{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R77",
  "content_hash": null,
  "slug": "bsh-claim-non-apwenian-distinction",
  "type": "claim",
  "title": "Non-apwenian does not mean that a determinant vanishes",
  "summary": "Exact integer elimination certifies \\(H_n\\ne0\\) for \\(1\\le n\\le110\\), and a self-reported audit modulo \\(100000007\\) finds nonzero residues through \\(n=4999\\); integer nonvanishing for every \\(n\\ge5000\\) remains open.",
  "relevance": "For Nonvanishing of Baum-Sweet Hankel determinants, record bsh-claim-non-apwenian-distinction (“Non-apwenian does not mean that a determinant vanishes”) records a bound, answer, status fact, or structural consequence. The record states: Exact integer elimination certifies \\(H_n\\ne0\\) for \\(1\\le n\\le110\\), and a self-reported audit modulo \\(100000007\\) finds nonzero residues through \\(n=4999\\); integer nonvanishing for every \\(n\\ge5000\\) remains open.",
  "relevance_source": "recorded",
  "body": "An apwenian binary sequence has every normalized Hankel determinant odd. Guo and Han's Example 11 classifies Baum-Sweet as non-apwenian. This agrees with \\(H_3=-2\\), but it leaves integer nonvanishing open. The classical function-field continued fraction also concerns a different object. Neither result resolves the candidate's claim.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "the parity pattern of all Baum-Sweet Hankel determinants"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://irma.math.unistra.fr/~guoniu/papers/p120autoapw.pdf",
      "locator": "Yining Guo and Guo-Niu Han, On a family of automatic apwenian sequences, Discrete Mathematics 348 (2025), Example 11"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://irma.math.unistra.fr/~guoniu/papers/p120autoapw.pdf",
    "locator": "Yining Guo and Guo-Niu Han, On a family of automatic apwenian sequences, Discrete Mathematics 348 (2025), Example 11"
  },
  "relations": [
    {
      "slug": "R76",
      "title": "The candidate uses the classical Baum-Sweet convention",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R73",
      "title": "Exact integer determinant sweep through order 110",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "baum-sweet-hankel-nonvanishing",
      "title": "baum sweet hankel nonvanishing",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}