Kahn's five-value range conjecture for Hamming-cube homomorphisms

Let $Q_d$ be the $d$-dimensional Hamming cube and let ${\cal F}$ be the set of graph homomorphisms $f\colon V(Q_d)\to{\bf Z}$ satisfying $f(\underline{0})=0$. For $f\in{\cal F}$, write $R(f)=\{f(v)\colon v\in V(Q_d)\}$, and use the uniform probability measure on ${\cal F}$.

Kahn's five-value range conjecture.

$${\bf P}(|R|>5)=e^{-\Omega(d)}\qquad\text{and}\qquad {\bf P}(|R|=5)=\Omega(1).$$

The conjecture gives the expected sharp typical range size after Kahn's earlier constant-range bound. It is proved in the present paper by asymptotically counting homomorphisms according to their range size.