{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R430",
  "content_hash": null,
  "slug": "kmmp-attempt-literature-audit",
  "type": "attempt",
  "title": "The literature supplies a cycle-statistics model rather than this finite maximum",
  "summary": "Roberts and Vivaldi treat reversible Hénon reductions over finite fields; the focused search found no published decomposition at p=1000003.",
  "relevance": "For Longest cycle of a nonlinear area-preserving map over F_1000003, record kmmp-attempt-literature-audit (“The literature supplies a cycle-statistics model rather than this finite maximum”) documents a concrete method, search boundary, or failed route. The record states: Roberts and Vivaldi treat reversible Hénon reductions over finite fields; the focused search found no published decomposition at p=1000003.",
  "relevance_source": "recorded",
  "body": "Roberts and Vivaldi's 2005 paper studies reductions of polynomial automorphisms of the plane over finite fields, with detailed experiments for the area-preserving Hénon map. Their 2009 paper proves a universal cycle-length distribution for the corresponding random-involution model and explains why reversible maps differ from random permutations.\n\nThe affine conjugacy in this fixture places the present map directly in that setting. The papers give asymptotic statistics and experiments at other primes. The focused title, formula, and citation search found no decomposition or longest-cycle value for this exact map at \\(p=1{,}000{,}003\\). The global maximum therefore remains open in this entry.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "published finite-field cycle results relevant to the exact Hénon-form map and the modulus p=1000003",
    "bounds": {
      "prime": {
        "min": 1000003,
        "max": 1000003
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.1088/0951-7715/18/5/015",
      "locator": "Roberts and Vivaldi, Nonlinearity 18 (2005), 2171-2192; Roberts and Vivaldi, Nonlinearity 22 (2009), 1965-1982"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1088/0951-7715/18/5/015",
    "locator": "Roberts and Vivaldi, Nonlinearity 18 (2005), 2171-2192; Roberts and Vivaldi, Nonlinearity 22 (2009), 1965-1982"
  },
  "relations": [
    {
      "slug": "R432",
      "title": "The map is a reversible area-preserving Hénon map",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R431",
      "title": "A certified cycle has length 11,656,512",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "kicked-map-million-prime-cycle",
      "title": "kicked map million prime cycle",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}