{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R171",
  "content_hash": null,
  "slug": "cst101-claim-exact-maximum",
  "type": "claim",
  "title": "The exact maximum has 97 digits",
  "summary": "An exact sweep of 42,376 signed-unit orbits proves that S={1,15,18,22,27} uniquely maximizes the tree count up to Cayley isomorphism.",
  "relevance": "For Most spanning trees in a 10-regular circulant on 101 vertices, record cst101-claim-exact-maximum (“The exact maximum has 97 digits”) records a bound, answer, status fact, or structural consequence. The record states: An exact sweep of 42,376 signed-unit orbits proves that S={1,15,18,22,27} uniquely maximizes the tree count up to Cayley isomorphism.",
  "relevance_source": "recorded",
  "body": "Let\n\\[\nS_*=\\{1,15,18,22,27\\}.\n\\]\nExact enumeration gives\n\\[\n\\max_{|S|=5}\\tau(G_S)=\n2708274425686971438523646018073797918175286202944373217379948692092215150697249050994964490229261.\n\\]\nMultiplication by 4 modulo 101, followed by replacing residues above 50 with their negatives, sends \\(S_*\\) to\n\\[\n\\{4,7,13,29,41\\}.\n\\]\nThus the sampled incumbent in the candidate record already attained the maximum.\n\nThe group \\((\\mathbf Z/101\\mathbf Z)^\\times/\\{\\pm1\\}\\) has order 50 and acts on the five-element step sets. The \\(\\binom{50}{5}=2{,}118{,}760\\) sets split into 42,376 orbits: 42,375 orbits of size 50 and one orbit of size 10. The exact search finds one maximizing orbit. The next orbit is represented by \\(\\{7,15,16,18,28\\}\\), with tree count\n\\[\n2705790677807473137809613684625587492780256938436447544415672932035143210140285775448737562551029.\n\\]\nEvery step is nonzero modulo the prime 101, so each graph in the search is connected.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all five-element subsets S of {1,...,50} and the associated simple 10-regular Cayley graphs on Z/101Z",
    "bounds": {
      "vertices": {
        "min": 101,
        "max": 101
      },
      "degree": {
        "min": 10,
        "max": 10
      },
      "step_set_size": {
        "min": 5,
        "max": 5
      },
      "step_sets": {
        "min": 2118760,
        "max": 2118760
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1711.00175",
      "locator": "Exact reconstruction in cst101-artifact-exact-orbit-sweep, independently checked at the winner by cst101-artifact-bareiss-cofactor"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1711.00175",
    "locator": "Exact reconstruction in cst101-artifact-exact-orbit-sweep, independently checked at the winner by cst101-artifact-bareiss-cofactor"
  },
  "relations": [
    {
      "slug": "R172",
      "title": "Six finite-field spectra determine every tree count exactly",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R169",
      "title": "Exact orbit sweep with modular Matrix-Tree products",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R168",
      "title": "Independent exact Laplacian-cofactor check",
      "object_type": "artifact",
      "relation": "independently_verifies",
      "direction": "incoming"
    },
    {
      "slug": "R170",
      "title": "The finite order-101 maximum was not located in the sources checked",
      "object_type": "attempt",
      "relation": "contextualizes",
      "direction": "incoming"
    },
    {
      "slug": "circulant-spanning-trees-101-degree10",
      "title": "circulant spanning trees 101 degree10",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}