In our previous project, ConANN [1], we introduced the first framework to provide formal, distribution-free error guarantees for Inverted File (IVF) search. However, the broader Approximate Nearest Neighbor (ANN) landscape is now dominated by graph-based (HNSW) and hash-based (LSH) architectures. These state-of-the-art indexes lack formal mechanisms to control accuracy, forcing practitioners to rely on manual, heuristic tuning. This research seeks to generalize Conformal Risk Control to these complex structures, providing rigorous False Negative Rate (FNR) guarantees across all major index types without requiring restrictive data assumptions. [1] Horchidan S, Zeiher F, Boström H, Carbone P. ConANN: Conformal Approximate Nearest Neighbor Search. Proceedings of the VLDB Endowment. 2025 Sep 1;19(1):29-42.