Distribution-independent SQ learning does not imply low dimension complexity

Does distribution-independent statistical-query learning imply low dimension complexity? We give a negative answer with a sharp exponential separation. Classes on 2N³ points admit O_ε(log N) queries of constant tolerance, yet have ordinary, exact probabilistic, and expected-error dimension Θ(N) at fixed approximation error below 1/2. The lower bound persists when the feature law may depend on a target prior and discard any fixed fraction of targets below one. These classes have constant classical SQ dimension, giving an explicit counterexample to a claimed bound of Karchmer and Malach. The construction combines the incidence flips of Hatami, Hatami, Pires, Tao, and Zhao with a rectangle-based learner and a sign-pattern count restricted to incidences. The fixed-tolerance query order is optimal; transcript representations give complementary upper bounds. The learner can be proper, deterministic, and polynomial-time in the explicit table. Version 2, dated 4 October 2026, corrects the personal-communication attribution, simplifies the query bounds, clarifies the randomized proper learner and restriction argument, strengthens the stated tail bound, and updates the bibliography link. The source bundle includes the bibliography audit and build checks.