Risk Identification and Evidence Complexity for Verifier-Guided Selection

A fixed verifier-guided selector can have different risks under routing laws with identical low-order statistics. We study the cost of estimating its worst compatible risk when each paid query restores a retained state and returns a fresh bounded loss. The feasible domain contains all nonnegative routing laws with the specified moments. For top-$b$ selection and moments through order $r$, the worst-case record count scales as $k^b$ while the unseen mass is bounded by $O(k^{-d})$, where $d=\lfloor r/2\rfloor$. We construct one positive law with prescribed heavy records at every geometric scale, uniformly in the number of scales and bounded product biases. The resulting acquisition rates match away from $b=2d$ and at critical breadth when $\log (1/\delta )$ dominates $\Lambda =1+\log (1/ϵ)$. At fixed confidence, the general critical bounds remain $\Omega ({ϵ}^{-2}{\Lambda }^2)$ and $O({ϵ}^{-2}{\Lambda }^3)$; the balanced top-two lower bound improves by a factor $\log \Lambda$. The critical gap separates a scalar-output lower bound from an upper algorithm that learns a near-optimal law; a specified mean family admits cheaper scalar estimation at fixed confidence. We also characterize the full space of exactly identified losses. A supplied distance bound to this space reduces the effective query depth even when the approximating loss is unknown, under the same paid observation interface.