{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R475",
  "content_hash": null,
  "slug": "lshac-claim-sixteen-cell-upper-bound",
  "type": "claim",
  "title": "An explicit half-set has long-shift supremum 3/16",
  "summary": "The union of cells 0,1,2,3,4,5,7,10 has boundary-shift intersection counts 3,3,3,3,2.",
  "relevance": "For Sharp long-shift autocorrelation for half-measure circle sets, record lshac-claim-sixteen-cell-upper-bound (“An explicit half-set has long-shift supremum 3/16”) records a bound, answer, status fact, or structural consequence. The record states: The union of cells 0,1,2,3,4,5,7,10 has boundary-shift intersection counts 3,3,3,3,2.",
  "relevance_source": "recorded",
  "body": "Partition \\(\\mathbb T\\) into the half-open cells\n\\[\nI_i=[i/16,(i+1)/16),\\qquad i\\in\\mathbb Z/16\\mathbb Z,\n\\]\nand take\n\\[\nA=I_0\\cup I_1\\cup I_2\\cup I_3\\cup I_4\\cup I_5\\cup I_7\\cup I_{10}.\n\\]\nThen \\(\\mu(A)=8/16=1/2\\). At the grid shifts \\(j/16\\), for \\(j=4,5,6,7,8\\), exact cyclic intersection counts are\n\\[\n3,3,3,3,2.\n\\]\nThus the corresponding overlaps are those counts divided by 16.\n\nFor a general shift \\(t=(j+u)/16\\), where \\(j\\) is an integer and \\(0\\leq u\\leq1\\), each shifted cell overlaps cells at the two neighboring grid positions. Consequently\n\\[\nR_A(t)=\\frac{(1-u)c_j+u c_{j+1}}{16},\n\\]\nwhere \\(c_j\\) is the cyclic intersection count at shift \\(j\\). The autocorrelation is linear between consecutive grid shifts. Its maximum on \\([1/4,1/2]\\) occurs at a grid endpoint and equals \\(3/16\\). Hence \\(C\\leq3/16\\).",
  "status": "established",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the union of the 16 equal half-open cells indexed by 0,1,2,3,4,5,7,10",
    "bounds": {
      "cell_count": {
        "min": 16,
        "max": 16
      },
      "selected_cells": {
        "min": 8,
        "max": 8
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/0711.0572",
      "locator": "Explicit construction and exact replay in lshac-artifact-cell-and-grid-enumeration"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/0711.0572",
    "locator": "Explicit construction and exact replay in lshac-artifact-cell-and-grid-enumeration"
  },
  "models": [],
  "relations": [
    {
      "slug": "R473",
      "title": "The constant lies between 1/6 and 3/16",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R471",
      "title": "Exact cell construction and finite-grid enumeration",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "long-shift-autocorrelation-half-set",
      "title": "long shift autocorrelation half set",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}