{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1523",
  "content_hash": null,
  "slug": "eventual-categoricity-aecs-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 important tame or universal settings and under strong large-cardinal assumptions. Exact unresolved remainder: The general ZFC theorem for arbitrary AECs remains open.",
  "relevance": "This is the dated publication status for the canonical target Shelah's eventual categoricity conjecture for AECs.",
  "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 important tame or universal settings and under strong large-cardinal assumptions.\n\nThe exact unresolved remainder is: The general ZFC theorem for arbitrary AECs remains open.\n\nA complete resolution must meet the following acceptance conditions:\n- Prove the stated tail transfer in ZFC with an explicit or definable threshold H(κ).\n- Or construct an AEC with arbitrarily large models categorical once above the proposed threshold but failing categoricity later.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://shelah.logic.at/papers/842/",
      "locator": "S. Shelah and S. Vasey, Categoricity and multidimensional diagrams, Sh:842 (2024). eventual categoricity theorem under strongly compact cardinals"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://shelah.logic.at/papers/842/",
    "locator": "S. Shelah and S. Vasey, Categoricity and multidimensional diagrams, Sh:842 (2024). eventual categoricity theorem under strongly compact cardinals"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R1522",
      "title": "Strongest checked neighboring result",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R1520",
      "title": "Dated source and duplicate audit",
      "object_type": "attempt",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R1521",
      "title": "Work at the unresolved boundary",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "eventual-categoricity-aecs",
      "title": "eventual categoricity aecs",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}