{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1617",
  "content_hash": null,
  "slug": "kaplansky-sixth-semisimple-hopf-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: The MathOverflow answers prove special cases and cite the conjecture as open. The checked survey and geometric-invariant-theory work do not supply a proof for all finite-dimensional semisimple Hopf algebras or a counterexample. Exact unresolved remainder: Prove that \\(\\dim_k V\\mid\\dim_k H\\) for every triple \\((k,H,V)\\) satisfying the displayed hypotheses. Alternatively, give explicit Hopf structure maps for a finite-dimensional semisimple counterexample, identify a simple module \\(V\\), prove simplicity, and verify the failed divisibility.",
  "relevance": "For Kaplansky's sixth conjecture for semisimple Hopf algebras, 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: The MathOverflow answers prove special cases and cite the conjecture as open. The checked survey and geometric-invariant-theory work do not supply a proof for all finite-dimensional semisimple Hopf algebras or a counterexample.\n\nExact unresolved remainder: Prove that \\(\\dim_k V\\mid\\dim_k H\\) for every triple \\((k,H,V)\\) satisfying the displayed hypotheses. Alternatively, give explicit Hopf structure maps for a finite-dimensional semisimple counterexample, identify a simple module \\(V\\), prove simplicity, and verify the failed divisibility.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1506.00314",
      "locator": "abstract and theorem giving algebraic integrality of the invariant family as a sufficient condition for Kaplansky's sixth conjecture"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1506.00314",
    "locator": "abstract and theorem giving algebraic integrality of the invariant family as a sufficient condition for Kaplansky's sixth conjecture"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R1329",
      "title": "Current checked status and unresolved remainder",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "outgoing",
      "metadata": {
        "reason": "Replaces unreadable status prose with the dated review from 2026-08-01."
      }
    },
    {
      "slug": "kaplansky-sixth-semisimple-hopf",
      "title": "kaplansky sixth semisimple hopf",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}