Long-running language-model systems accumulate interaction history that outgrows the context window, so they must continually evict. When an eviction policy drops a task-critical detail, for example an access token issued at login or a path the next call needs, the action fails. We present LRE (Learned Relevance Eviction), a kilobyte-scale, CPU-only, language-model-free scorer that learns which units of history are task-critical and keeps them by verbatim extraction. Under a matched-budget comparison, in our experiment, no baseline dominates LRE on the accuracy-cost plane. On agents, LRE recovers 93% of the aggregate accuracy of keeping the entire history (41.1 vs. 44.0) and exceeds it by 27% on the simplest tasks, while requiring zero compressor calls and cutting the worst-case peak prompt by 52%. A controlled study trace shows LRE completes tasks where the others loop, finishing one such task in 37% fewer calls than keeping everything and solving 14 tasks where no other run policy does. On conversational memory, LRE outranks dense and token-pruning encoders at zero neural cost while being 295-1569x smaller in size. In downstream evaluation, LRE gives the best budgeted answer quality on LoCoMo reading 68% fewer tokens. Its supervision can also be annotation-free: training only on the system's own behavior recovers 95% of the supervised scorer's effectiveness. We argue that, because memory eviction in LLM agents is a fidelity problem, it requires a deployable proactive policy where the future query is unavailable and exact state is decisive, and that cheap learned relevance can be sufficient.

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