The Price of Scale in Proactive Memory Recall

Yichuan Chen, Xin Wang

A long-lived assistant that brings up memories without being asked must decide at every turn whether anything in its store is worth an interruption. We show that this decision is a multiple-testing problem whose difficulty grows with the store size $N$. On streaming replays of LongMemEval and LoCoMo, a similarity threshold tuned to 5% false alarms at $N=8$ reaches 87-100% at $N=2048$, across embedding models and for explicit and implicit cues; LLM judges show the same growth from a lower base. We prove that a gate with false-alarm rate $α$ operates at a per-item false-positive rate of about $-\ln(1-α)/N$, which pins its hit rate to the far left of the single-comparison ROC curve and, under regular variation, makes it decay as a power of $N$ with an explicit constant. The resulting prediction matches the hit rate of a per-$N$ oracle threshold to within one percentage point. The analysis explains why lexical and semantic channels need thresholds growing like $\ln N$ and $\sqrt{\ln N}$, and why per-cue normalisation roughly doubles the hit rate at matched false alarms. A cue-swap conformal gate, which compares the current cue with the user's other cues on the same store, needs no…