{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R254",
  "content_hash": null,
  "slug": "ffe-attempt-literature-audit",
  "type": "attempt",
  "title": "Fibonacci iterative equations are studied over the reals",
  "summary": "The located papers discuss continuous or analytic iterative equations; no finite-field orbit-cover count was found.",
  "relevance": "For Power-of-two solution counts for a finite-field functional equation, record ffe-attempt-literature-audit (“Fibonacci iterative equations are studied over the reals”) documents a concrete method, search boundary, or failed route. The record states: The located papers discuss continuous or analytic iterative equations; no finite-field orbit-cover count was found.",
  "relevance_source": "recorded",
  "body": "Greenfield and Nussbaum study Fibonacci and quadratic-Fibonacci recurrences through planar dynamics. Their discussion on pages 80-81 uses the real golden-ratio map \\(V\\) satisfying\n\\[\nV(V(x))=V(x)+x\n\\]\nand constructs real analytic solutions related to the Fibonacci recurrence. Tang and Zhang study continuous solutions of a broader second-order iterative equation on \\(\\mathbb R\\).\n\nThese works support the iterate-recurrence viewpoint. Their domains, regularity assumptions, and goals differ from counting arbitrary functions on a prime field. Targeted searches for the exact equation, finite fields, permutation functional equations, Fibonacci-map cycles, and solution counts found no source giving \\(N(p)\\), the orbit-cover classification, or the power-of-two assertion.\n\nThis is a bounded literature audit. It supplies no novelty proof.",
  "status": "inconclusive",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "literature directly concerning the Fibonacci iterative equation and finite-field solution counts",
    "bounds": {
      "search_date": {
        "min": 20260724,
        "max": 20260724
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://sites.math.rutgers.edu/~nussbaum/Pubs/dynamicsJDE.pdf",
      "locator": "Stephen J. Greenfield and Roger D. Nussbaum, Dynamics of a Quadratic Map in Two Complex Variables, Journal of Differential Equations 169 (2001), 57-141, DOI 10.1006/jdeq.2000.3895, pages 80-81; search performed 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://sites.math.rutgers.edu/~nussbaum/Pubs/dynamicsJDE.pdf",
    "locator": "Stephen J. Greenfield and Roger D. Nussbaum, Dynamics of a Quadratic Map in Two Complex Variables, Journal of Differential Equations 169 (2001), 57-141, DOI 10.1006/jdeq.2000.3895, pages 80-81; search performed 2026-07-24"
  },
  "relations": [
    {
      "slug": "R258",
      "title": "The power-of-two claim is verified below 500 and unresolved in general",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "fibonacci-functional-equation-prime-count",
      "title": "fibonacci functional equation prime count",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}