{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R367",
  "content_hash": null,
  "slug": "hcc4-attempt-literature-audit",
  "type": "attempt",
  "title": "The grid-transfer literature gives the method, while this coefficient question appears unsettled",
  "summary": "The located primary sources support the transfer construction and the claw-free real-rootedness boundary; none reports this cylindrical width-four log-concavity sweep.",
  "relevance": "For Hard-core coefficient log-concavity on the first ten thousand four-cycle strips, record hcc4-attempt-literature-audit (“The grid-transfer literature gives the method, while this coefficient question appears unsettled”) documents a concrete method, search boundary, or failed route. The record states: The located primary sources support the transfer construction and the claw-free real-rootedness boundary; none reports this cylindrical width-four log-concavity sweep.",
  "relevance_source": "recorded",
  "body": "Calkin and Wilf build independent sets in a rectangular grid by adjoining compatible column masks. Their Section 1 defines the mask set and the zero-intersection compatibility matrix. The transfer in this record adds a fugacity \\(z^{|m|}\\) for each mask and uses cyclic adjacency inside the four-vertex column. Their paper studies total counts and the hard-square constant, rather than coefficient log-concavity for the cylindrical width-four strip.\n\nChudnovsky and Seymour prove that every claw-free graph has a real-rooted independence polynomial, which implies log-concavity by Newton's inequalities. For \\(n\\geq2\\), \\(C_4\\mathbin{\\square}P_n\\) contains an induced claw centered at a vertex with two cycle neighbors and one path neighbor. The theorem therefore supplies no certificate here. The obstruction is visible at \\(n=2\\):\n\\[\nI_2(z)=1+8z+16z^2+8z^3+2z^4\n\\]\nhas discriminant \\(-26624\\), hence one conjugate pair of nonreal roots.\n\nFocused searches for the exact Cartesian product, cylindrical grid independence polynomials, and grid-strip coefficient log-concavity found the two methodological sources listed here and no published result covering the stated finite range. Publication novelty remains unverified.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.1137/S089548019528993X",
      "locator": "Neil J. Calkin and Herbert S. Wilf, The Number of Independent Sets in a Grid Graph, SIAM Journal on Discrete Mathematics 11(1) (1998), 54-60, Section 1; Maria Chudnovsky and Paul Seymour, The Roots of the Independence Polynomial of a Clawfree Graph, Journal of Combinatorial Theory, Series B 97(3) (2007), 350-357, DOI 10.1016/j.jctb.2006.06.001"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1137/S089548019528993X",
    "locator": "Neil J. Calkin and Herbert S. Wilf, The Number of Independent Sets in a Grid Graph, SIAM Journal on Discrete Mathematics 11(1) (1998), 54-60, Section 1; Maria Chudnovsky and Paul Seymour, The Roots of the Independence Polynomial of a Clawfree Graph, Journal of Combinatorial Theory, Series B 97(3) (2007), 350-357, DOI 10.1016/j.jctb.2006.06.001"
  },
  "relations": [
    {
      "slug": "R368",
      "title": "Every four-cycle strip through length 3000 has a log-concave independence sequence",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "hard-core-c4-strip-log-concavity",
      "title": "hard core c4 strip log concavity",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}