{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R72",
  "content_hash": null,
  "slug": "bpe8c-claim-sufficient-and-obstructing-families",
  "type": "claim",
  "title": "Two diagonals and 64 crosses span, while each vacant outer line is stable",
  "summary": "These finite families turn monotonicity into certified lower and upper counts.",
  "relevance": "For Exact spanning-set count for two-neighbor bootstrap percolation on the eight grid, record bpe8c-claim-sufficient-and-obstructing-families (“Two diagonals and 64 crosses span, while each vacant outer line is stable”) records a bound, answer, status fact, or structural consequence. The record states: These finite families turn monotonicity into certified lower and upper counts.",
  "relevance_source": "recorded",
  "body": "For the main diagonal, every cell at diagonal distance \\(d>0\\) has two neighbors at distance \\(d-1\\). Induction fills the board. Reflection gives the other long diagonal.\n\nIf row \\(r\\) and column \\(c\\) are initially occupied, a cell away from their union has two neighbors whose coordinate distances to that union are each one step smaller. Induction fills all four rectangular regions. Every superset of any of these 66 patterns therefore spans.\n\nIf an outer line is vacant and all other cells are occupied, each vertex of that line has at most one occupied neighbor. The line stays vacant. Removing more initially occupied vertices cannot create a second occupied neighbor, so every initial set disjoint from that line fails to span.",
  "status": "established",
  "evidence_grade": "mathematical_argument",
  "scope": {
    "kind": "bounded",
    "statement": "the two long diagonals, the 64 row-column crosses, and the four outer lines of the open-boundary 8 by 8 grid",
    "bounds": {
      "spanning_base_patterns": {
        "min": 66,
        "max": 66
      },
      "stable_vacant_lines": {
        "min": 4,
        "max": 4
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1088/0305-4470/21/19/017",
      "locator": "Elementary induction and stability argument, replayed on every base pattern by 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": "Elementary induction and stability argument, replayed on every base pattern by bpe8c-artifact-closure-and-bound-verifier"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R71",
      "title": "The spanning-set count lies between 177,024,301,925,259,284 and 18,161,310,923,858,378,752",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R69",
      "title": "Deterministic closure verifier and inclusion-exclusion certificate",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "bootstrap-percolation-eight-count",
      "title": "bootstrap percolation eight count",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}