Bounded Total Regret at Critical Smoothness: Noise-Free Optimization on an Interval

We study noise-free sequential optimization on an interval when every function differs from its affine chord on each subinterval by at most a known constant times the interval length. An explicit deterministic dyadic refinement policy has uniformly bounded total regret. The condition is equivalent to order-one Zygmund regularity and covers the full one-dimensional Matérn reproducing kernel Hilbert space with smoothness one, including non-Lipschitz functions. Matching lower bounds give constant-order minimax cumulative regret, while terminal simple regret decreases inversely with the horizon. A spatial gap-packing proof explains this separation. Under arbitrary adaptive evaluation errors of unknown magnitude, the same policy attains the minimax order B + min(epsilon, B)T without knowing the error bound or horizon. A continuous counterexample shows that a one-sided chord certificate alone cannot ensure uniformly bounded total regret. The results concern one dimension and a supplied regularity bound; the higher-dimensional critical case is stated as a conjecture.