{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1766",
  "content_hash": null,
  "slug": "standard-map-positive-metric-entropy-claim-status-20260801",
  "type": "claim",
  "title": "Current status and exact unresolved remainder",
  "summary": "OPEN as checked on 2026-08-01. Strongest checked neighboring result: Positive entropy can be created by nearby conservative perturbations, and hyperbolic sets of zero area are known for many parameters. Exact unresolved remainder: Positive Lebesgue metric entropy for the exact standard map at any K remains unproved.",
  "relevance": "This is the dated publication status for the canonical target Positive metric entropy for the standard map.",
  "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: Positive entropy can be created by nearby conservative perturbations, and hyperbolic sets of zero area are known for many parameters.\n\nThe exact unresolved remainder is: Positive Lebesgue metric entropy for the exact standard map at any K remains unproved.\n\nA complete resolution must meet the following acceptance conditions:\n- Exhibit K≠0 and prove h_Leb(T_K)>0.\n- Or prove h_Leb(T_K)=0 for every real K.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://people.math.harvard.edu/~knill/books/KnillDissertation.pdf",
      "locator": "O. Knill, dissertation discussion of the standard-map entropy problem. standard map open problem discussion"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://people.math.harvard.edu/~knill/books/KnillDissertation.pdf",
    "locator": "O. Knill, dissertation discussion of the standard-map entropy problem. standard map open problem discussion"
  },
  "relations": [
    {
      "slug": "R1765",
      "title": "Strongest checked neighboring result",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1763",
      "title": "Dated source and duplicate audit",
      "object_type": "attempt",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R1764",
      "title": "Work at the unresolved boundary",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "standard-map-positive-metric-entropy",
      "title": "standard map positive metric entropy",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}