When Feedback Is Not Reward: Active Search under Pool-Dependent Utility

Active search typically assumes that querying an item reveals the label used by the search objective. In many scientific settings, however, a query returns a local measurement, while whether the queried item counts as a discovery depends on a shared latent pool. We formalize this setting as active search with pool-dependent utility. Under additive terminal utility and equal query costs, the one-step Bayes action ranks candidates by posterior terminal-success probability. With two queries remaining, an exact decomposition shows that lookahead improves the first action only when observation-induced decision value offsets the posterior-utility gap from leaving a higher-ranked candidate. Repeated greedy search can be arbitrarily suboptimal in the worst case, while a weak-posterior-movement condition bounds its loss. On 230 MatPES PBE-to-r2SCAN systems, using complete-pool rather than history-visible membership to define terminal success improves first-query utility by $0.657$ confirmations and budget-curve area by $6.617$. Once the terminal event is fixed, posterior integration provides little additional utility over a deterministic complete-pool score. An anchored two-step rollout selects a different action in $91.7\%$ of systems at $B=6$, but improves terminal utility by only $0.039$ confirmations per system at $46.9\times$ the measured wall-clock cost. These results separate the effect of specifying the terminal event from the value of deeper planning.