= Prime-power modular intersection bound
{title2=$|\mathcal F|\le\binom n{\min(r,m-1)}$}
Let $m=p^a$ and let an $r$-<uniform set family> have no distinct pair whose intersection is congruent to $r$ modulo $m$. Its size is bounded as displayed; the customary nontrivial form $\binom n{m-1}$ requires $m-1\le r$. The <Lucas theorem> makes $\binom{z-r+m-1}{m-1}$ a residue detector modulo $p$, equal to one exactly when $z\equiv r\pmod m$. Its Boolean evaluation functions have degree at most $m-1$ and a diagonal evaluation matrix on the family. The <low-degree evaluation rank on a uniform layer> supplies the sharp dimension bound. Without the size qualification, three singletons with $n=3,r=1,m=4$ contradict the customary printed form, since $3>\binom33$.
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