{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R685",
  "content_hash": null,
  "slug": "rsh-artifact-modular-certificates-5000",
  "type": "artifact",
  "title": "Modular nonvanishing certificates through order 5,000",
  "summary": "Five prime fields certify every order through 5,000 except the known zeros \\(2,5,8\\).",
  "relevance": "For Nonvanishing of Rudin-Shapiro Hankel determinants, record rsh-artifact-modular-certificates-5000 (“Modular nonvanishing certificates through order 5,000”) supplies evidence or a replay used to check the packet. The record states: Five prime fields certify every order through 5,000 except the known zeros \\(2,5,8\\).",
  "relevance_source": "recorded",
  "body": "Berlekamp-Massey computations over \\(\\mathbb F_p\\), for \\(p=3,5,7,11,13\\), certify nonzero determinants through order 5,000. The first witnessing-prime counts were 3: 3,290 orders; 5: 1,357; 7: 306; 11: 39; 13: 5. The only orders with no prime witness were 2, 5, and 8, whose integer determinants vanish. This computation was performed during the status audit; a standalone repository script has yet to be retained.",
  "status": "reported",
  "evidence_grade": "self_reported",
  "scope": {
    "kind": "bounded",
    "statement": "every determinant order from 1 through 5000",
    "bounds": {
      "n": {
        "min": 1,
        "max": 5000
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "artifact",
    "citation": {
      "url": "https://doi.org/10.1016/j.aim.2016.08.013",
      "locator": "TheoremDB entry audit on 2026-07-24; Berlekamp-Massey linear-complexity certificates over five primes"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/j.aim.2016.08.013",
    "locator": "TheoremDB entry audit on 2026-07-24; Berlekamp-Massey linear-complexity certificates over five primes"
  },
  "relations": [
    {
      "slug": "R684",
      "title": "Exact signed determinant sweep through order 110",
      "object_type": "artifact",
      "relation": "strengthens",
      "direction": "outgoing"
    },
    {
      "slug": "R686",
      "title": "Settle the remaining orders congruent to 2 modulo 3",
      "object_type": "attempt",
      "relation": "uses",
      "direction": "incoming"
    },
    {
      "slug": "rudin-shapiro-hankel-nonvanishing",
      "title": "rudin shapiro hankel nonvanishing",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}