{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R401",
  "content_hash": null,
  "slug": "hs16-claim-certified-radius-bracket",
  "type": "claim",
  "title": "A certified radius bracket and an isolated real zero for the sixteen grid",
  "summary": "The zero radius lies between 27/256 and 0.122005371541038. A simple real zero is isolated near -0.122005371541037445, while its global least-modulus status remains open.",
  "relevance": "For Nearest hard-square partition-function zero for the sixteen grid, record hs16-claim-certified-radius-bracket (“A certified radius bracket and an isolated real zero for the sixteen grid”) records a bound, answer, status fact, or structural consequence. The record states: The zero radius lies between 27/256 and 0.122005371541038.",
  "relevance_source": "recorded",
  "body": "Let \\(\\rho_{16}=\\min\\{|z|:Z_{16}(z)=0\\}\\). Exact row-mask transfer gives all 129 integer coefficients of \\(Z_{16}\\). An exact rational Rouché calculation isolates a simple real zero in\n\\[\n[-0.122005371541038,-0.122005371541037]+i[-10^{-15},10^{-15}].\n\\]\nThe rectangle has diameter below \\(3\\mathbin{\\cdot}10^{-15}\\).\n\nThe grid has maximum degree four. The Shearer disk bound, in the independent-polynomial form developed by Scott and Sokal, gives\n\\[\nZ_{16}(z)\\ne0\\qquad\\text{for }|z|<\\frac{(4-1)^{4-1}}{4^4}=\\frac{27}{256}.\n\\]\nConsequently,\n\\[\n\\frac{27}{256}\\leq\\rho_{16}\\leq0.122005371541038.\n\\]\n\nThe local Rouché disk proves that the displayed zero is simple. It does not count the other 127 zeros or exclude a nonreal zero in the remaining annulus. The requested least-modulus identification remains open in this fixture.",
  "status": "partial",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the independence polynomial of the single graph P16 Cartesian-product P16, with a local certificate for one zero and a universal maximum-degree zero-free disk",
    "bounds": {
      "grid_side": {
        "min": 16,
        "max": 16
      },
      "vertices": {
        "min": 256,
        "max": 256
      },
      "polynomial_degree": {
        "min": 128,
        "max": 128
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1007/s10955-004-2055-4",
      "locator": "Exact transfer in hs16-artifact-exact-polynomial and rational Rouché certificate in hs16-artifact-root-enclosure"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1007/s10955-004-2055-4",
    "locator": "Exact transfer in hs16-artifact-exact-polynomial and rational Rouché certificate in hs16-artifact-root-enclosure"
  },
  "relations": [
    {
      "slug": "R398",
      "title": "Exact 2,584-mask transfer polynomial",
      "object_type": "artifact",
      "relation": "reproduces",
      "direction": "incoming"
    },
    {
      "slug": "R399",
      "title": "Exact rational Rouché enclosure for one simple zero",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R400",
      "title": "Primary literature supports the model, transfer, and zero-free disk",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "hard-square-sixteen-zero-radius",
      "title": "hard square sixteen zero radius",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}