When Familiarity Is Not Enough: Geometry and Regret in Dual-Process Memory Routing

Searching again can recover a missing memory, but a router must decide whether to incur the cost before seeing the results. RF-Mem makes this decision from initial similarity summaries, even though Recollection changes the query direction. We show that this mismatch creates an information barrier: corpora with identical complete indexed scores can require opposite optimal actions, with exact Bayes and minimax regret gaps. The separation persists under published Bag semantics across explicit single-beam parameter ranges, including arbitrarily many finite rounds. We trace the missing information to memory–memory geometry and ask which geometric observations can improve the decision before Recollection begins. In a constructed family, even the initial Top-$K$ Gram block is uninformative, yet seed–candidate inner products yield an exact acquisition-cost versus regret tradeoff. Small observation budgets can leave the optimal action unchanged; a cached centroid resolves it through one scalar probe whose precision requirement tightens with candidate count. A continuous Bag model further connects stable retrieval and probabilistic relevance to a learned-probe guarantee under explicit sampling conditions. Together, these results link RF-Mem's routing barrier to the observations, costs, and precision needed to overcome it.