{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R211",
  "content_hash": null,
  "slug": "dfb100-claim-exact-maximum",
  "type": "claim",
  "title": "The longest trajectory in the 100 box has 13 steps",
  "summary": "The exact maximum is 13; 2,816 ordered quadruples attain it.",
  "relevance": "For Longest four-number Ducci trajectory in the 100 box, record dfb100-claim-exact-maximum (“The longest trajectory in the 100 box has 13 steps”) records a bound, answer, status fact, or structural consequence. The record states: The exact maximum is 13; 2,816 ordered quadruples attain it.",
  "relevance_source": "recorded",
  "body": "Write \\(\\tau(x)\\) for the first time at which \\(D^\\tau(x)=0\\). Exact enumeration gives\n\\[\n\\max_{x\\in\\{0,\\ldots,100\\}^4}\\tau(x)=13.\n\\]\nOne witness is \\((0,7,20,44)\\). Its trajectory is\n\\[\n\\begin{aligned}\n&(0,7,20,44),(7,13,24,44),(6,11,20,37),(5,9,17,31),\\\\\n&(4,8,14,26),(4,6,12,22),(2,6,10,18),(4,4,8,16),\\\\\n&(0,4,8,12),(4,4,4,12),(0,0,8,8),(0,8,0,8),\\\\\n&(8,8,8,8),(0,0,0,0).\n\\end{aligned}\n\\]\nThe candidate witness \\((57,81,37,44)\\) belongs to the same class: subtracting 37 gives \\((20,44,0,7)\\), a rotation of the displayed witness.\n\nThe number of ordered starting tuples at each stopping time \\(0,1,\\ldots,13\\) is\n\\[\n(1,100,20100,1353400,53030200,17977176,22886192,5951312,2093072,540976,165040,31136,8880,2816).\n\\]\nThese counts sum to \\(101^4=104060401\\).",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "all ordered quadruples in {0,...,100}^4 under the four-number Ducci map",
    "bounds": {
      "coordinate": {
        "min": 0,
        "max": 100
      },
      "ordered_quadruples": {
        "min": 104060401,
        "max": 104060401
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://oeis.org/A065677",
      "locator": "Independent exhaustive artifact dfb100-artifact-symmetry-quotient-enumeration; OEIS A065677 gives the same bounded maximum sequence"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://oeis.org/A065677",
    "locator": "Independent exhaustive artifact dfb100-artifact-symmetry-quotient-enumeration; OEIS A065677 gives the same bounded maximum sequence"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R209",
      "title": "Symmetry-quotient exhaustive certificate",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R210",
      "title": "Twelve primitive dihedral classes give every maximizer",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R212",
      "title": "OEIS already records the bounded maximum sequence",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R1500",
      "title": "Direct answer and proof for Longest four-number Ducci trajectory in the 100 box",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "incoming",
      "metadata": {
        "reason": "Adds the direct answer and human-readable proof reviewed on 2026-08-01."
      }
    },
    {
      "slug": "ducci-four-box-100",
      "title": "ducci four box 100",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}