For each prime \(p<500\), the exact orbit-cover computation determines \(N(p)\). Forty-six counts are positive and forty-nine vanish. Every positive count in this range is a power of two. The observed positive values have distribution \[ 1^1,\quad2^{30},\quad4^7,\quad8^3,\quad32^3,\quad512^1,\quad1024^1, \] where exponents record the number of primes attaining each value. The largest is \(N(461)=1024\).

The structural reduction in `ffe-claim-exact-orbit-cover` turns \(N(p)\) into a finite exact-cover count. Exact-cover counts can have odd factors in general. This audit found no additional involution or component theorem forcing the present counts to be powers of two for every prime. The universal question therefore remains unresolved in this entry.

Suppose \[ f(f(x))=f(x)+x \] for 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 \(f(0)=0\).

Define the invertible Fibonacci map \[ T_p(u,v)=(v,u+v) \] on \(\mathbb F_p^2\), and let \[ G_f=\{(x,f(x)):x\in\mathbb F_p\}. \] The functional equation gives \[ T_p(x,f(x))=(f(x),x+f(x))=(f(x),f(f(x)))\in G_f. \] Thus \(G_f\) 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.

Conversely, 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.

Therefore \(N(p)\) 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.

For \(f_a(x)=ax\), substitution gives \[ a^2x=(a+1)x \] for every \(x\), so the necessary and sufficient condition is \[ a^2-a-1=0. \] At \(p=5\), this polynomial has the repeated root \(a=3\), giving one linear solution. For odd \(p\neq5\), it has two roots exactly when 5 is a quadratic residue modulo \(p\), equivalently when \(p\equiv1\) or \(4\pmod5\). It has no roots at \(p=2\).

The exact census gives \(N(29)=4\). Since \(29\equiv4\pmod5\), precisely two of these functions are scalar-linear. The other two are nonlinear. Every smaller prime has either zero solutions, the single solution at 5, or the two scalar-linear solutions. Thus \(p=29\) is the first prime with a nonlinear solution.

The exact computation enumerates all cycles of \(T_p\) on the full set of \(p^2\) ordered pairs. It retains a cycle precisely when its first coordinates are distinct and exclude zero, then recursively counts disjoint projection covers of \(\mathbb F_p^*\). A minimum-option branching rule changes search order only; memoization keys the uncovered field elements exactly.

Selected results are \[ N(5)=1,\ N(11)=2,\ N(29)=4,\ N(139)=8, \] \[ N(199)=512,\ N(211)=32,\ N(281)=32,\ N(461)=1024. \] The canonical record stores, for each prime, its count, eligible-cycle count, total \(T_p\)-cycle count, and number of memoized exact-cover states. Its SHA-256 digest is `c8ef45793aebcc55c45cb1ee99d55c67d9e418b867c5b5d6939f92e9493d2e37`.

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 \[ V(V(x))=V(x)+x \] and constructs real analytic solutions related to the Fibonacci recurrence. Tang and Zhang study continuous solutions of a broader second-order iterative equation on \(\mathbb R\).

These 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.

This is a bounded literature audit. It supplies no novelty proof.

Each pair \((u,v)\) is encoded by the integer \(pu+v\). A byte array marks all \(p^2\) states while the program follows the invertible map \((u,v)\mapsto(v,u+v)\). Eligible projection sets are Python integer bit masks. The recursive counter selects an uncovered element with the fewest currently compatible cycles and sums over every such choice.

The code asserts the number of tested primes, the power-of-two property in the bounded range, and the complete count distribution. It prints selected counts and the digest of the full canonical record. The seven-line output has SHA-256 digest `01469652117bd7508b75c6764cbd07fe2a1a5d5a17e5b8a6376f14759946f1d5`.