{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R478",
  "content_hash": null,
  "slug": "lwf101-claim-exact-maximum-12",
  "type": "claim",
  "title": "The exact maximum number of roots is 12",
  "summary": "A complete enumeration of all 2^30 sign vectors finds two polynomials with 12 roots and none with more.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "The lexicographically first maximizing sign vector under the artifact's mask order is\n\\[\n(-1,1,1,-1,-1,-1,1,-1,-1,-1,-1,1,-1,-1,-1,-1,-1,1,-1,-1,-1,-1,1,-1,-1,-1,1,1,-1,1).\n\\]\nIts distinct roots in \\(\\mathbb F_{101}\\) are\n\\[\n\\{20,30,43,47,63,64,66,72,75,93,94,96\\}.\n\\]\nThe exhaustive calculation splits the 30 signs into two blocks of 15. It evaluates all 32,768 assignments for each block at all 101 field elements, then checks every one of the \\(32{,}768^2=2^{30}\\) pairs. The complete root-count histogram has two vectors with 12 roots, 50 with 10 roots, and no vector with 11 roots or more than 12. The exhaustive artifact supplies the full histogram and source.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "degree-30 Littlewood polynomials over F_101 with constant coefficient 1",
    "bounds": {
      "field_order": {
        "min": 101,
        "max": 101
      },
      "degree": {
        "min": 30,
        "max": 30
      },
      "sign_vectors": {
        "min": 1073741824,
        "max": 1073741824
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1112/S1461157016000334",
      "locator": "Complete executable enumeration in lwf101-artifact-exhaustive-mitm"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1112/S1461157016000334",
    "locator": "Complete executable enumeration in lwf101-artifact-exhaustive-mitm"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R476",
      "title": "Complete 15-plus-15 meet-in-the-middle root sweep",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R477",
      "title": "Finite-field root bounds give context rather than this fixed value",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "littlewood-f101-max-roots-30",
      "title": "littlewood f101 max roots 30",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}