The Benjamini--Hochberg Procedure Can Fail to Control the FDR for Correlated Two-Sided Gaussian Tests

We show that the Benjamini--Hochberg procedure can fail to control the false discovery rate (FDR) at its nominal level for correlated two-sided Gaussian $p$-values.

We construct a factor model for which, at level $\alpha=0.01$, a rigorous interval-arithmetic certificate proves $FDR>0.0104$ for all sufficiently large numbers of hypotheses.

This disproves a conjecture widely believed to be true for twenty years.

Monte Carlo experiments are consistent with the theoretical result.

The proof was obtained by GPT-5.6 Pro and carefully checked by the author.