{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R714",
  "content_hash": null,
  "slug": "sdcm-claim-certified-24-to-25",
  "type": "claim",
  "title": "The optimal cyclic win count is 24 or 25",
  "summary": "An explicit partition attains 24 wins on every cyclic edge, while the sharp universal random-variable bound excludes 26 wins.",
  "relevance": "For Largest cyclic winning margin for six disjoint six-sided dice, record sdcm-claim-certified-24-to-25 (“The optimal cyclic win count is 24 or 25”) records a bound, answer, status fact, or structural consequence. The record states: An explicit partition attains 24 wins on every cyclic edge, while the sharp universal random-variable bound excludes 26 wins.",
  "relevance_source": "recorded",
  "body": "Write\n\\[\nM=36\\max_{D_0,\\ldots,D_5}\\min_i\\Pr(D_i>D_{i+1}).\n\\]\nThe certified interval is\n\\[\n\\boxed{24\\leq M\\leq25},\n\\qquad\n\\boxed{\\frac23\\leq\\max\\min_i\\Pr(D_i>D_{i+1})\\leq\\frac{25}{36}}.\n\\]\nThe lower bound is attained by\n\\[\n\\begin{aligned}\nD_0&=\\{3,4,5,32,33,34\\},&\nD_1&=\\{1,2,28,29,30,31\\},\\\\\nD_2&=\\{22,23,24,25,26,27\\},&\nD_3&=\\{16,17,18,19,20,21\\},\\\\\nD_4&=\\{10,11,12,13,14,15\\},&\nD_5&=\\{6,7,8,9,35,36\\}.\n\\end{aligned}\n\\]\nIts cyclic win counts are \\([24,24,36,36,24,24]\\).\n\nKomisarski proves that every cycle of six independent random variables with pairwise tie probability zero has some cyclic winning probability strictly below\n\\[\n1-\\frac{1}{4\\cos^2(\\pi/8)}=\\frac{1}{\\sqrt2}.\n\\]\nThe theorem applies directly to fair rolls of these dice. A count of 26 would give probability \\(26/36=13/18\\), and\n\\[\n\\left(\\frac{13}{18}\\right)^2=\\frac{169}{324}>\\frac12.\n\\]\nThus \\(13/18>1/\\sqrt2\\), so six counts of at least 26 are impossible. Integrality gives \\(M\\leq25\\).\n\nThe remaining question is whether a partition with all six counts at least 25 exists. No such partition or nonexistence certificate was produced in this research pass.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all partitions of the labels 1 through 36 into six labeled dice with six distinct faces each",
    "bounds": {
      "dice": {
        "min": 6,
        "max": 6
      },
      "faces_per_die": {
        "min": 6,
        "max": 6
      },
      "labels": {
        "min": 36,
        "max": 36
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1080/00029890.2021.1889921",
      "locator": "Andrzej Komisarski, Nontransitive Random Variables and Nontransitive Dice, American Mathematical Monthly 128 (2021), 423-434, sharp max-min bound for cycles of n independent random variables; exact witness replay in sdcm-artifact-witness-verifier"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1080/00029890.2021.1889921",
    "locator": "Andrzej Komisarski, Nontransitive Random Variables and Nontransitive Dice, American Mathematical Monthly 128 (2021), 423-434, sharp max-min bound for cycles of n independent random variables; exact witness replay in sdcm-artifact-witness-verifier"
  },
  "relations": [
    {
      "slug": "R712",
      "title": "Exact verifier for the 24-win construction",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R713",
      "title": "The universal cycle bound leaves one integer case",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "six-dice-cyclic-margin",
      "title": "six dice cyclic margin",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}