{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R432",
  "content_hash": null,
  "slug": "kmmp-claim-henon-conjugacy",
  "type": "claim",
  "title": "The map is a reversible area-preserving Hénon map",
  "summary": "An affine change of coordinates sends the kicked map to (a,b) -> (a^2+1-b,a).",
  "relevance": "For Longest cycle of a nonlinear area-preserving map over F_1000003, record kmmp-claim-henon-conjugacy (“The map is a reversible area-preserving Hénon map”) records a bound, answer, status fact, or structural consequence. The record states: An affine change of coordinates sends the kicked map to (a,b) -> (a^2+1-b,a).",
  "relevance_source": "recorded",
  "body": "Set\n\\[\na=x+1,\\qquad b=x-y+1.\n\\]\nThen the kicked map becomes\n\\[\nH(a,b)=(a^2+1-b,a),\n\\]\nthe quadratic area-preserving Hénon form. Its Jacobian determinant is one. The coordinate swap \\(R(a,b)=(b,a)\\) is a reversing involution because\n\\[\nRHR=H^{-1},\\qquad H^{-1}(a,b)=(b,b^2+1-a).\n\\]\nEquivalently, \\(H=(HR)R\\) is a product of two involutions. Each involution has \\(p\\) fixed points over \\(\\mathbb F_p\\) when \\(p\\) is odd.\n\nRoberts and Vivaldi study finite-field reductions of this reversible Hénon class and the resulting cycle statistics. Their asymptotic random-involution model has natural period scale \\(p\\) in this case. The certified period here is about \\(11.656477p\\). That model supplies context and no finite upper bound for this particular prime.",
  "status": "established",
  "evidence_grade": "mathematical_identity",
  "scope": {
    "kind": "family",
    "statement": "the kicked map over any field 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 affine conjugacy above; Roberts and Vivaldi, Signature of time-reversal symmetry in polynomial automorphisms over finite fields, 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 affine conjugacy above; Roberts and Vivaldi, Signature of time-reversal symmetry in polynomial automorphisms over finite fields, Nonlinearity 18 (2005), 2171-2192"
  },
  "relations": [
    {
      "slug": "R430",
      "title": "The literature supplies a cycle-statistics model rather than this finite maximum",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "kicked-map-million-prime-cycle",
      "title": "kicked map million prime cycle",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}