Expert-Query Complexity under Linear State-Value Realizability

A short decision horizon can reduce the expert advice needed to learn with a high-dimensional value representation. We study deterministic experts whose state-value functions are exactly linear in known features. The learner observes rewards and has restart and one-step reset access. Ordinary calls cover training and one expert-free deployment, including paid lookahead and fresh committed transitions. We give an algorithm with O((1 + B²H²/ε²) log(4B/ε)) expert queries and polynomial ordinary cost, where B bounds the expert's parameter norm. This expert budget is independent of feature dimension. A constant-norm task-library construction gives a lower bound of q ≥ c min{d/log²(L+2), H²/(ε² log⁴(L+2))} expected expert queries for a prescribed ordinary allowance L, with logarithmic action capacity and explicit dimension and horizon thresholds. Together with DELPHI, the bounds match in their leading dimension, horizon, and accuracy powers under a common polynomial ordinary budget. The upper algorithm turns progress toward a value target into a bound on expert calls. The lower construction separates the number of secrets stored in the representation from the number of tasks visited in an episode; scaling rewards and task visits together preserves constant norms. These results identify when expert-query complexity follows the horizon-to-accuracy scale and when dimension determines its leading power. Version 0.1. Submitted to ALT 2027 on 28 September 2026. This deposit contains the same PDF as the conference submission.