Sub-Quadratic Bisimulation Metrics via Approximate Nearest Neighbors: Coverage-Augmented Guarantees and Computable Two-Sided Certificates

Bisimulation metrics quantify behavioral similarity in Markov decision processes, but their Wasserstein fixed-point operator updates every state pair and incurs quadratic pairwise work. We give a certificate-carrying sub-quadratic method for MDPs with bounded transition support and a useful low-dimensional indexing representation: an approximate-nearest-neighbor index selects the pairs updated by the exact restricted operator, while monotone lower and upper runs enclose the exact metric at every sweep. The main analytical result is a coverage-augmented anytime bound: local index quality alone cannot control global error, because uncovered pairs retain their initialization gap. The limiting error is at most \max(ρ,\eop/(1-γ)), and with exact covered backups the lower arm satisfies |\dann-d|_\infty=ρ. Because ρ depends on the unknown exact metric, the algorithm returns the observable sandwich width instead; agreement of the induced lower and upper clusterings certifies exact recovery of the covered aggregation. A reward-oblivious lower bound shows sub-quadratic index-first coverage cannot remove the coverage term, while a separate adaptive lower bound requires Ω(|\Scal|) pair evaluations. Exact-operator experiments verify the identity and enclosure in every seeded run, and timing experiments recover quadratic versus sub-quadratic scaling under both cheap and full Wasserstein backups. On the grouped |\Scal|=64 benchmark, exact restricted refinement reaches the exact-metric skyline once retrieval covers roughly half of all pairs, while independently trained MICo and DBC baselines stay 22-33\times above that skyline at every retrieval budget. Taxi shows the certificate abstaining under an uninformative embedding, while a 2500-state gridworld improves over a reward-only metric by 28.6% using 12.8% of one quadratic sweep.