In fact, for sufficiently large $s$, $E_s$ is identical to $E$; unfortunately, this $s$ will depend on $n$. Due to uniformity issues, we will instead fix $s$ and later show that, even when $s$ is finite but large, the approximation is good.

The major benefit of the $s$-graded model is that its edge count is very simple to express in terms of $X_I$, the number of points in each $A_I$: $$\begin{aligned}
|E_s| &= \displaystyle \sum_{I \in T} \biggl[{X_I \choose 2}  + \frac{1}{2} \sum_{J\, : \, 0 < d(I,J) \leq s} X_I X_J \biggr]\\
&=  \frac{1}{2}\sum_{I \in T} X_I \left[\biggl(\displaystyle \sum_{J\, : \, d(I,J) \leq s} X_J\biggr) -1\right].\end{aligned}$$ This random variable is defined in terms of i.i.d. random variables, which eases the analysis greatly. Furthermore, the ambient geometry of the torus is encoded completely by the metric $d$. Finally, each $X_I$ only appears in finitely many terms in this expression. The “finite range" nature of the representation will play a major role in the proof presented.

We quantify this fact as follows: for any $I \in T$, let $N_I := \{J: d(I,J) \leq s\}$. Thanks to translation invariance of $d$, the cardinality of this set is independent of the choice of 