{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R473",
  "content_hash": null,
  "slug": "lshac-claim-certified-interval",
  "type": "claim",
  "title": "The constant lies between 1/6 and 3/16",
  "summary": "A three-translate argument gives the universal lower bound, while an eight-cell union gives the upper bound.",
  "relevance": "For Sharp long-shift autocorrelation for half-measure circle sets, record lshac-claim-certified-interval (“The constant lies between 1/6 and 3/16”) records a bound, answer, status fact, or structural consequence. The record states: A three-translate argument gives the universal lower bound, while an eight-cell union gives the upper bound.",
  "relevance_source": "recorded",
  "body": "For\n\\[\nR_A(t)=\\mu(A\\cap(A+t)),\n\\qquad\nM(A)=\\sup_{1/4\\leq t\\leq1/2}R_A(t),\n\\]\nthe two certified bounds are\n\\[\n\\frac16\\leq C=\\inf_{\\mu(A)=1/2}M(A)\\leq\\frac3{16}.\n\\]\nThe lower bound follows already at \\(t=1/3\\). The upper bound is attained by the displayed 16-cell set in `lshac-claim-sixteen-cell-upper-bound`. Their gap is\n\\[\n\\frac3{16}-\\frac16=\\frac1{48}.\n\\]\nThis falls short of the candidate's requested \\(10^{-8}\\) gap. The exact value remains unresolved in this entry.",
  "status": "supported",
  "evidence_grade": "mathematical_identity",
  "scope": {
    "kind": "universal",
    "statement": "all measurable half-measure subsets of the circle and the explicit 16-cell construction"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/0711.0572",
      "locator": "Independent proof and exact construction recorded in the two supporting claims, 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/0711.0572",
    "locator": "Independent proof and exact construction recorded in the two supporting claims, 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R474",
      "title": "The shift 1/3 always has overlap at least 1/6",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R475",
      "title": "An explicit half-set has long-shift supremum 3/16",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R471",
      "title": "Exact cell construction and finite-grid enumeration",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "R472",
      "title": "The located covariogram literature does not give this interval extremum",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "long-shift-autocorrelation-half-set",
      "title": "long shift autocorrelation half set",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}