= Benjamini-Hochberg leave-two-out identity
{c}
{title2=$R^{(ij)}=R_{-ij}+2$}
Replace two <p-values> $P_i,P_j$ by zeros in the <Benjamini-Hochberg procedure> at level $\alpha$, keeping the original denominator $m$, and let $R^{(ij)}$ be the resulting rejection count. Equivalently, remove the two values and apply the step-up critical values $\alpha(k+2)/m$ to the remaining ordered <p-values>; their rejection count is $R_{-ij}=R^{(ij)}-2$. For $r\ge2$,
$$
\{P_i\le\alpha r/m,\ P_j\le\alpha r/m,\ R=r\}
=\{P_i\le\alpha r/m,\ P_j\le\alpha r/m,\ R^{(ij)}=r\}.
$$
If both original values are rejected, lowering them changes no ordered rank greater than $r$, so the rejection count stays $r$. Conversely, if the modified count is $r$ and both original values meet the rank-$r$ threshold, reinserting them leaves at least $r$ qualifying values. Monotonicity under lowering gives at most $r$ original rejections. This identity is deterministic and does not require <independence> or a distributional assumption.
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