= Cross-intersection bound from cube separation
{title2=$\mu_m(\mathcal F)\mu_m(\mathcal G)\leq e^{-w^2/m}$}
If two nonempty <set families> on $m\geq1$ coordinates have every cross-intersection greater than an integer $w\geq0$, their density product obeys the displayed bound. Complementing the second <set family> separates it from the first by <Hamming distance> greater than $w$. <Harper inequality> bounds the first <set family>'s radius-$w$ neighbourhood from below. If $S_m(a-1)<|\mathcal F|\leq S_m(a)$, disjointness yields $|\mathcal F||\mathcal G|\leq S_m(a)S_m(m-a-w)$. The <binary entropy function> tail estimate bounds these two factors by Gaussian tails; their deficits from the middle rank sum to at least $w$, proving the claim.
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