= Modular intersection polynomial
{title2=$P_A(x)=\prod_{\lambda\in L}(\sum_{i\in A}x_i-\lambda)$}
If member sizes modulo a <prime number> $p$ avoid $L$ while distinct intersections lie in $L$, evaluation at <characteristic vectors of sets> gives a diagonal matrix with nonzero diagonal. After <multilinear reduction on the Boolean cube>, these <polynomials> remain <linearly independent> and have degree at most $|L|$. Counting square-free <monomials> proves $|\mathcal F|\leq\sum_{j=0}^{|L|}\binom nj$. On a fixed uniform layer, the <low-degree evaluation rank on a uniform layer> sharpens this to $\binom n{|L|}$ when $|L|$ does not exceed the layer rank.
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