{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R247",
  "content_hash": null,
  "slug": "fcptp-claim-three-cube-exact-probability",
  "type": "claim",
  "title": "The exact three-cube tunnel probability is 4,355 over 16,384",
  "summary": "Exactly 35,676,160 of the 2^27 voxel subsets have nonzero first homology over the two-element field.",
  "relevance": "For Exact tunnel probability for site percolation on a four by four by four cubical box, record fcptp-claim-three-cube-exact-probability (“The exact three-cube tunnel probability is 4,355 over 16,384”) records a bound, answer, status fact, or structural consequence. The record states: Exactly 35,676,160 of the 2^27 voxel subsets have nonzero first homology over the two-element field.",
  "relevance_source": "recorded",
  "body": "For the same closed-union convention on a \\(3\\times3\\times3\\) box, the exhaustive transfer gives\n\\[\n\\#\\{X:\\beta _1(X)>0\\}=35{,}676{,}160.\n\\]\nThe exact probability is therefore\n\\[\n\\frac{35{,}676{,}160}{134{,}217{,}728}\n=\\frac{4{,}355}{16{,}384}\n=0.26580810546875.\n\\]\nThe complete distribution is\n\\[\n\\begin{array}{c|rrrrrr}\n\\beta _1&0&1&2&3&4&5\\\\ \\hline\n\\#X&98{,}541{,}568&27{,}030{,}016&7{,}812{,}864&805{,}376&27{,}648&256.\n\\end{array}\n\\]\nThese six integers sum to \\(2^{27}\\).",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "all subsets of the 27 unit cubes in a fixed 3 by 3 by 3 cubical box, with every subset equally likely",
    "bounds": {
      "side_length": {
        "min": 3,
        "max": 3
      },
      "unit_cubes": {
        "min": 27,
        "max": 27
      },
      "configurations": {
        "min": 134217728,
        "max": 134217728
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1910.12803",
      "locator": "Exact symbolic enumeration in fcptp-artifact-three-cube-frontier-transfer"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1910.12803",
    "locator": "Exact symbolic enumeration in fcptp-artifact-three-cube-frontier-transfer"
  },
  "models": [],
  "relations": [
    {
      "slug": "R245",
      "title": "Exact two-connectivity frontier transfer for the three-cube box",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "four-cube-site-percolation-tunnel-probability",
      "title": "four cube site percolation tunnel probability",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}