{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R172",
  "content_hash": null,
  "slug": "cst101-claim-spectral-certificate",
  "type": "claim",
  "title": "Six finite-field spectra determine every tree count exactly",
  "summary": "Matrix-Tree eigenvalue products modulo six 61-bit primes reconstruct every integer count without floating-point comparisons.",
  "relevance": "For Most spanning trees in a 10-regular circulant on 101 vertices, record cst101-claim-spectral-certificate (“Six finite-field spectra determine every tree count exactly”) records a bound, answer, status fact, or structural consequence. The record states: Matrix-Tree eigenvalue products modulo six 61-bit primes reconstruct every integer count without floating-point comparisons.",
  "relevance_source": "recorded",
  "body": "For \\(\\zeta=e^{2\\pi i/101}\\), the nonzero Laplacian eigenvalues are\n\\[\n\\lambda_k=10-\\sum_{s\\in S}(\\zeta^{ks}+\\zeta^{-ks}),\n\\qquad 1\\leq k\\leq100.\n\\]\nKirchhoff's theorem gives\n\\[\n\\tau(G_S)=\\frac1{101}\\prod_{k=1}^{100}\\lambda_k.\n\\]\nSince \\(\\lambda_{101-k}=\\lambda_k\\), it suffices to multiply the first 50 values and square.\n\nThe executable replay performs the same calculation in six prime fields. Each modulus is congruent to 1 modulo 101 and comes with an explicit element of order 101. It reconstructs \\(\\tau(G_S)\\) by the Chinese remainder theorem. The combined modulus has 366 bits and exceeds \\(10^{110}\\). The arithmetic-geometric mean inequality applied to the 100 nonzero Laplacian eigenvalues gives\n\\[\n\\tau(G_S)\\leq\\frac{10.1^{100}}{101}<10^{99}.\n\\]\nThe reconstructed residue is therefore the unique integer tree count for every orbit representative. This exact comparison avoids any ordering decision based on floating-point logarithms.",
  "status": "established",
  "evidence_grade": "mathematical_identity",
  "scope": {
    "kind": "bounded",
    "statement": "every graph G_S in the order-101 degree-10 candidate family",
    "bounds": {
      "vertices": {
        "min": 101,
        "max": 101
      },
      "degree": {
        "min": 10,
        "max": 10
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1711.00175",
      "locator": "Mednykh and Mednykh, The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic, equations for the even-valent circulant Laplacian and Theorem 3; exact finite-field implementation in cst101-artifact-exact-orbit-sweep"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1711.00175",
    "locator": "Mednykh and Mednykh, The number of spanning trees in circulant graphs, its arithmetic properties and asymptotic, equations for the even-valent circulant Laplacian and Theorem 3; exact finite-field implementation in cst101-artifact-exact-orbit-sweep"
  },
  "relations": [
    {
      "slug": "R171",
      "title": "The exact maximum has 97 digits",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R169",
      "title": "Exact orbit sweep with modular Matrix-Tree products",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "circulant-spanning-trees-101-degree10",
      "title": "circulant spanning trees 101 degree10",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}