arXiv:2607.14812v4 Announce Type: replace Abstract: We study the worst-case false discovery rate (FDR) of the Benjamini-Hochberg procedure for both one- and two-sided Gaussian tests with general correlation matrices. For each setting and given target FDR level $q$, we construct a family of finite Gaussian models whose FDR divided by $q$ diverges as $q\downarrow0$. This disproves any universal multiplicative FDR bound. Moreover, we establish new FDR lower bounds that are independent of the number of hypotheses. For two-sided tests, the worst-case FDR over the number of hypotheses, mean vector, and correlation matrix is at least an explicit ${\ell_{=}}(q)>q$: \[ \ell_{=}(q)=\frac{q\sqrt{\log(1/q)}}{2\sqrt{\pi}}+c_\ell q+o(q), \qquad c_\ell=0.6492828\ldots. \] For one-sided tests, we show a lower bound $\ell_{\le}(q)>q$: \[ \ell_{\le}(q)=\frac{q\sqrt{\log(1/q)}}{\sqrt\pi} +\frac q2+o(q). \] Finally, we prove the leading constants are sharp for a class of correlation matrices for both one- and two-sided tests.