Question about false discovery rate with Benjamini-Hochberg-adjusted p-values and classifier false positive rate

Suppose one is testing multiple hypotheses and, for a given FDR, computes Benjamini-Hochberg adjusted p-values. Furthermore, consider the classifier defined by $$f(p; t) = \{\text{True if } p \ge t; \text{else False}\}$$ where $$p$$ is the p-value from the statistical test and the classes are "null hypothesis is rejected" and "not rejected".

My question is: at the adjusted p-value threshold corresponding to the FDR, would we expect the classifier's false positive rate to equal the FDR?

• I'm struggling a bit to understand this scenario. Is the idea that all assumptions of the underlying statistical tests are met, you compute the p-values for each test, feed them into your p-value adjustment machine, and then use your classifier to reject anything with p ≥t and "accept?" when p<t, and your "false positive" means "reject" when null is true? Commented Dec 28, 2023 at 16:17
• Are you assuming anything about the true proportion of time the null hypothesis is actually true? Commented Dec 28, 2023 at 16:22