{
  "schema_version": 1,
  "problem_number": "AMR-050-0068",
  "title": "Equal Focal Inverted Pedal Areas Can Vanish in Elliptic Billiards",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For an elliptic billiard with a nondegenerate confocal elliptic caustic and effective even period, we prove that the pedals of the two unit-focus-inverted orbit polygons are congruent and have equal ordered signed areas. Equality does not imply nonvanishing. We construct a continuous family of genuine ten-periodic billiards of winding three and certify opposite signs of the common area at two rational major semiaxes. Exact rational interval arithmetic and the intermediate value theorem give a member for which both areas are zero. The construction satisfies the reflection law, has ten distinct vertices and a nondegenerate confocal elliptic caustic. As a contrast, we give an explicit positive formula for the counterclockwise four-periodic case. The focal area ratio remains one wherever its denominator is nonzero; the common-zero example obstructs an everywhere-defined literal quotient, not that ratio identity on its natural domain. Self-audited, unrefereed preprint prepared with AI assistance. Classical symmetry is credited; application novelty is undetermined after bounded literature searches. No independent human review, formal proof-assistant verification or absolute-priority claim is made. The source identity is AMR-050-0068 (raw ID 5100068), invariant k908,a in the frozen ulamai/UnsolvedMath v1.6.0 dataset. This is not a hyperbolic-caustic or whole-source-list closure. PDF, English LaTeX source and standard-library exact reproducibility checkers are included. Author: Alper Ferudun, Mercury Software GmbH.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "keywords": [
    "elliptic billiards",
    "focal inversion",
    "pedal polygon",
    "signed area",
    "confocal elliptic caustic",
    "primitive ten-periodic orbit",
    "certified rational intervals",
    "k908,a"
  ],
  "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-050-0068/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0068/paper.pdf?v=089aa71ead94",
  "doi": "10.5281/zenodo.23073120",
  "zenodo_record_url": "https://zenodo.org/records/23073120",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every effective even-period elliptic-caustic billiard orbit, the paired focal inverted-pedal polygons are congruent and have equal ordered signed areas. There exists a primitive ten-periodic orbit of winding three in a noncircular ellipse with a nondegenerate confocal elliptic caustic for which both signed areas are zero. Thus a universally defined quotient is obstructed, while ratio 1 wherever the common area is nonzero remains true. A separate explicit positive formula proves four-periodic nonvanishing; no hyperbolic-caustic, minimum-zero-period or priority claim. Natural-domain ratio 1 is not refuted; no full source-record closure. Classical symmetry is 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": "089aa71ead9430429ba054e3091b78c3790fa407b95327c9a51fffad97f5bde5"
    },
    "source.zip": {
      "sha256": "ec565d0cdece2b805412b2210808af90b245ede79b7ff352a333a2242b76582b"
    },
    "verification_report.md": {
      "sha256": "ff339314744b5adf220b83d4bce48031c22574922071277c3897321934574634"
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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."
}