{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R789",
  "content_hash": null,
  "slug": "tlef15-claim-certified-bounds-33-56",
  "type": "claim",
  "title": "The current certified interval is 33 through 56",
  "summary": "An explicit 33-point set proves the lower bound, while a parity decomposition and an exhaustive T_7 calculation prove the upper bound 56.",
  "relevance": "For Largest equilateral-triangle-free subset of the fifteen-row triangular lattice, record tlef15-claim-certified-bounds-33-56 (“The current certified interval is 33 through 56”) records a bound, answer, status fact, or structural consequence. The record states: An explicit 33-point set proves the lower bound, while a parity decomposition and an exhaustive T_7 calculation prove the upper bound 56.",
  "relevance_source": "recorded",
  "body": "Write \\(M_{15}\\) for the requested maximum. The displayed construction gives \\(M_{15}\\geq33\\).\n\nFor the upper bound, partition \\(T_{15}\\) by the two coordinate parities. After division by two, the even-even class is a copy of \\(T_8\\). Each other class is a copy of \\(T_7\\). Exhaustive search gives \\(\\alpha(T_7)=12\\). The inner seven rows of \\(T_8\\) form \\(T_7\\), and its remaining diagonal has eight vertices, so \\(\\alpha(T_8)\\leq12+8=20\\). Every triangle-free subset therefore has size at most\n\\[\n20+3\\cdot12=56.\n\\]\nThus \\(33\\leq M_{15}\\leq56\\). The exact value remains open in this entry.",
  "status": "partial",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "subsets of T_15 with no three vertices of an equilateral triangle in either orientation and at any lattice scale",
    "bounds": {
      "rows": {
        "min": 15,
        "max": 15
      },
      "vertices": {
        "min": 120,
        "max": 120
      },
      "equilateral_triangles": {
        "min": 2380,
        "max": 2380
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/2405.12321",
      "locator": "Construction and parity certificate reproduced by tlef15-artifact-construction-and-upper-bound"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/2405.12321",
    "locator": "Construction and parity certificate reproduced by tlef15-artifact-construction-and-upper-bound"
  },
  "models": [],
  "relations": [
    {
      "slug": "R790",
      "title": "A 33-point equilateral-triangle-free set",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R787",
      "title": "Executable construction and parity upper-bound certificate",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R788",
      "title": "The nearby literature counts and colors the triangles",
      "object_type": "attempt",
      "relation": "contextualizes",
      "direction": "incoming"
    },
    {
      "slug": "triangular-lattice-15-equilateral-free",
      "title": "triangular lattice 15 equilateral free",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}