\ball_{L_k}(u)$ and assume that $d(x,y)=R \ge L_{k-1}^{1+\rho} = L_{k-1}^{7/6}$, $\rho = 1/6$.

By assumption, either $\ball_{L_k}(u)$ contains no $(E_j,m)$-S $L_{k-1}$-ball, or there is a ball $\ball_{L_{k-1}}(w)$ such that any ball $\ball_{L_{k-1}}(v)$ disjoint with $\ball_{L_{k-1}(w)}$ is $(E_j,m)$-NS. We consider first the latter case and prove the assertion, avoiding the ball $\ball_{L_{k-1}}(w)$. To this end, set $r' = d(x, w)$, $r'' = d(y, w)$. Then the function $(x',x'')\mapsto \psi_j(x') \psi_j(x'')$ is $(\ell,q)$-subharmonic in $\ball_{r'}(x)\times\ball_{r''}(y)$. Therefore, one can apply the bound , with $r'+r'' \ge R - 2L_{k-1}$, and write, with the convention $-\ln 0 = +\infty$: \[eq:proof.lem.NT.implies.nloc.IID\] -|\_i(x) (y) | & -( e\^[-(m,L\_[k-1]{})L\_[k-1]{}]{})\^[ ]{} A direct comparison of Eqn with Eqn shows that the RHS of the latter is bigger, owing to the absence of the factor $e^{L_{k-1}^\beta} >1$, thus it admits the same lower bound as in Eqn .

\[rem:universality.IID\] Assertions of Lemma \[lem:few.nloc.and.few.R.imply.nloc.IID\] and Lemma \[lem:NT.implies.nloc.IID\] are deterministic and do not rely upon a particular structure of the potential. In other words, these statements are valid for arbitrary LSO, including multi-particle operators.

\[lem:induct.IID\] Let $p, b>0$ satisfy \[eq:cond.p.b\] p &gt; , 0 &lt; 3b - , Suppose that for some $k\ge 0$ and all $0 \le k' \le k$ the 