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

arXiv: https://arxiv.org/abs/2607.12208 · HTML: https://mlsanity.com/paper/2607.12208/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 28 references that are in the map.
- First posted on arXiv 13 Jul 2026.

## Builds on (most cited first, 6 of 6)

- [Conditional calibration for false discovery rate control under dependence](https://mlsanity.com/paper/2007.10438/) (2020)
- [Point Convergence of Nesterov's Accelerated Gradient Method: An AI-Assisted Proof](https://mlsanity.com/paper/2510.23513/) (2025)
- [Solving a Research Problem in Mathematical Statistics with AI Assistance](https://mlsanity.com/paper/2511.18828/) (2025)
- [G"odel Test: Can Large Language Models Solve Easy Conjectures?](https://mlsanity.com/paper/2509.18383/) (2025)
- [Accelerating mathematical research with language models: A case study of an interaction with GPT-5-Pro on a convex analysis problem](https://mlsanity.com/paper/2510.26647/) (2025)
- [Dependence-Aware False Discovery Rate Control in Two-Sided Gaussian Mean Testing](https://mlsanity.com/paper/2511.19960/) (2025)

## Abstract (arXiv)

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 α=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.