{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1720",
  "content_hash": null,
  "slug": "purely-cosmetic-surgery-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: Floer, Casson, hyperbolic, and computational methods settle broad knot classes, including all knots through substantial crossing ranges. Exact unresolved remainder: The universal statement for nontrivial knots in S³ remains open.",
  "relevance": "This is the dated publication status for the canonical target Purely cosmetic surgery conjecture.",
  "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: Floer, Casson, hyperbolic, and computational methods settle broad knot classes, including all knots through substantial crossing ranges.\n\nThe exact unresolved remainder is: The universal statement for nontrivial knots in S³ remains open.\n\nA complete resolution must meet the following acceptance conditions:\n- Prove no nontrivial knot in S³ admits a purely cosmetic pair.\n- Or exhibit a nontrivial knot, distinct slopes, and an orientation-preserving homeomorphism of the surgeries.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1515/crll.1990.405.181",
      "locator": "S. Boyer and D. Lines, Surgery formulae for Casson's invariant and extensions to homology lens spaces, J. Reine Angew. Math. 405 (1990). cosmetic-surgery consequences"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1515/crll.1990.405.181",
    "locator": "S. Boyer and D. Lines, Surgery formulae for Casson's invariant and extensions to homology lens spaces, J. Reine Angew. Math. 405 (1990). cosmetic-surgery consequences"
  },
  "relations": [
    {
      "slug": "R1719",
      "title": "Strongest checked neighboring result",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1717",
      "title": "Dated source and duplicate audit",
      "object_type": "attempt",
      "relation": "evidences",
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    },
    {
      "slug": "R1718",
      "title": "Work at the unresolved boundary",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "purely-cosmetic-surgery-conjecture",
      "title": "purely cosmetic surgery conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}