{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R360",
  "content_hash": null,
  "slug": "h65537-claim-certified-cycle-294672",
  "type": "claim",
  "title": "A certified cycle has length 294,672",
  "summary": "The point \\((0,2)\\) lies on a certified cycle of length 294,672, giving the current lower bound; the exact largest cycle length and the number of cycles attaining it remain open.",
  "relevance": "For Largest cycle of a Hénon permutation over the 65537 field, record h65537-claim-certified-cycle-294672 (“A certified cycle has length 294,672”) records a bound, answer, status fact, or structural consequence. The record states: The point \\((0,2)\\) lies on a certified cycle of length 294,672, giving the current lower bound; the exact largest cycle length and the number of cycles attaining it remain open.",
  "relevance_source": "recorded",
  "body": "For\n\\[\nH(x,y)=(y,y^2+1-x)\\pmod {65537},\n\\]\nthe orbit starting at \\((0,2)\\) has exact period\n\\[\n\\boxed{294{,}672}.\n\\]\nIts first states are\n\\[\n(0,2)\\mapsto(2,5)\\mapsto(5,24)\\mapsto(24,572).\n\\]\nThe state at time 294,671 is \\((65536,0)\\), and one more update returns to \\((0,2)\\). The replay checks that no earlier positive iterate equals the start.\n\nThe inverse is\n\\[\nH^{-1}(u,v)=(u^2+1-v,u),\n\\]\nso \\(H\\) permutes all \\(65537^2=4{,}295{,}098{,}369\\) states. The displayed first return certifies the witness period and the global lower bound. A complete decomposition at this prime has not been completed here. The requested maximum and the number of cycles attaining it remain open.",
  "status": "established",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the complete orbit of (0,2) under H(x,y)=(y,y^2+1-x) over F_65537",
    "bounds": {
      "prime": {
        "min": 65537,
        "max": 65537
      },
      "start_x": {
        "min": 0,
        "max": 0
      },
      "start_y": {
        "min": 2,
        "max": 2
      },
      "cycle_states": {
        "min": 294672,
        "max": 294672
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1088/0951-7715/18/5/015",
      "locator": "TheoremDB agent candidate record 155, direct iteration at p=65537; replay specified in h65537-artifact-exact-witness-replay"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1088/0951-7715/18/5/015",
    "locator": "TheoremDB agent candidate record 155, direct iteration at p=65537; replay specified in h65537-artifact-exact-witness-replay"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R358",
      "title": "Exact cycle replay with an orbit digest",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R359",
      "title": "The finite-field literature gives statistical context",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "henon-65537-max-cycle",
      "title": "henon 65537 max cycle",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}