    },
    "source.zip": {
      "sha256": "6680bf0a3fd4cade6fabaf2061f88941c9d57776fe65f28bfb9e69acb8b39876"
    },
    "verification_report.md": {
      "sha256": "c84cf16d7a46801a6a438228924d069f93bda9a7c8242875435fa777c6afd8f5"
    }
  },
  "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."
}