{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R210",
  "content_hash": null,
  "slug": "dfb100-claim-complete-extremal-classification",
  "type": "claim",
  "title": "Twelve primitive dihedral classes give every maximizer",
  "summary": "All 2,816 maximizers come from twelve listed primitive representatives by scaling, translation, rotation, and reflection.",
  "relevance": "For Longest four-number Ducci trajectory in the 100 box, record dfb100-claim-complete-extremal-classification (“Twelve primitive dihedral classes give every maximizer”) records a bound, answer, status fact, or structural consequence. The record states: All 2,816 maximizers come from twelve listed primitive representatives by scaling, translation, rotation, and reflection.",
  "relevance_source": "recorded",
  "body": "The primitive representatives, chosen lexicographically within their dihedral orbits, are\n\\[\n\\begin{gathered}\n(0,7,20,44),(0,10,29,64),(0,12,34,75),(0,13,37,81),\\\\\n(0,13,38,84),(0,15,43,95),(0,24,37,44),(0,35,54,64),\\\\\n(0,41,63,75),(0,44,68,81),(0,46,71,84),(0,52,80,95).\n\\end{gathered}\n\\]\nEvery ordered maximizer has the unique form\n\\[\nc\\mathbf 1+g\\sigma(p),\n\\]\nwhere \\(p\\) is on this list, \\(\\sigma(p)\\) is one of its eight distinct rotations and reflections, \\(1\\leq g\\leq\\lfloor100/\\max(p)\\rfloor\\), and \\(0\\leq c\\leq100-g\\max(p)\\). Conversely, every tuple of this form has stopping time 13.\n\nFor a representative with maximum coordinate \\(m\\), the number contributed is\n\\[\n8\\sum_{g=1}^{\\lfloor100/m\\rfloor}(101-gm).\n\\]\nThe representative maxima are \\(44,64,75,81,84,95\\), each occurring twice. Their contributions sum to 2,816.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all ordered quadruples in {0,...,100}^4 with Ducci stopping time 13",
    "bounds": {
      "coordinate": {
        "min": 0,
        "max": 100
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://oeis.org/A065677",
      "locator": "Complete classification emitted and checked by dfb100-artifact-symmetry-quotient-enumeration"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://oeis.org/A065677",
    "locator": "Complete classification emitted and checked by dfb100-artifact-symmetry-quotient-enumeration"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R209",
      "title": "Symmetry-quotient exhaustive certificate",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R211",
      "title": "The longest trajectory in the 100 box has 13 steps",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "ducci-four-box-100",
      "title": "ducci four box 100",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}