Let an interval $I\subset \DR$ be given, and suppose that a ball $\ball_{L_0}(u)$ is $(m,I)$-NT. Then it is also .

\[cor:CT.cor.loc\] Under the assumptions of Lemma \[lem:CT.prob\], there exist $\tilde L^{(4)}\in\DN$ and $C, c>0$ such that, for any $L_1\ge \tilde L^{(4)}$ and for some $\eta(L_1)>0$, $m=m(L_1) \ge CL_1^{-1/3}$, $$\pr{ \ball_{L_1}(0) \text{ is \Inloc\, } } \leq e^{-c L_1^{d/6}}.$$

First, we choose $L_0$ as in Corollary \[cor:CT.prob\], and set $L_1 = \big[L_0^{\alpha}\big]+1 = \big[L_0^{4/3} \big]+1$, so that $L_0 \approx L_1^{3/4}$. Owing to , with probability not smaller than $$1 - |\ball_{L_1}(0)|^2 e^{-2cL_0^{d/4}} \ge 1 - e^{-cL_1^{3d/16}}$$ there is no pair of disjoint $(E,m)$-singular balls of radius $L_0$ inside $\ball_{L_1}(0)$. Now the claim follows from Lemma \[lem:CT.cor.loc\].

Corollary \[cor:CT.cor.loc\] allows to establish uniform bounds on eigenfunction correlators. However, the main technical tool of the scale induction at “extreme” energies becomes the following analog of Lemma \[lem:NT.implies.nloc.IID\] for the Green functions:

\[lem:CT.ind.step\] For any $C>0$ there exists $L^*(C)>0$ such that for any $L_k\ge L^*(C)$, if a ball $\ball_{L_k}$ is $E$-NR and $(m,I)$-NT with $m\ge CL_{k}^{-1/2}$, then it is $(E,m)$-NS.

The proof is quite similar to the proof of Lemma \[lem:NT.implies.nloc.IID\], with minor modifications. Actually, the sufficiency of the lower bound $m\ge CL_{k}^{-1/2}$ on 