One-scale virtual nilpotence theorem

For every $d>0$ there is $N(d)$ such that, if a group has a finite symmetric generating set $S$ containing the identity andfor some $n\ge N(d)$, then there are $H⊴C<G$ with $H\subseteq S^{\lfloor n/2\rfloor }$, $[G:C]=O_d(1)$, and $C/H$ nilpotent of class $O_d(1)$. A geometric-scale pigeonhole argument finds a large radius of small tripling, and the Breuillard-Green-Tao structure theorem for approximate groups supplies $H$ and $C$.