Finite-Resolution Decision Sufficiency for Linear Optimization

Exact optimal-decision recovery can be stronger than required when decisions are evaluated at a finite regret tolerance. This paper studies finite-resolution sufficiency under partial linear observations and bounded measurement error. A design is sufficient when every noisy observation fiber admits a single feasible decision with regret at most a prescribed tolerance for all costs in that fiber. We prove a complete finite certificate for insufficiency: an indistinguishable hard tuple of costs with no common low-regret decision. For linear-optimization regret, the required tuple size is controlled by projected cost dimension after quotienting directions that are invisible to regret. For finite cost libraries, this certificate gives exact fixed-design verification and separation. Under coordinatewise noise and a finite query library, query design is exactly set cover over hard tuples; this reduction transfers the standard set-cover greedy guarantee and hardness barrier, and supports an exact delayed-constraint method under exact verification and exact restricted-master solves. Structured cases reduce to polynomial formulations through separability, nested one-dimensional coverage, and interval incidence.