{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R420",
  "content_hash": null,
  "slug": "ising6-claim-certified-gap-interval",
  "type": "claim",
  "title": "A certified rational interval for the six-torus heat-bath gap",
  "summary": "The gap is at least 1/170005193383307227693056 and at most 48265367094829605687889/36363194510128970638045284, approximately 0.00132731372325897517.",
  "relevance": "For Exact heat-bath spectral gap on the six by six Ising torus, record ising6-claim-certified-gap-interval (“A certified rational interval for the six-torus heat-bath gap”) records a bound, answer, status fact, or structural consequence. The record states: The gap is at least 1/170005193383307227693056 and at most 48265367094829605687889/36363194510128970638045284, approximately 0.00132731372325897517.",
  "relevance_source": "recorded",
  "body": "Write \\(\\gamma\\) for the discrete-time spectral gap under the convention in the problem. The certified interval is\n\\[\n\\frac{1}{170005193383307227693056}\n\\leq \\gamma \\leq\n\\frac{48265367094829605687889}{36363194510128970638045284}\n=0.00132731372325897517\\ldots.\n\\]\nThe lower endpoint equals \\(1/(36\\,2^{72})\\). The upper endpoint is an exact Rayleigh quotient for total magnetization \\(M=\\sum_v\\sigma_v\\). This trial function lies in the spin-flip-odd, translation-invariant symmetry sector. Its projected transition eigenvalue is\n\\[\n\\frac{36314929143034141032357395}{36363194510128970638045284}.\n\\]\n\nFor the upper bound, reversibility and the heat-bath projection identity give\n\\[\n\\langle M,(I-P)M\\rangle_\\pi\n=\\mathbb E_\\pi\\!\\left[\\operatorname{Var}(\\sigma_0\\mid\\sigma_{V\\setminus\\{0\\}})\\right].\n\\]\nThe exact transfer calculation finds\n\\[\n\\sum_\\sigma 2^{a(\\sigma)}M(\\sigma)^2\n=273793464546853425980576256,\n\\]\nwhere \\(a(\\sigma)\\) is the number of agreeing edges. With common conditional-variance scale \\(7225=25\\cdot17^2\\), it also finds\n\\[\n\\sum_\\sigma 2^{a(\\sigma)}\\,7225\\operatorname{Var}(\\sigma_0\\mid\\sigma_{V\\setminus\\{0\\}})\n=2625635969958730549421161600.\n\\]\nTheir quotient gives the stated upper endpoint.\n\nFor the lower bound, compare with independent random-scan heat bath under the uniform measure. If \\(m\\) and \\(M_0\\) are the minimum and maximum density ratios of the Ising measure against uniform measure, edge conductances give \\(\\mathcal E_P\\geq m\\mathcal E_0\\), while \\(\\operatorname{Var}_\\pi f\\leq M_0\\operatorname{Var}_0f\\). The independent chain has gap \\(1/36\\). The Gibbs weights are proportional to \\(2^{a(\\sigma)}\\), and the bipartite torus realizes both \\(a=0\\) and \\(a=72\\). Hence \\(m/M_0=2^{-72}\\), proving the lower endpoint.\n\nThe calculation supplies rigorous bounds. Determining the exact second eigenvalue still requires a certificate that covers every symmetry sector.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the random-scan single-site heat-bath chain for the zero-field ferromagnetic Ising model on C6 Cartesian-product C6 at beta=(log 2)/2",
    "bounds": {
      "side_length": {
        "min": 6,
        "max": 6
      },
      "vertices": {
        "min": 36,
        "max": 36
      },
      "states": {
        "min": 68719476736,
        "max": 68719476736
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1007/s00222-012-0404-5",
      "locator": "Exact transfer certificate in ising6-artifact-row-transfer-rayleigh and comparison proof in this record"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1007/s00222-012-0404-5",
    "locator": "Exact transfer certificate in ising6-artifact-row-transfer-rayleigh and comparison proof in this record"
  },
  "relations": [
    {
      "slug": "R418",
      "title": "Exact 64-state row-transfer certificate",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R419",
      "title": "Literature gives the general framework, while the exact 6 by 6 value remains unlocated",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "ising-six-torus-heat-bath-gap",
      "title": "ising six torus heat bath gap",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}