 Benjamini-Hochberg leave-two-out identity

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 $r\ge 2$,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.