{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1391",
  "content_hash": null,
  "slug": "birch-and-swinnerton-dyer-rank-conjecture-status-packet-quality-20260801",
  "type": "claim",
  "title": "Dated status and exact unresolved remainder",
  "summary": "Unresolved in this packet after the dated source check. Strongest checked result: For elliptic curves over Q, Gross-Zagier and Kolyvagin, together with modularity, prove equality of algebraic and analytic rank and finiteness of the Tate-Shafarevich group when the analytic rank is 0 or 1. The rank equality for analytic rank at least 2 remains open in general. Exact unresolved remainder: Prove equality of algebraic and analytic rank for every elliptic curve over Q, or exhibit a curve for which both ranks are rigorously determined and unequal.",
  "relevance": "For Birch and Swinnerton-Dyer rank conjecture, this successor gives readable dated status prose and the exact remaining research boundary.",
  "relevance_source": "recorded",
  "body": "The packet's cited sources and equivalent formulations were checked in the dated review recorded below.\n\nStrongest checked result: For elliptic curves over Q, Gross-Zagier and Kolyvagin, together with modularity, prove equality of algebraic and analytic rank and finiteness of the Tate-Shafarevich group when the analytic rank is 0 or 1. The rank equality for analytic rank at least 2 remains open in general.\n\nExact unresolved remainder: Prove equality of algebraic and analytic rank for every elliptic curve over Q, or exhibit a curve for which both ranks are rigorously determined and unequal.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/",
      "locator": "official problem description and unsolved classification"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/",
    "locator": "official problem description and unsolved classification"
  },
  "relations": [
    {
      "slug": "R936",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "outgoing"
    },
    {
      "slug": "birch-and-swinnerton-dyer-rank-conjecture",
      "title": "birch and swinnerton dyer rank conjecture",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}