{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R368",
  "content_hash": null,
  "slug": "hcc4-claim-log-concavity-through-3000",
  "type": "claim",
  "title": "Every four-cycle strip through length 3000 has a log-concave independence sequence",
  "summary": "Exact transfer verifies log-concavity for every \\(C_4\\mathbin{\\square}P_n\\) with \\(1\\le n\\le3000\\); the cases \\(3001\\le n\\le10000\\) remain unchecked.",
  "relevance": "For Hard-core coefficient log-concavity on the first ten thousand four-cycle strips, record hcc4-claim-log-concavity-through-3000 (“Every four-cycle strip through length 3000 has a log-concave independence sequence”) records a bound, answer, status fact, or structural consequence. The record states: Exact transfer verifies log-concavity for every \\(C_4\\mathbin{\\square}P_n\\) with \\(1\\le n\\le3000\\); the cases \\(3001\\le n\\le10000\\) remain unchecked.",
  "relevance_source": "recorded",
  "body": "Let\n\\[\nI_n(z)=\\sum_{k=0}^{2n}i_{n,k}z^k\n\\]\nbe the independence polynomial of \\(C_4\\mathbin{\\square}P_n\\). Exact integer computation gives\n\\[\ni_{n,k}^2\\geq i_{n,k-1}i_{n,k+1}\n\\]\nfor every \\(1\\leq n\\leq3000\\) and \\(1\\leq k\\leq2n-1\\). The sweep checks exactly\n\\[\n\\sum_{n=1}^{3000}(2n-1)=9{,}000{,}000\n\\]\ninequalities. All are strict.\n\nThe independent masks in one four-cycle column are\n\\[\n0,1,2,4,5,8,10.\n\\]\nCycle symmetry reduces their terminal polynomials to \\(a_n\\) for the empty mask, \\(b_n\\) for each singleton, and \\(c_n\\) for each opposite pair. Starting with\n\\[\n(a_1,b_1,c_1)=(1,z,z^2),\n\\]\nthe exact transfer is\n\\[\n\\begin{aligned}\na_{n+1}&=a_n+4b_n+2c_n,\\\\\nb_{n+1}&=z(a_n+3b_n+c_n),\\\\\nc_{n+1}&=z^2(a_n+2b_n+c_n),\n\\end{aligned}\n\\qquad I_n=a_n+4b_n+2c_n.\n\\]\nThe program checks this reduction against the full seven-state transfer through \\(n=20\\).\n\nAt \\(n=1\\), the least margin is 14 at \\(k=1\\). For every \\(2\\leq n\\leq3000\\), the least margin occurs at \\(k=2n-1\\), where the adjacent triple is\n\\[\n\\bigl(4(n^2+n-2),4n,2\\bigr)\n\\]\nand the margin is \\(8n^2-8n+16\\). The requested lengths \\(3001\\) through \\(10000\\) remain unchecked.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "every integer strip length 1 <= n <= 3000 and every internal coefficient index 1 <= k <= 2n-1 for the independence polynomial of C4 Cartesian-product Pn",
    "bounds": {
      "strip_length": {
        "min": 1,
        "max": 3000
      },
      "inequalities_checked": {
        "min": 9000000,
        "max": 9000000
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1137/S089548019528993X",
      "locator": "Exact transfer and replay certificate in hcc4-artifact-exact-transfer-sweep"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1137/S089548019528993X",
    "locator": "Exact transfer and replay certificate in hcc4-artifact-exact-transfer-sweep"
  },
  "relations": [
    {
      "slug": "R366",
      "title": "Replayable exact transfer sweep through strip length 3000",
      "object_type": "artifact",
      "relation": "reproduces",
      "direction": "incoming"
    },
    {
      "slug": "R367",
      "title": "The grid-transfer literature gives the method, while this coefficient question appears unsettled",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "hard-core-c4-strip-log-concavity",
      "title": "hard core c4 strip log concavity",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}