crBH: Conformal Top-$k$ Selection with False Discovery Rate Control

Selecting candidates that truly belong to the top-k of a large pool is important in scientific screening and information retrieval, but their outcomes are unavailable at selection time. Existing conformal selection rules address fixed outcome thresholds; top-k membership instead depends on the other items in the test pool. Calibration and test ranks also live on different scales when their pool sizes differ. We formulate inductive top-k selection with false discovery rate (FDR) as its error criterion and propose crBH to address this rank-space mismatch. An exact rank projection supplies calibration-item probabilities of falling outside the test topk; projected conformal statistics combine these probabilities with trained ranking scores. A null-proportion-adjusted BH step-up rule then selects a score-ordered subset. We characterize the asymptotic FDR and power of crBH and connect its power to the quality and ordering induced by the ranking model. Although the guarantee is asymptotic, controlled and application benchmarks show empirical FDR close to the nominal level at the studied finite sample sizes when the method makes substantial discoveries, while retaining substantial power. For applications requiring an exact finite-sample guarantee, we additionally give a conservative certified variant under joint calibration/test exchangeability.