Modular layer vanishing lemma

Let $p$ be a prime number, $0\le a<p$, $1\le s<p$, and $n\ge a+s$. A multilinear polynomial over ${\mathbb{F}}_p$ of degree less than $s$ that vanishes on every Boolean vertex whose weight is not congruent to $a$ modulo $p$ vanishes everywhere. Alternating sums over intervals of $s$ free coordinates eliminate the possible nonzero weight levels one at a time. This proves independence of the auxiliary functions used for the uniform Frankl-Wilson theorem bound.