{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R361",
  "content_hash": null,
  "slug": "h65537-claim-reversible-henon-form",
  "type": "claim",
  "title": "The map is a reversible area-preserving Hénon map",
  "summary": "Coordinate swap reverses H, placing its finite-field cycles in the Roberts and Vivaldi setting.",
  "relevance": "For Largest cycle of a Hénon permutation over the 65537 field, record h65537-claim-reversible-henon-form (“The map is a reversible area-preserving Hénon map”) records a bound, answer, status fact, or structural consequence. The record states: Coordinate swap reverses H, placing its finite-field cycles in the Roberts and Vivaldi setting.",
  "relevance_source": "recorded",
  "body": "The Jacobian determinant of \\(H\\) is one. For the coordinate swap \\(R(x,y)=(y,x)\\), direct substitution gives\n\\[\nRHR=H^{-1},\\qquad H^{-1}(u,v)=(u^2+1-v,u).\n\\]\nThus \\(H\\) is a reversible quadratic Hénon map. Both reversing involutions have 65,537 fixed points at the target prime: \\(R\\) fixes the diagonal, and \\(HR(x,y)=(x,x^2+1-y)\\) fixes \\(2y=x^2+1\\).\n\nRoberts and Vivaldi use fixed sets of reversing involutions to study cycle statistics for finite-field reductions of this class. Their model predicts distributions and typical scales. It supplies no finite upper bound for the longest cycle at \\(p=65537\\).",
  "status": "established",
  "evidence_grade": "mathematical_identity",
  "scope": {
    "kind": "family",
    "statement": "the maps H(x,y)=(y,y^2+1-x) over fields of odd characteristic",
    "family": "quadratic area-preserving polynomial automorphisms"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1088/0951-7715/18/5/015",
      "locator": "Direct identities above; Roberts and Vivaldi, Nonlinearity 18 (2005), 2171-2192"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1088/0951-7715/18/5/015",
    "locator": "Direct identities above; Roberts and Vivaldi, Nonlinearity 18 (2005), 2171-2192"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R359",
      "title": "The finite-field literature gives statistical context",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "henon-65537-max-cycle",
      "title": "henon 65537 max cycle",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}