{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R431",
  "content_hash": null,
  "slug": "kmmp-claim-certified-cycle-11656512",
  "type": "claim",
  "title": "A certified cycle has length 11,656,512",
  "summary": "The state \\((343233,119429)\\) lies on a certified cycle of length 11,656,512, giving the current lower bound; the exact maximum cycle length over \\(\\mathbb F_{1000003}^2\\) remains open.",
  "relevance": "For Longest cycle of a nonlinear area-preserving map over F_1000003, record kmmp-claim-certified-cycle-11656512 (“A certified cycle has length 11,656,512”) records a bound, answer, status fact, or structural consequence. The record states: The state \\((343233,119429)\\) lies on a certified cycle of length 11,656,512, giving the current lower bound; the exact maximum cycle length over \\(\\mathbb F_{1000003}^2\\) remains open.",
  "relevance_source": "recorded",
  "body": "For \\(p=1{,}000{,}003\\), direct iteration starting at\n\\[\n(x_0,y_0)=(343233,119429)\n\\]\nfirst returns to that state after\n\\[\n\\boxed{11{,}656{,}512}\n\\]\nupdates. The preceding state is \\((223804,39277)\\). It maps to the start, while no earlier positive iterate in the replay equals the start. The cycle begins\n\\[\n(343233,119429)\\mapsto(1524,658294)\\mapsto(982388,980864).\n\\]\nThis improves the candidate record's lower bound of 2,732,283 by more than a factor of four.\n\nThe map is bijective because\n\\[\nx=x'-y',\\qquad y=y'-(x'-y')^2.\n\\]\nEvery starting state therefore lies on a cycle. A first return certifies the exact period of this witness without storing its 11,656,512 states. The ordered cycle trace, encoding each \\((x,y)\\) as two little-endian unsigned 64-bit integers, has FNV-1a digest `e1b26cc098a16596`.\n\nThis is a lower bound for the requested maximum. The phase space contains 1,000,006,000,009 states, and the 4,096-start search does not exclude a longer cycle elsewhere.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the permutation T(x,y)=(x+y+x^2,y+x^2) on the full state space F_p squared at p=1000003",
    "bounds": {
      "prime": {
        "min": 1000003,
        "max": 1000003
      },
      "states_in_phase_space": {
        "min": 1000006000009,
        "max": 1000006000009
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1088/0951-7715/18/5/015",
      "locator": "Exact 4096-start C search and full witness replay in kmmp-artifact-search-and-cycle-replay"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1088/0951-7715/18/5/015",
    "locator": "Exact 4096-start C search and full witness replay in kmmp-artifact-search-and-cycle-replay"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R428",
      "title": "Exact 4,096-start search and record-cycle replay",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R430",
      "title": "The literature supplies a cycle-statistics model rather than this finite maximum",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "kicked-map-million-prime-cycle",
      "title": "kicked map million prime cycle",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}