{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R151",
  "content_hash": null,
  "slug": "cm3ih-claim-rational-lower-bound",
  "type": "claim",
  "title": "An exact three-interval construction gives H_3 at least 61/20",
  "summary": "The packet certifies \\(61/20\\le H_3\\le(11+\\sqrt{61})/4\\); the exact value of \\(H_3\\) remains open.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Put\n\\[\nA=[0,1/5]\\cup[3/4,21/20]\\cup[11/5,27/10].\n\\]\nIts component lengths are \\(1/5,3/10,1/2\\), so \\(|A|=1\\). For a consecutive block of components from \\([a_i,b_i]\\) through \\([a_j,b_j]\\), let \\(m_{ij}\\) be its total occupied length. Every point in\n\\[\nK_{ij}=[b_j-3m_{ij}/2,\\ a_i+3m_{ij}/2]\n\\]\nhas a centered interval containing that block whose average of \\(\\mathbf 1_A\\) is at least \\(1/3\\). The six block intervals are\n\\[\n[-1/10,3/10],\\ [3/10,3/4],\\ [3/5,6/5],\\ [6/5,3/2],\\ [3/2,39/20],\\ [39/20,59/20].\n\\]\nTheir union is \\([-1/10,59/20]\\), of length \\(61/20\\). Hence\n\\[\nH_3\\geq 61/20=3.05.\n\\]\nThis is a certified lower bound. It does not establish that the displayed interval is the entire halo.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the centered Hardy-Littlewood halo at threshold 1/3 for one explicit union of three bounded intervals",
    "bounds": {
      "interval_count": {
        "min": 3,
        "max": 3
      },
      "threshold_numerator": {
        "min": 1,
        "max": 1
      },
      "threshold_denominator": {
        "min": 3,
        "max": 3
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/math/0311452",
      "locator": "Exact rational argument and executable replay in cm3ih-artifact-rational-construction"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/math/0311452",
    "locator": "Exact rational argument and executable replay in cm3ih-artifact-rational-construction"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R148",
      "title": "Exact rational replay of the 61/20 construction",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R149",
      "title": "The exact value of H_3 remains unresolved in this record",
      "object_type": "attempt",
      "relation": "constrains",
      "direction": "outgoing"
    },
    {
      "slug": "centered-maximal-three-interval-halo",
      "title": "centered maximal three interval halo",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}