Choose independently and uniformly a label $f(i)$ from a nonempty finite set for every coordinate $i$. If two real predicates $B,C$ have mean zero, then $\mathbb E_f\bigl[\prod_{i\in S}B(f(i))\prod_{i\in T}C(f(i))\bigr]$ is zero for $S\ne T$, and is $(\mathbb E_z B(z)C(z))^{|S|}$ for $S=T$.

This is the independence and cancellation argument from equation (6) to equation (7) in the proof of Lemma 4.5. For the application, labels are $s$-bit words and the predicates are balanced sign-valued functions.

For a finite family of balanced predicates, the second moment of a weighted sum over supports therefore contains only equal-support terms, weighted by the squares of their coefficients. This yields the exact expression in equation (7) before the correlation estimates are applied. For supports of size at least $\ell$ and coefficients of total squared mass at most one, this second moment is bounded by the sum of the absolute correlations to the power $\ell$.