When Is One Accurate Gradient Query Optimal? Exact Deterministic Bounds under Precision Costs

We study smooth convex minimization using gradient queries with adversarial absolute errors. A query with absolute error tolerance b > 0 costs 1/b. Algorithms must terminate on every legal instance and satisfy a pathwise budget, but no uniform bound on their number of queries is imposed. In every fixed dimension n ≥ 2, we determine the exact minimax function-gap risk of one deterministic query for every positive error tolerance. At most one query is minimax optimal among all deterministic adaptive strategies for every budget 0 ≤ B ≤ 10; the same values hold for randomized strategies when B ≤ 2. For B ≤ 1, no query is needed. The lower bound uses reflected smooth convex functions that admit a common reply at one expensive query while concealing all cheaper queries through a global gradient bound. An explicit rational certificate shows that two steps of ordinary gradient descent strictly outperform the best deterministic at-most-one-query strategy whenever B ≥ 63/5. The stated conclusions remain valid for continuous stateless oracles without a shared continuity modulus. The one-query optimum also survives a fixed per-query startup cost over a shifted budget interval. Restricted error tolerances and other prices give further consequences. One-query optimality for 10 < B < 63/5 and the exact minimax values for B > 10 remain open.