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

This paper claims to prove that the Benjamini–Hochberg (BH) procedure can exceed its nominal false discovery rate under correlated two-sided Gaussian p-values. The authors construct a factor model and provide a rigorous interval-arithmetic certificate showing that, at nominal level α=0.01, the FDR is greater than 0.0104 for all sufficiently large numbers of hypotheses. Monte Carlo experiments are reported as consistent with the theoretical result. The result disproves a conjecture that had been widely believed for roughly two decades. The abstract also states that GPT-5.6 Pro obtained the proof, which was checked by the author.