Prefix caching is a key latency optimization for autoregressive LLM serving, yet existing systems assume dense per-token key/value reuse. State-space models change the structure of the problem: a recurrent layer can resume from a single stored state rather than requiring the entire token history. This asymmetry opens a new design point between no reuse and dense caching: store exact recurrent states at a sparse set of checkpoint positions and, on a cache hit, resume from the deepest stored checkpoint and recompute the remaining suffix exactly. We formalize sparse prefix caching as checkpoint placement under a distribution over overlap depths, yielding an exact O(NM) dynamic program. For use cases where requests share a non-trivial prefix (e.g. asking different questions about a single long document), we show that our method consistently improves the Pareto frontier traced by standard heuristics on real-world data. Across QuALITY and System Prompts, distribution-aware placement dominates every fixed-budget baseline on the measured layer-group Pareto frontier and matches or outperforms the strongest heuristic (block caching) while typically using substantially fewer checkpoints, with the largest gains at low checkpoint budgets where the overlap distribution is most non-uniform. The method is most relevant when many requests share a substantial but not identical prefix within a retained cache entry. It preserves exact outputs, does not change the recurrent computation itself or require new recurrent update kernels, applies to recurrent/SSM layers whose hidden state can be extracted and restored exactly, and for hybrid models can be combined with existing KV-cache compression techniques.