{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R159",
  "content_hash": null,
  "slug": "cpcrt-claim-exact-classification",
  "type": "claim",
  "title": "The candidate odd threshold is n(n-1)/2",
  "summary": "Exact search proves the formula through n=19; a general matching lower bound remains open in this record.",
  "relevance": "For Reset threshold of the cyclic pair-compression automaton, record cpcrt-claim-exact-classification (“The candidate odd threshold is n(n-1)/2”) records a bound, answer, status fact, or structural consequence. The record states: Exact search proves the formula through n=19; a general matching lower bound remains open in this record.",
  "relevance_source": "recorded",
  "body": "The construction in cpcrt-claim-odd-reset-upper-bound proves \\(\\operatorname{rt}(A_n)\\leq n(n-1)/2\\) for every odd \\(n\\). A general fiber argument gives the rigorous lower bound \\(\\operatorname{rt}(A_n)\\geq\\lceil\\log_2 n\\rceil\\): the letter \\(a\\) is a permutation, every fiber of \\(b\\) has size at most two, and a word containing \\(k\\) copies of \\(b\\) has fibers of size at most \\(2^k\\). A reset word has a fiber of size \\(n\\). Exact breadth-first search gives equality with the quadratic upper bound at every odd \\(n\\leq19\\). Separately weighted searches find that every reset word in this tested range uses at least \\(n-1\\) copies of \\(b\\) and at least \\((n-1)(n-2)/2\\) copies of \\(a\\). These two sharper inequalities would prove the formula in general, but this record does not supply a universal proof of them.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "universal",
    "statement": "the reset threshold of A_n for every odd integer n at least 3"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.4213/rm10005e",
      "locator": "Exact subset-automaton checks through n=19 in cpcrt-artifact-exact-sweep; general upper bound and fiber lower bound proved in this record"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.4213/rm10005e",
    "locator": "Exact subset-automaton checks through n=19 in cpcrt-artifact-exact-sweep; general upper bound and fiber lower bound proved in this record"
  },
  "models": [],
  "relations": [
    {
      "slug": "R160",
      "title": "Every odd order has a quadratic reset word",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R156",
      "title": "Exact subset-automaton sweep and weighted lower checks",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "R157",
      "title": "Focused literature search found neighboring classes",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "cyclic-pair-compression-reset-threshold",
      "title": "cyclic pair compression reset threshold",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}