Vector search now runs as an operator inside database systems, where a missed neighbor makes a query answer wrong, yet approximate nearest neighbor (ANN) indexes are tuned only for a target average recall. Tuned to an average recall of 0.99, indexes left 5-19% of in-distribution queries and, without a query log, up to 99.6% of out-of-distribution queries below it (more under updates) and changed a top-5-tag aggregate over the results for up to 27% of queries. In the spirit of approximate query processing, we present PQR-Search, a $k$-ANN operator that guarantees $\Pr[\mathrm{Recall}(q) < τ] \le δ$ for every query and, for $τ< 1$, $\mathbb E[\mathrm{Recall}(q)] \ge τ$. The probability is over a random rotation inside the index alone, so no distributional assumption is needed, only that queries and updates do not depend on it, and the guarantee holds after insertions and deletions without retuning. PQR-Search prunes 1-bit codes of rotated IVF residuals with exact error quantiles and predicts its cost without data-fitted constants. On 20 datasets of up to 100M vectors, failure rates stayed at most $δ$ in every audit, including the NeurIPS'23 streaming runbook, and the aggregate changed in at most 0.04% of trials. The guarantee costs 2.5$\times$ the single-query latency of HNSW tuned as above and 1.3$\times$ lower batch throughput (geometric means). PQR-Search was 26$\times$ faster than FAISS exhaustive search in batch, faster than IVF with $n_{\rm probe}$ chosen afterwards to meet the target on every test query, and 27-76$\times$ faster than PUFFINN, which also guarantees per-query recall.

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