Lemma: Uniform local interaction count

For every $x\in \Lambda$, the number of allowed supports in (eq:H) containing $x$ is at most ${\nu }_r$, and

Proof. Every such support lies in the graph ball of radius $r$ about $x$. That ball is contained in an ambient ${\ell }^1$ ball with $v_r$ lattice sites. There are at most $2^{v_r-1}$ subsets containing the centre. Some are not allowed supports, which only improves the bound.