{
  "schema_version": 1,
  "problem_number": "AMR-050-0041",
  "title": "A Telescoping Formula for Evolute Areas in Elliptic Billiards",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For a noncircular elliptic billiard with a nondegenerate confocal elliptic caustic, we derive closed formulas for the signed area ratios of the caustic contact polygon and the boundary tangent polygon to their discrete evolutes. The formulas hold for every physical periodic orbit of least period greater than four, including star trajectories. Unit complex contact parameters satisfy a biquadratic relation. The consecutive circumcenters have a rational expression whose local area term differs from a constant multiple of the contact-area term by an explicit rational coboundary. A polynomial certificate and cyclic summation prove the inner formula. Conic polarity and a fixed-conic circumcenter transformation yield the outer formula. The exceptional coefficients force least periods three or four, which proves that the stated quotients have nonzero denominators. Exact examples also show why repeating primitive four-cycles does not extend the theorem to a list-length interpretation of period. The results address invariants k703 and k702 of Reznik, Garcia and Koiller in one manuscript, corresponding to AMR-050-0041 and AMR-050-0040 in the frozen ulamai/UnsolvedMath v1.6.0 dataset.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "keywords": [
    "elliptic billiards",
    "Poncelet polygons",
    "confocal ellipses",
    "discrete evolute",
    "signed area",
    "circumcenter",
    "telescoping identity",
    "rational coboundary"
  ],
  "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-050-0041/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0041/paper.pdf?v=032dab58294c",
  "doi": "10.5281/zenodo.23061834",
  "zenodo_record_url": "https://zenodo.org/records/23061834",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "One manuscript for the linked inner and outer evolute-area assertions k703 and k702, corresponding to AMR-050-0041 and AMR-050-0040. The formulas concern physical orbits of least period N>4 in a noncircular elliptic billiard with a nondegenerate confocal elliptic caustic and use signed shoelace areas, including for star trajectories. The source's printed N>4 is clarified as least period, not quoted as an explicit source definition; repeated primitive four-cycles refute only the broader list-length extension. No hyperbolic-caustic or unsigned-region-area conclusion is asserted. 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": "032dab58294cd206ca11d294a112ba99c6ddc095e381f4d55790a08480eeea6d"
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    "source.zip": {
      "sha256": "ad9896aedcc065ada05cffd47e3559c7696dcc447b950357ab14a5c33737ea7b"
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    "verification_report.md": {
      "sha256": "e18075a7bb2f8324d891684b0f8d1913ef30910775422f2b5b245a579eea773e"
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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."
}