Query-Pool Geometry and Bayes Risk in Noisy Affine Prediction

We show that query-pool geometry improves optimal adaptive prediction while preserving the optimal limiting risk of fixed acquisition. We study two regions: one follows a uniformly drawn noisy binary affine rule, and the other has independent fair labels. Let $r$ be the parameter dimension, $0<\eta <1/2$ the noise rate, $\lambda$ the budget ratio, and $c=[1-h_2(\eta ){]}^{-1}$ the recovery cost, where $h_2$ is binary entropy. At budget $\lfloor \lambda r\rfloor$ with $c<\lambda <c+1$, uniform random pools of size $r^2$ give the same optimal limiting Bayes Brier risk for fixed, two-batch, and fully adaptive acquisition, including full-pool inspection. A prescribed subcube with the same pool cardinality preserves the fixed-design limit and lets two batches attain the risk when the affine region is known. For the lower bound, we combine uniform early-evidence control with prediction bounds at outcome-dependent counts. The structured policy uses $o(r)$ verification labels before recovery. We study the verification-continuation tradeoff through exact acquisition studies and paired experiments, and establish the separation between acquisition classes under positive noise through finite-risk inequalities. The implementation is available at https://anonymous.4open.science/r/LATC-74F7.