are $(E,m)$-S} } \le f(L).$$ Then for any $x,y\in\DZ^d$ with $\rd(x,y)> 2L+1$, any connected subset $\Lam\supset\ball_L(x) \cup \ball_L(y)$ and any Borel function $\phi\in\csB_1(I)$ \[eq:thm.MSA.to.DL\] CL\^d \^[-mL]{} + f(L).

Fix points $x,y\in\DZ^d$ with $\rd(x,y)> 2L+1$ and a finite connected graph $\Lam\supset\ball_L(x) \cup \ball_L(y)$. The operator $H_\Lam(\om)$ has a finite orthonormal eigenbasis $\{\psi_i\}$ with respective eigenvalues $\{\lam_i\}$. Set $S = \pt \ball_L(x) \cup \pt \ball_L(y)$ (recall: this is a set of *pairs* $(u,u')$). Suppose that for some $\om$, for each $i$ there is $z\in \{x,y\}$ such that $\ball_L(z_i)$ is $\lam_i,m)$-NS; let $\{v_i \}= \{x,y\}\setminus \{z_i\}$. Denote $\mu_{x,y}(\phi) = \langle \big|\operatorname{\mathbf{1}}_{x} | \phi(H_\Lam(\om)) | \operatorname{\mathbf{1}}_{y} \rangle$, with $\big|\mu_{x,y}(\phi)\big|\le 1$. Then by the GRI for eigenfunctions, and by Bessel’s inequality used at the last stage of derivation, $$\bal
 \big|\mu_{x,y}(\phi)\big|
& \le \|\phi\|_\infty \, \sum_{\lam_i \in I} |\psi_i(x) \psi_i(y)|
\le \sum_{\lam_i \in I} |\psi_i(z_i) \psi_i(v_i)|
\\
& \le \sum_{\lam_i \in I} |\psi_i(v_i)| {\mathrm{e}}^{-mL}
      \sum_{(u,u')\in\pt \ball_L(z_i)} |\psi_i(u)|
\qquad\qquad\qquad\qquad\qquad\qquad
\\
$$ & \^[-mL]{} \_[\_i I]{} \_[(u,u’)S]{} |\_i(u)| (|\_i(x)| + |\_i(y)| ) $$$$ &\^[-mL]{} |S| \_[u]{} \_[\_i I]{} ( |\_i(u)|\^2 + |\_i(x)|\^2 + |\_i(y)|\^2 )\
& \^[-mL]{} \_[u]{} ( 2\_u\^2 + \_x\^2 + \_y\^2 ) = \^[-mL]{} |S| 2 $$ where $|S|\le C L^d$. Denote $\CN_L = \myset{\exists\, 