{
  "schema_version": 1,
  "problem_number": "AMR-050-0029",
  "title": "Inner Steiner Pedal Ratios and Exceptional Zero-Area Families",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "For an odd-periodic elliptic billiard with a fixed nondegenerate confocal elliptic caustic, consider the polygon of caustic contact points and its own Steiner curvature centroid. We prove that this centroid is finite at every real phase and that the signed area of its pedal polygon is a phase-independent real multiple R of the contact polygon's area. Thus the reciprocal area ratio proposed as invariant k503 is constant whenever defined. Its denominator need not be nonzero: we construct a genuine convex five-periodic family in which the stationary pedal area vanishes identically, specifying its nondegenerate ellipse and caustic by a uniquely isolated algebraic root of degree 18. This is a correction to universal definedness, not phase variation of a defined ratio. The result includes odd coprime star windings under an explicit angle convention, odd repetitions, and a separate circle case. The analytic proof uses the pedal-area quadratic, elliptic-function poles and characters, and an odd quotient torus. Portable exact quadratic-field, symbolic-identity, and rational Bernstein certificates accompany the exceptional family. 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",
    "Poncelet polygons",
    "caustic contact polygon",
    "Steiner curvature centroid",
    "pedal area",
    "Jacobi elliptic functions",
    "zero-area family",
    "exact algebraic certificate"
  ],
  "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-050-0029/",
  "pdf_url": "https://eulersolve.org/papers/amr-050-0029/paper.pdf?v=daa7701789f6",
  "doi": "10.5281/zenodo.23090907",
  "zenodo_record_url": "https://zenodo.org/records/23090907",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "For every fixed nondegenerate confocal elliptic-caustic billiard family of odd listed length, the caustic-contact polygon has a finite own Steiner curvature centroid and its own-centroid signed pedal area equals a phase-independent real multiple R of its positive oriented polygon area. Hence exact k503 is constant on its defined domain: R nonzero makes the reciprocal ratio defined everywhere, while R zero makes it undefined everywhere in that family. A genuine convex least-period-five zero-area family is explicitly specified by an isolated degree18 algebraic root. This refutes universal definedness, not phase invariance of a defined ratio. All admitted odd coprime stars under the explicit internal-ray angle convention, odd repetitions and the separate circle case are covered. No k502, whole invariant-list, hyperbolic/degenerate, absolute-priority, independent-human-review or formal-verification claim. AI-assisted, self-audited, unrefereed preprint. No independent human review, proof-assistant formalization or absolute-priority certification is claimed.",
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    "source.zip": {
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    "verification_report.md": {
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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.",
  "original_problem_resolved": true,
  "theorem_scope_resolved": true,
  "whole_source_record_resolved": true,
  "retained_public_get_count": 5,
  "retained_metadata_stable_before_after": true
}