When a Known Answer Unlocks Search: Split–Localize Control for Structured One-Sided Search

A query whose answer is already known can still be essential for efficient search. We study structured one-sided search in which targets must be explicitly localized before they leave the active query system. On an anchor family, exact one-step information gain and exact discovery gain require $\Theta (n)$ expected queries, while executing a zero-information localization first enables $\Theta (\log n)$ search. This motivates *split-localize control*, whose exact-posterior form admits an $O(H({\mu }_0)+{\bar{k}}_0)$ scored-query bound and a constant-factor total-query guarantee after certification costs are included. We introduce Fractional-SL as a scalable realization based on entropy-regularized action weights. Under fixed nested-prefix geometry, exact optimization, and order-aligned proposal spread $R=\exp (\Delta /\tau )$, Fractional-SL uses $O_{\lambda }(RK\log (en))$ total queries. A one-target family forces Fractional-SL to use $\Omega (K\log (n/K))$ queries, exposing its sensitivity to a loose target-count upper bound. With 10,000 tail candidates, Fractional-SL requires 19.63 queries on average versus 5,004.5 for the exact myopic policies. In informative-score multi-target experiments, enforcing the certified positive-set constraint saves 3.55-7.75 queries on average under a same-selector ablation.