MUVERA: Why Query and Document Are Encoded Asymmetrically

The trick that makes MUVERA's fixed vectors approximate Chamfer similarity is that queries and documents are encoded differently. A document has ~100 token vectors, so most buckets are occupied: each bucket takes the mean of its tokens, and any empty bucket is filled from the nearest non-empty bucket (by Hamming distance) — so every possible query token is guaranteed a document-side neighbor to match against. A query has only ~10–30 token vectors, so most buckets are empty: each bucket takes the sum of its tokens, and empty buckets stay zero — an absent query token must contribute nothing. Written out, FDE(q) · FDE(d) = sum over buckets of (sum of query tokens) x (mean of document tokens): summing on the query side adds one term per query token, which is the shape of Chamfer / MaxSim, where each query token is scored against an estimate of its nearest document token. Doc-side fill guarantees that estimate exists; query-side zeros guarantee nothing spurious is added.