{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1592",
  "content_hash": null,
  "slug": "hilberts-tenth-problem-over-the-rationals-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: Koenigsmann gives a universal first-order definition of Z in Q and proves that the universal-existential theory of Q is undecidable. The purely existential decision problem asked by Hilbert's tenth problem over Q remains open. Exact unresolved remainder: Give a terminating algorithm deciding rational solvability of every integer-coefficient polynomial equation, or prove that no such algorithm exists.",
  "relevance": "For Hilbert's tenth problem over the rationals, 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: Koenigsmann gives a universal first-order definition of Z in Q and proves that the universal-existential theory of Q is undecidable. The purely existential decision problem asked by Hilbert's tenth problem over Q remains open.\n\nExact unresolved remainder: Give a terminating algorithm deciding rational solvability of every integer-coefficient polynomial equation, or prove that no such algorithm exists.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.4007/annals.2016.183.1.2",
      "locator": "Theorem 1 and Corollary 3 on pages 74-75"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.4007/annals.2016.183.1.2",
    "locator": "Theorem 1 and Corollary 3 on pages 74-75"
  },
  "models": [],
  "relations": [
    {
      "slug": "R1080",
      "title": "Current status and unresolved remainder",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "outgoing"
    },
    {
      "slug": "hilberts-tenth-problem-over-the-rationals",
      "title": "hilberts tenth problem over the rationals",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}