This paper demonstrates that the Benjamini-Hochberg procedure can fail to control the false discovery rate (FDR) for correlated two-sided Gaussian $p$-values. A factor model is constructed such that, at a significance level of $\alpha=0.01$, a rigorous interval-arithmetic certificate proves that the FDR is 0.0104 for all sufficiently large numbers of hypotheses.
This result disproves a conjecture that has been widely believed to be true for twenty years. Monte Carlo experiments are consistent with the theoretical outcome. The proof was obtained by GPT-5.6 Pro and carefully verified by the author.
Blogger's Review: This article highlights the limitations of the Benjamini-Hochberg procedure under specific conditions, challenging a long-held consensus. By combining theory and empirical evidence, the author effectively illustrates the complexities of controlling false discovery rates in high-dimensional data analysis, warranting deeper contemplation and attention from the statistical community.