{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R149",
  "content_hash": null,
  "slug": "cm3ih-attempt-close-exact-gap",
  "type": "attempt",
  "title": "The exact value of H_3 remains unresolved in this record",
  "summary": "Current certified bounds are 61/20 <= H_3 <= (11+sqrt(61))/4; proving an exact optimum requires a finite global analysis of the possible halo configurations.",
  "relevance": "This records a line of attack on the target problem, including what it establishes and where it stops.",
  "relevance_source": "derived",
  "body": "The search reduces each rational test set to halo intervals associated with consecutive blocks of the three components. It found the exact construction recorded above. A proof of optimality would have to control every overlap order among the six candidate block intervals, including cases where the full halo has several components. That case analysis has not been completed. The present entry therefore records\n\\[\n\\frac{61}{20}\\leq H_3\\leq\\frac{11+\\sqrt{61}}4\n\\]\nas rigorous bounds and leaves the exact constant open.",
  "status": "open_strategy",
  "evidence_grade": "proposed",
  "scope": {
    "kind": "universal",
    "statement": "all unions of at most three bounded intervals at threshold 1/3"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/math/0311452",
      "locator": "The upper endpoint is Melas's theorem; the lower endpoint is cm3ih-claim-rational-lower-bound"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/math/0311452",
    "locator": "The upper endpoint is Melas's theorem; the lower endpoint is cm3ih-claim-rational-lower-bound"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R151",
      "title": "An exact three-interval construction gives H_3 at least 61/20",
      "object_type": "claim",
      "relation": "constrains",
      "direction": "incoming"
    },
    {
      "slug": "R150",
      "title": "Melas's sharp weak-type theorem gives H_3 at most (11+sqrt(61))/4",
      "object_type": "claim",
      "relation": "constrains",
      "direction": "incoming"
    },
    {
      "slug": "centered-maximal-three-interval-halo",
      "title": "centered maximal three interval halo",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}