{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R618",
  "content_hash": null,
  "slug": "q12ns794-claim-certified-interval",
  "type": "claim",
  "title": "The maximum lies between 6,456,734,424 and 7,623,232,012",
  "summary": "The binary initial segment supplies the lower endpoint. Walsh Parseval, Harper's edge bound, and an exact secant inequality supply the upper endpoint.",
  "relevance": "For Maximum half-noise stability of a 794-set in the twelve cube, record q12ns794-claim-certified-interval (“The maximum lies between 6,456,734,424 and 7,623,232,012”) records a bound, answer, status fact, or structural consequence. The record states: The binary initial segment supplies the lower endpoint.",
  "relevance_source": "recorded",
  "body": "Let \\(M_{12,794}\\) denote the requested maximum. The certified result is\n\\[\n6{,}456{,}734{,}424\\leq M_{12,794}\\leq7{,}623{,}232{,}012.\n\\]\nThe lower endpoint is attained by the explicit initial segment in q12ns794-claim-lex-witness.\n\nFor the upper bound, put \\(f=1_A\\), \\(\\mu=794/4096\\), and use the normalized Walsh coefficients\n\\[\n\\widehat f(S)=2^{-12}\\sum_x f(x)(-1)^{\\sum_{i\\in S}x_i}.\n\\]\nThe product kernel has Walsh eigenvalue \\(4^{12-|S|}2^{|S|}\\), hence\n\\[\nE(A)=8^{12}\\sum_{S\\subseteq[12]}2^{-|S|}\\widehat f(S)^2.\n\\]\nParseval gives \\(R:=\\sum_{S\\ne\\varnothing}\\widehat f(S)^2=\\mu-\\mu^2=655447/4194304\\). If \\(b(A)\\) is the undirected edge boundary, then\n\\[\nD:=\\sum_{S\\ne\\varnothing}|S|\\widehat f(S)^2=\\frac{b(A)}{2\\cdot4096}.\n\\]\nHarper's edge-isoperimetric theorem says that an initial binary segment maximizes the internal edges. At size 794 it has 3,693 internal edges, so every such \\(A\\) has \\(b(A)\\geq12\\cdot794-2\\cdot3693=2142\\) and \\(D\\geq1071/4096\\).\n\nConvexity gives, for each integer \\(1\\leq k\\leq12\\),\n\\[\n2^{-k}\\leq\\frac{12-k}{11}\\,2^{-1}+\\frac{k-1}{11}\\,2^{-12}.\n\\]\nSumming this inequality against the nonnegative Fourier weights and inserting the lower bound on \\(D\\) yields\n\\[\nE(A)\\leq\\frac{83{,}855{,}552{,}164}{11}=7{,}623{,}232{,}014+\\frac{10}{11}.\n\\]\nFor even \\(|A|\\), the objective is divisible by four: modulo four, the diagonal contributes \\(|A|\\) and the paired off-diagonal terms contribute \\(2\\binom{|A|}{2}\\), whose sum is \\(|A|^2\\). The largest multiple of four below the rational bound is 7,623,232,012.\n\nThe endpoints do not match. The exact maximum and the cube-automorphism orbits of its maximizers remain open in this certificate.",
  "status": "open",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all subsets A of the labeled twelve-dimensional binary cube having exactly 794 vertices, for the ordered-pair objective sum 3^(12-d_H(x,y))",
    "bounds": {
      "dimension": {
        "min": 12,
        "max": 12
      },
      "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": "Harper's edge-isoperimetric theorem combined with the exact verifier 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": "Harper's edge-isoperimetric theorem combined with the exact verifier q12ns794-artifact-fourier-edge-verifier"
  },
  "models": [],
  "relations": [
    {
      "slug": "R620",
      "title": "The binary initial segment has objective 6,456,734,424",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R616",
      "title": "Exact pair-sum, Walsh-transform, and edge-bound verifier",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R619",
      "title": "The radius-four Hamming ball has objective 5,884,957,476",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R617",
      "title": "Classical inequalities give a certified gap; the exact weighted optimum was not located",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "q12-noise-stability-794",
      "title": "q12 noise stability 794",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}