{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R619",
  "content_hash": null,
  "slug": "q12ns794-claim-hamming-ball-comparison",
  "type": "claim",
  "title": "The radius-four Hamming ball has objective 5,884,957,476",
  "summary": "The symmetric ball has the required size and trails the initial segment by 571,776,948.",
  "relevance": "For Maximum half-noise stability of a 794-set in the twelve cube, record q12ns794-claim-hamming-ball-comparison (“The radius-four Hamming ball has objective 5,884,957,476”) records a bound, answer, status fact, or structural consequence. The record states: The symmetric ball has the required size and trails the initial segment by 571,776,948.",
  "relevance_source": "recorded",
  "body": "The family\n\\[\nH_4=\\{x\\in\\{0,1\\}^{12}:|x|\\leq4\\}\n\\]\nhas size \\(\\sum_{j=0}^4\\binom{12}{j}=794\\). Direct summation and the Walsh calculation give\n\\[\nE(H_4)=5{,}884{,}957{,}476.\n\\]\nThus \\(E(A_L)-E(H_4)=571{,}776{,}948\\). A vertex-isoperimetric or small-diameter preference for Hamming balls cannot settle this weighted objective. The ball has 2,784 internal cube edges and boundary 3,960, while the initial segment has 3,693 internal edges.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the radius-four Hamming ball centered at zero in the labeled twelve-dimensional binary cube",
    "bounds": {
      "dimension": {
        "min": 12,
        "max": 12
      },
      "radius": {
        "min": 4,
        "max": 4
      },
      "cardinality": {
        "min": 794,
        "max": 794
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/S0021-9800(66)80059-5",
      "locator": "Exact evaluations in q12ns794-artifact-fourier-edge-verifier"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/S0021-9800(66)80059-5",
    "locator": "Exact evaluations in q12ns794-artifact-fourier-edge-verifier"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R616",
      "title": "Exact pair-sum, Walsh-transform, and edge-bound verifier",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R618",
      "title": "The maximum lies between 6,456,734,424 and 7,623,232,012",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "q12-noise-stability-794",
      "title": "q12 noise stability 794",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}