{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R713",
  "content_hash": null,
  "slug": "sdcm-attempt-literature-and-search-audit",
  "type": "attempt",
  "title": "The universal cycle bound leaves one integer case",
  "summary": "Published work settles the unrestricted max-min threshold, while the fixed six-face partition problem remains unresolved after a focused search.",
  "relevance": "For Largest cyclic winning margin for six disjoint six-sided dice, record sdcm-attempt-literature-and-search-audit (“The universal cycle bound leaves one integer case”) documents a concrete method, search boundary, or failed route. The record states: Published work settles the unrestricted max-min threshold, while the fixed six-face partition problem remains unresolved after a focused search.",
  "relevance_source": "recorded",
  "body": "Komisarski studies cycles of arbitrary independent random variables and proves the sharp threshold\n\\[\n1-\\frac{1}{4\\cos^2(\\pi/(n+2))}.\n\\]\nAt \\(n=6\\) this is \\(1/\\sqrt2\\). Since the theorem bounds the minimum probability around the cycle, it applies without a balance assumption. It supplies the upper half of the interval in this record.\n\nRooney characterizes the rational winning probabilities attainable by balanced nontransitive \\(n\\)-tuples when the number of faces may vary. Schaefer and Schweig construct balanced triples and discuss larger sets. Booth and Goff use SAT to search finite nonstandard-dice problems. These sources give context and useful methods, though none reports the exact optimum for a partition of \\(1,\\ldots,36\\) into six six-sided dice on a prescribed directed cycle.\n\nA direct finite SMT model was also tried. It used one die-membership variable for each ordered label, six exact-cardinality constraints, and one sum of pair indicators for each cyclic edge. The target of 25 wins per edge did not resolve within the allotted run. This failed run is not used as evidence. A conclusive next pass could encode the color word with cardinality networks and publish either a satisfying word or a checked unsatisfiability proof for threshold 25.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.1080/00029890.2021.1889921",
      "locator": "Focused searches for six-die cyclic winning margin, balanced nontransitive dice, max-min stochastic precedence cycles, and SAT searches for nonstandard dice, completed 2026-07-25"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1080/00029890.2021.1889921",
    "locator": "Focused searches for six-die cyclic winning margin, balanced nontransitive dice, max-min stochastic precedence cycles, and SAT searches for nonstandard dice, completed 2026-07-25"
  },
  "models": [],
  "relations": [
    {
      "slug": "R714",
      "title": "The optimal cyclic win count is 24 or 25",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "six-dice-cyclic-margin",
      "title": "six dice cyclic margin",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}