{
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  "ref": "R1395",
  "content_hash": null,
  "slug": "birkhoff-billiard-conjecture-claim-status-20260801",
  "type": "claim",
  "title": "Current status and exact unresolved remainder",
  "summary": "OPEN as checked on 2026-08-01. Strongest checked neighboring result: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses. Exact unresolved remainder: The global classification of all smooth strictly convex integrable tables remains open.",
  "relevance": "This is the dated publication status for the canonical target Birkhoff conjecture for integrable convex billiards.",
  "relevance_source": "recorded",
  "body": "The problem was checked as open on 2026-08-01.\n\nThe strongest neighboring result found in the cited sources is: Local rigidity is known near many ellipses and under symmetry or perturbative hypotheses.\n\nThe exact unresolved remainder is: The global classification of all smooth strictly convex integrable tables remains open.\n\nA complete resolution must meet the following acceptance conditions:\n- Prove every table satisfying the invariant-caustic hypothesis is an ellipse.\n- Or construct a nonelliptic smooth strictly convex table with the full caustic foliation.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1007/s00222-025-01397-y",
      "locator": "Illya Koval, “Local strong Birkhoff conjecture and local spectral uniqueness of almost every ellipse,” Inventiones Mathematicae 244(1) (2026), 221–298. main local strong Birkhoff theorem"
    },
    "missing": [
      "source",
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    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1007/s00222-025-01397-y",
    "locator": "Illya Koval, “Local strong Birkhoff conjecture and local spectral uniqueness of almost every ellipse,” Inventiones Mathematicae 244(1) (2026), 221–298. main local strong Birkhoff theorem"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1394",
      "title": "Strongest checked neighboring result",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1392",
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      "object_type": "attempt",
      "relation": "evidences",
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    },
    {
      "slug": "R1393",
      "title": "Work at the unresolved boundary",
      "object_type": "attempt",
      "relation": "addresses",
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    {
      "slug": "birkhoff-billiard-conjecture",
      "title": "birkhoff billiard conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
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}