MultiHit principle for learned sparse retrieval: deterministic bounds on candidate volume and Top-$K$ survival

Learned sparse retrieval such as SPLADE supports semantic search while retaining the efficiency of inverted indexes, but semantic expansion can make a query touch a large fraction of the corpus. We study whether the overlap structure across posting lists can be used as a filter to remove background documents before document scoring. We identify a MultiHit separation: as the required number of matched coordinates increases, the candidate set shrinks approximately exponentially while many highly ranked documents survive. This motivates an $a$-of-$B$ filter that retains documents matching at least $a$ of the $B$ highest-weight query coordinates. Since SPLADE rankings are determined by weighted inner products rather than hit counts, however, it is nontrivial whether top-ranked documents still survive the filter. We characterize both sides of this separation using finite-corpus statistics. Intersection counts bound the candidate-set size without any distributional assumptions, while score-margin moments give lower bounds on overall Top-$K$ survival. We further formulate threshold selection as finite-sample risk control for future queries from the same distribution. Our end-to-end implementation on MS MARCO, DBpedia, and FiQA achieves $1.47$–$3.14\times$ speedups over Naive, with substantially larger speedups in single-query latency over PISA, BMP, and Seismic when using 16 threads, while maintaining over 99% mean Top-1000 survival.