For a family of supports $S_j$ and independently sampled labels $f(i)$, the expectation of $\prod_j\prod_{i\in S_j}B_j(f(i))$ factors over coordinates $i$. If the predicates are balanced and a coordinate occurs in exactly one support, the expectation vanishes. Thus only double covers contribute to the higher-moment expansion in equation (12).

A double cover uses every point of its union at least twice, so $2|\bigcup_jS_j|\leq\sum_j|S_j|$. This is the union-size bound used in the reduction from double covers to even covers in Lemma 4.15. Whenever $2|\bigcup_jS_j|\leq r\leq m$, one can retain exactly $r$ positions that still cover each point at least twice. This is the subcover extraction used in Lemma 4.16.