What does this equation mean?

The fix is hybrid retrieval: combine a lexical score, which is exact-match strong and semantically blind, with a dense score, which is semantically strong and exact-match weak. The complication is that the two scores are not on comparable scales. Pinecone’s documentation states the problem directly: dense vectors scored by inner product against unit-normalized embeddings fall roughly in the range [-1, 1] , while BM25-style sparse scores are unbounded positive values that grow with term frequency, document length, and vocabulary rarity, so that “without explicit weighting, the sparse component dominates the combined score” [ 4 ] . The documented fix is a convex combination of normalized scores, governed by a single weighting parameter: $s_{\mathrm{hybrid}}(d) = \alpha \, \hat{s}_{\mathrm{dense}}(d) + (1-\alpha)\, \hat{s}_{\mathrm{sparse}}(d), \qquad \alpha \in [0, 1]$. where the hats denote scores each rescaled onto a comparable range before combination. At $\alpha$ = 1 the ranking is purely semantic; at $\alpha$ = 0 it collapses to lexical search; $\alpha$ = 0.5 weights the two signals equally.