{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R667",
  "content_hash": null,
  "slug": "rnfc9-attempt-literature-audit",
  "type": "attempt",
  "title": "The primary literature gives asymptotics rather than this finite table",
  "summary": "The checked random flag-complex papers define the model and prove threshold or limit theorems; no nine-vertex exact distribution was located.",
  "relevance": "This records a line of attack on the target problem, including what it establishes and where it stops.",
  "relevance_source": "derived",
  "body": "Kahle defines \\(X(n,p)\\) as the clique complex of \\(G(n,p)\\), proves vanishing and nonvanishing regimes for higher homology, and gives estimates for expected Betti numbers. Kahle and Meckes prove Poisson and normal approximation theorems for random clique-complex Betti numbers. Kahle's later paper sharpens cohomology-vanishing thresholds. These results concern asymptotic regimes as \\(n\\) grows.\n\nThe audit checked the exact title phrase, `nine vertex random flag complex`, `exact Betti distribution random clique complex`, and combinations of \\(n=9\\), \\(p=1/2\\), \\(\\beta_1\\), graph enumeration, and clique complexes. It also checked the three primary papers cited here. No exact finite numerator or beta histogram for these parameters was found. This leaves novelty of the table unverified, while the value itself is settled by the included exhaustive computation.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "focused primary-source search for random clique-complex homology and finite beta-one distributions relevant to n=9 and p=1/2",
    "bounds": {
      "target_vertices": {
        "min": 9,
        "max": 9
      },
      "target_edge_probability_numerator": {
        "min": 1,
        "max": 1
      },
      "target_edge_probability_denominator": {
        "min": 2,
        "max": 2
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://arxiv.org/abs/math/0605536",
      "locator": "Matthew Kahle, Topology of random clique complexes, Discrete Mathematics 309 (2009), 1658-1671; Matthew Kahle and Elizabeth Meckes, Limit theorems for Betti numbers of random simplicial complexes, Homology, Homotopy and Applications 15 (2013), 343-374, arXiv:1009.4130, with erratum arXiv:1501.03759; Matthew Kahle, Sharp vanishing thresholds for cohomology of random flag complexes, Annals of Mathematics 179 (2014), 1085-1107, arXiv:1207.0149; Brendan McKay and Adolfo Piperno, Practical Graph Isomorphism, II, Journal of Symbolic Computation 60 (2014), 94-112, DOI 10.1016/j.jsc.2013.09.003"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/math/0605536",
    "locator": "Matthew Kahle, Topology of random clique complexes, Discrete Mathematics 309 (2009), 1658-1671; Matthew Kahle and Elizabeth Meckes, Limit theorems for Betti numbers of random simplicial complexes, Homology, Homotopy and Applications 15 (2013), 343-374, arXiv:1009.4130, with erratum arXiv:1501.03759; Matthew Kahle, Sharp vanishing thresholds for cohomology of random flag complexes, Annals of Mathematics 179 (2014), 1085-1107, arXiv:1207.0149; Brendan McKay and Adolfo Piperno, Practical Graph Isomorphism, II, Journal of Symbolic Computation 60 (2014), 94-112, DOI 10.1016/j.jsc.2013.09.003"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R669",
      "title": "Exactly 50,766,340,113 graphs have nonzero first homology",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "random-nine-flag-complex-beta-one",
      "title": "random nine flag complex beta one",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}