{
  "schema_version": 1,
  "problem_number": "AMR-050-0027",
  "title": "Steiner-Centroid Pedal Area Ratios in Elliptic Billiards",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For an odd-periodic billiard in an ellipse with a fixed confocal elliptic caustic, let C be the Steiner curvature centroid: the weighted average of the vertices with weights sin(2 theta_i), where theta_i is the internal reflection angle. Let A be the signed area of the original chord polygon and B_C the signed area of its pedal polygon with respect to C. We prove that C is always defined, that B_C is nonzero, and that A/B_C is invariant throughout the family. The result covers every coprime odd winding and odd repeated traversal, with an explicit signed-star convention; circular billiards are treated separately. This resolves invariant k501 in Table 6 of Reznik, Garcia and Koiller's Fifty New Invariants of N-Periodics in the Elliptic Billiard. The proof combines the classical pedal-area quadratic with reflection trace identities, complementary elliptic-function divisors, a character argument on an odd quotient torus, and a strict odd-grid theta inequality. The quotient-torus argument also identifies the moving centroid's axis-aligned elliptic locus. Exact standard-library checks accompany the proof but are not used as a substitute for the general analytic argument. This is an unrefereed, AI-assisted preprint; no absolute priority, independent human review, or formal proof-assistant verification is asserted.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "math.DS",
    "math.MG"
  ],
  "keywords": [
    "elliptic billiards",
    "odd periodic orbit",
    "confocal elliptic caustic",
    "Steiner curvature centroid",
    "pedal polygon",
    "signed area ratio",
    "nonvanishing",
    "meromorphic trace",
    "odd quotient torus",
    "theta inequality",
    "k501"
  ],
  "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-050-0027/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0027/paper.pdf?v=2d6e66f9bba1",
  "doi": "10.5281/zenodo.23089778",
  "zenodo_record_url": "https://zenodo.org/records/23089778",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every fixed ellipse a>b>0 and nondegenerate confocal elliptic caustic 0<lambda<b^2, every closed billiard family of odd listed length N>=3 has a finite original-polygon Steiner curvature centroid C and a nonzero original-sideline pedal signed area B_C; the literal ratio A/B_C is phase-independent and positive in the positive orientation. All admissible coprime star windings under the explicitly stated internal-ray angle convention, odd repetitions, and the directly proved circle case are included. Only exact retained k501 of AMR-050-0027 is resolved; no k502/k503, whole invariant-list, hyperbolic/degenerate-caustic, absolute-priority, independent-human-review or formal-verification claim. Classical pedal-area, billiard, elliptic-function and theta methods and the originating invariant of Reznik, Garcia and Koiller are credited. AI-assisted, self-audited, unrefereed preprint; no independent human review, formalization or absolute-priority certification.",
  "files": {
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      "sha256": "2d6e66f9bba1931747e5a2a5d860eb79c585db9473bfe8121a64445bb30fbc7b"
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    "source.zip": {
      "sha256": "6aadcce697537b4861001d34fe280dd34470d67514750562ce7e5a3cb9b4d30d"
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    "verification_report.md": {
      "sha256": "a4a9926f8f08a1869f2fb7faeb3d7b6fd88b9e3ee7b3477f8578aa8ecbc4185c"
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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."
}