Finite equal-load theorem — two-sided bound

For every natural number $d$, there exists a nonnegative real $C$ such that for every positive real $\varepsilon$, there exists a natural number $N$ such that, for every natural number $n\ge N$, the uniform probability that the maximum load differs from $\log\log(n+1)/\log(d+2)$ by at most $C$ is at least $1-\varepsilon$. The process allocates $n+1$ balls sequentially to $n+1$ initially empty bins, each ball making $d+2$ independent uniform choices with replacement and joining the earliest minimum-load choice. The constant $C$ may depend on $d$ but not on $\varepsilon$ or $n$; $N$ may depend on $d$ and $\varepsilon$.