{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1424",
  "content_hash": null,
  "slug": "cabling-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: The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general. Exact unresolved remainder: The classification for arbitrary knots in S³ remains open.",
  "relevance": "This is the dated publication status for the canonical target Cabling conjecture for reducible Dehn surgery.",
  "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: The conjecture holds for many classes, including strongly invertible and recent thin-knot regimes, with strong restrictions in general.\n\nThe exact unresolved remainder is: The classification for arbitrary knots in S³ remains open.\n\nA complete resolution must meet the following acceptance conditions:\n- Prove the cable-and-slope conclusion for every nontrivial knot in S³.\n- Or give a noncable knot and slope whose surgery is reducible.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/0040-9383(95)00016-X",
      "locator": "C. McA. Gordon and J. Luecke, “Reducible manifolds and Dehn surgery,” Topology 35(2) (1996), 385–409. main reducible-surgery theorems"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/0040-9383(95)00016-X",
    "locator": "C. McA. Gordon and J. Luecke, “Reducible manifolds and Dehn surgery,” Topology 35(2) (1996), 385–409. main reducible-surgery theorems"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1423",
      "title": "Strongest checked neighboring result",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1421",
      "title": "Dated source and duplicate audit",
      "object_type": "attempt",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R1422",
      "title": "Work at the unresolved boundary",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "cabling-conjecture",
      "title": "cabling conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}