{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R634",
  "content_hash": null,
  "slug": "qorp-claim-general-quadratic-recurrence-bound",
  "type": "claim",
  "title": "Quadratic orbits satisfy a sublinear general recurrence bound",
  "summary": "Heath-Brown proves recurrence after O(q/log log q) steps for X^2+1 over odd finite fields.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Heath-Brown studies the directed functional graph of a quadratic polynomial over \\(\\mathbb F_q\\), separating each trajectory into its preperiodic tail and cycle. Corollary 1 gives an unconditional \\(O(q/\\log\\log q)\\) bound for the first recurrence of the critical orbit. Corollary 2 gives the corresponding cycle and preperiod bounds for reductions of fixed positive definite quadratics, including \\(X^2+1\\).\n\nThis theorem supplies asymptotic context for the finite search. The paper contains no finite record table identified during the focused search, so the witness and minimality claim above rest on the reproduced computation.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "family",
    "statement": "reductions of x^2+1 over finite fields of odd characteristic",
    "family": "the quadratic maps X^2+1 over odd finite fields"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1701.02707",
      "locator": "D. R. Heath-Brown, Iteration of Quadratic Polynomials Over Finite Fields, Corollaries 1-2 and their proofs in section 1; Mathematika 63 (2017), 1041-1059, DOI 10.1112/S0025579317000328"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1701.02707",
    "locator": "D. R. Heath-Brown, Iteration of Quadratic Polynomials Over Finite Fields, Corollaries 1-2 and their proofs in section 1; Mathematika 63 (2017), 1041-1059, DOI 10.1112/S0025579317000328"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R631",
      "title": "Exhaustive increasing-prime sweep",
      "object_type": "artifact",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "quadratic-orbit-record-prime-10m",
      "title": "quadratic orbit record prime 10m",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}