Parameterized Dense-Sparse Fusion for Hybrid Retrieval: Tuning a Rank-Score Mix on BEIR SciFact with Qdrant

We study a parameterized hybrid ranker that fuses a dense embedding list and a sparse lexical list. The method has a small, explicit parameter vector: a dense prior $α\in [0,1]$, a score-versus-rank mix $λ\in [0,1]$, an RRF smoothing parameter $κ> 0$, optional list-geometry coefficients that move $α$ per query, and a router margin $τ$ that can turn sparse search off. We grid-search those ranges on SciFact train (809 queries) and freeze the chosen values on SciFact test (300). The tuned rank-score mix ($α= 0.8$, $λ= 0.75$, $κ= 20$) reaches 0.753 nDCG@10 and 0.889 recall@10, outperforming dense BGE (0.742 / 0.871) and equal-weight RRF (0.707 nDCG@10) on that test split. A list-conditioned $α$ adds +0.0006 nDCG; a sparse-off router is rejected by the same train split (any $τ$ that skipped approximately 50% of queries lost nDCG). These coefficients are dataset-specific. Equal RRF with the same models does not beat dense on a nine-zip BEIR macro-average (0.479 vs. 0.519 nDCG@10). Repeating the same train-then-freeze sweep independently on all 20 indexed units beats equal RRF on 20/20 and dense on 16/20 (unit-mean nDCG@10 0.467 vs. 0.462 dense vs. 0.420 RRF). Other corpora should reuse the ranges, not a copy of the SciFact point.