Corollary: Bounded path law

For every fixed bounded scalar label,

For every fixed $a>0$,

Proof. In (eq:finitepath), use $x=\epsilon \sqrt{N}$ and $L=\lfloor C\sqrt{N}\rfloor$. The sum divided by ${\pi }_N$ tends to zero exponentially in $\sqrt{N}$, up to a polynomial factor. Then let $C$ increase, using (eq:codingconsequences). The relative statement follows by division by $a\sqrt{N}$.