{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R620",
  "content_hash": null,
  "slug": "q12ns794-claim-lex-witness",
  "type": "claim",
  "title": "The binary initial segment has objective 6,456,734,424",
  "summary": "With x encoded as the integer sum of 2^i x_i, take every integer from 0 through 793.",
  "relevance": "For Maximum half-noise stability of a 794-set in the twelve cube, record q12ns794-claim-lex-witness (“The binary initial segment has objective 6,456,734,424”) records a bound, answer, status fact, or structural consequence. The record states: With x encoded as the integer sum of 2^i x_i, take every integer from 0 through 793.",
  "relevance_source": "recorded",
  "body": "Index coordinates by \\(i=0,\\ldots,11\\) and encode \\(x\\in\\{0,1\\}^{12}\\) as \\(\\iota(x)=\\sum_i2^i x_i\\). The witness is\n\\[\nA_L=\\{x:0\\leq\\iota(x)<794\\}.\n\\]\nEquivalently, its 4,096-bit characteristic integer is \\(B_L=2^{794}-1\\), where bit \\(j\\) records membership of the vertex with integer label \\(j\\). Its canonical 512-byte little-endian encoding has SHA-256 digest\n\\[\n\\texttt{4f55e719c472514a6f2b2e657127780a0740471a7038205203f403f17988ba8a}.\n\\]\nDirect ordered-pair summation and an independent Walsh transform both give\n\\[\nE(A_L)=6{,}456{,}734{,}424.\n\\]\nThe family is also the disjoint union of binary subcubes corresponding to\n\\(794=512+256+16+8+2\\). Coordinate permutations and coordinate complements preserve the objective, so the full cube-automorphism orbit of this family gives witnesses with the same value.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the 794-vertex binary initial segment A_L in the labeled twelve-cube under the stated integer encoding",
    "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": "Exact construction and two independent 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 construction and two independent evaluations in q12ns794-artifact-fourier-edge-verifier"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R618",
      "title": "The maximum lies between 6,456,734,424 and 7,623,232,012",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R616",
      "title": "Exact pair-sum, Walsh-transform, and edge-bound verifier",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "q12-noise-stability-794",
      "title": "q12 noise stability 794",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}