{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R76",
  "content_hash": null,
  "slug": "bsh-claim-classical-cubic",
  "type": "claim",
  "title": "The candidate uses the classical Baum-Sweet convention",
  "summary": "Its generating series satisfies \\(B(z)=B(z^4)+zB(z^2)\\), and over \\(\\mathbb F_2\\) it is the Baum-Sweet cubic.",
  "relevance": "For Nonvanishing of Baum-Sweet Hankel determinants, record bsh-claim-classical-cubic (“The candidate uses the classical Baum-Sweet convention”) records a bound, answer, status fact, or structural consequence. The record states: Its generating series satisfies \\(B(z)=B(z^4)+zB(z^2)\\), and over \\(\\mathbb F_2\\) it is the Baum-Sweet cubic.",
  "relevance_source": "recorded",
  "body": "The recurrence is\n\\[\nb_0=1,\\qquad b_{2m+1}=b_m,\\qquad b_{4m}=b_m,\\qquad b_{4m+2}=0.\n\\]\nIt produces \\(1,1,0,1,1,0,0,1,0,1,0,0,1,0,0,1,\\ldots\\) and is equivalent to the candidate's binary-block definition. For \\(B(z)=\\sum b_nz^n\\), splitting indices gives \\(B(z)=B(z^4)+zB(z^2)\\). In characteristic two, this reduces to \\(B^3+zB+1=0\\). Baum and Sweet's original work and the later continued-fraction papers concern this algebraic series.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "the full Baum-Sweet coefficient sequence"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://annals.math.princeton.edu/1976/103-3/p12",
      "locator": "Leonard E. Baum and Melvin M. Sweet, Continued Fractions of Algebraic Power Series in Characteristic 2, Annals of Mathematics 103 (1976), pages 593-610"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://annals.math.princeton.edu/1976/103-3/p12",
    "locator": "Leonard E. Baum and Melvin M. Sweet, Continued Fractions of Algebraic Power Series in Characteristic 2, Annals of Mathematics 103 (1976), pages 593-610"
  },
  "relations": [
    {
      "slug": "R77",
      "title": "Non-apwenian does not mean that a determinant vanishes",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "baum-sweet-hankel-nonvanishing",
      "title": "baum sweet hankel nonvanishing",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}