{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R71",
  "content_hash": null,
  "slug": "bpe8c-claim-certified-count-interval",
  "type": "claim",
  "title": "The spanning-set count lies between 177,024,301,925,259,284 and 18,161,310,923,858,378,752",
  "summary": "Explicit spanning families give the lower bound, while four stable vacant boundary lines give the upper bound.",
  "relevance": "For Exact spanning-set count for two-neighbor bootstrap percolation on the eight grid, record bpe8c-claim-certified-count-interval (“The spanning-set count lies between 177,024,301,925,259,284 and 18,161,310,923,858,378,752”) records a bound, answer, status fact, or structural consequence. The record states: Explicit spanning families give the lower bound, while four stable vacant boundary lines give the upper bound.",
  "relevance_source": "recorded",
  "body": "Let \\(B_8\\) denote the requested number. The certified result is\n\\[\n177{,}024{,}301{,}925{,}259{,}284\n\\leq B_8\\leq\n18{,}161{,}310{,}923{,}858{,}378{,}752.\n\\]\n\nFor the lower bound, either long diagonal is a spanning set. A full occupied row together with a full occupied column also spans: each of the four resulting rectangles fills by induction on distance from the row-column cross. Inclusion-exclusion over these 66 sufficient patterns gives the displayed lower endpoint.\n\nFor the upper bound, an entirely vacant outer row or outer column remains vacant forever. Inclusion-exclusion over the four boundary lines shows that 285,433,149,851,172,864 initial sets have at least one such obstruction. Subtracting these from \\(2^{64}\\) gives the upper endpoint.\n\nThe exact value remains undetermined by this certificate.",
  "status": "open",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "all 2^64 initial occupied sets on the labeled open-boundary grid P_8 Cartesian P_8 under the two-neighbor bootstrap rule",
    "bounds": {
      "vertices": {
        "min": 64,
        "max": 64
      },
      "initial_sets": {
        "min": 18446744073709552000,
        "max": 18446744073709552000
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1088/0305-4470/21/19/017",
      "locator": "Certified constructions and inclusion-exclusion in bpe8c-artifact-closure-and-bound-verifier"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1088/0305-4470/21/19/017",
    "locator": "Certified constructions and inclusion-exclusion in bpe8c-artifact-closure-and-bound-verifier"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R72",
      "title": "Two diagonals and 64 crosses span, while each vacant outer line is stable",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R69",
      "title": "Deterministic closure verifier and inclusion-exclusion certificate",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R70",
      "title": "The audit found asymptotic theory and no published exact eight-board count",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "bootstrap-percolation-eight-count",
      "title": "bootstrap percolation eight count",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}