Closed Forms and Synthetic Twins: Predicting Approximate Nearest Neighbor Recall from Embedding Statistics

Abstract

Embedding models are trained and evaluated as if retrieval were exact; in production they serve behind approximate indexes -- HNSW, IVF, product quantization, or the fixed-dimensional encodings (FDEs) of late-interaction models -- whose behavior the encoder's benchmarks never see: one modern encoder recovers just 14% of its exact top-10 through its raw FDE index. Such failures surface only after an index is built, and the standard patches -- corpus-fitted transforms such as whitening -- must be fitted, stored, and refit as the corpus changes, and can silently rewrite what the encoder returns. This paper shows that index behavior is predictable before anything is built, from label-free statistics of the raw embeddings, through a ladder of instruments matched to what each index family consumes: (1) closed-form moment statistics for the fixed-grid quantizers (PQ, FDE); (2) simulation on a synthetic twin corpus -- cluster statistics made generative, on which any index, composed production systems included, can be built and tested -- for partition indexes; (3) size-extrapolated, lightly calibrated twins for graph indexes at million-document scale. Predictions land within 0.03 of measured recall on an unseen million-document corpus. The same geometry is trainable: targeting the one statistic no post-hoc transform can move -- the score margin -- lifts recall for every index family at once, at a small measured task cost. The result: index choice, correction pricing, and production recall forecast from one cheap measurement pass, on new corpora and new indexes alike; serving without per-corpus transform machinery, suited to continuously changing corpora; and a recall-compute frontier pushed by adapting encoders to geometry rather than coupling them to any single index.