{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R256",
  "content_hash": null,
  "slug": "ffe-claim-exact-orbit-cover",
  "type": "claim",
  "title": "Solutions are exactly projection-disjoint covers by Fibonacci-map cycles",
  "summary": "The graph of a solution is an invariant union of cycles of T(x,y)=(y,x+y), and the first-coordinate projections partition the field.",
  "relevance": "For Power-of-two solution counts for a finite-field functional equation, record ffe-claim-exact-orbit-cover (“Solutions are exactly projection-disjoint covers by Fibonacci-map cycles”) records a bound, answer, status fact, or structural consequence. The record states: The graph of a solution is an invariant union of cycles of T(x,y)=(y,x+y), and the first-coordinate projections partition the field.",
  "relevance_source": "recorded",
  "body": "Suppose\n\\[\nf(f(x))=f(x)+x\n\\]\nfor every \\(x\\in\\mathbb F_p\\). Equal values \\(f(x)=f(y)\\) give \\(x=y\\), so \\(f\\) is injective and hence a permutation. The equation at zero then gives \\(f(0)=0\\).\n\nDefine the invertible Fibonacci map\n\\[\nT_p(u,v)=(v,u+v)\n\\]\non \\(\\mathbb F_p^2\\), and let\n\\[\nG_f=\\{(x,f(x)):x\\in\\mathbb F_p\\}.\n\\]\nThe functional equation gives\n\\[\nT_p(x,f(x))=(f(x),x+f(x))=(f(x),f(f(x)))\\in G_f.\n\\]\nThus \\(G_f\\) is a union of complete \\(T_p\\)-cycles. Its first-coordinate projection is bijective. Each selected cycle consequently has distinct first coordinates, and the projection sets of the selected cycles partition \\(\\mathbb F_p\\). The fixed cycle \\(((0,0))\\) is forced.\n\nConversely, take any family of \\(T_p\\)-cycles whose first-coordinate projections are individually injective and jointly partition \\(\\mathbb F_p\\). Their union is the graph of a unique function \\(f\\). Invariance under \\(T_p\\) says that the graph contains \\((f(x),x+f(x))\\), so its value at \\(f(x)\\) is \\(f(f(x))=x+f(x)\\). This constructs every solution.\n\nTherefore \\(N(p)\\) is exactly the number of exact covers of \\(\\mathbb F_p\\) by the first-coordinate projections of eligible \\(T_p\\)-cycles, with cycles retained as distinct choices when their projections coincide.",
  "status": "established",
  "evidence_grade": "mathematical_identity",
  "scope": {
    "kind": "universal",
    "statement": "every prime p and every function from F_p to F_p satisfying the equation"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://sites.math.rutgers.edu/~nussbaum/Pubs/dynamicsJDE.pdf",
      "locator": "Independent graph-invariance and converse proof, 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://sites.math.rutgers.edu/~nussbaum/Pubs/dynamicsJDE.pdf",
    "locator": "Independent graph-invariance and converse proof, 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R255",
      "title": "Exact orbit-cover counts for every prime below 500",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "fibonacci-functional-equation-prime-count",
      "title": "fibonacci functional equation prime count",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}