= Modular layer vanishing lemma
Let $p$ be a <prime number>, $0\leq a<p$, $1\leq s<p$, and $n\geq a+s$. 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.
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