Let $B$ be uniform among predicates with exactly half of their $N$ inputs true. For $y\notin S$, the absolute mean of $B(y)\prod_{z\in S} B(z)$, using sign values, is at most $|S|/(N-|S|)$. This follows by multiplying the zero sum of all signs by the character on $S$, then using permutation symmetry outside $S$. The same bound holds with the character replaced by any function of absolute value at most one that depends only on the coordinates in $S$. This more general form also handles repeated predicate inputs.

For a fixed upper bound on $|S|$, this gives the $O(1/N)$ cancellation needed for the large-coefficient term in Lemma 4.8. It replaces the exact product formula of Lemma 4.9 with a sufficient bound.