Critical Circulations Govern Second-Order Complexity under Dynamic Access

First-order sequential-design theory replaces an adaptive policy by its long-run occupation measure. This forgets order. Under dynamic access constraints, where choosing an experiment changes what can be chosen next, order can determine the first unbounded correction to sample complexity. We characterize that correction. The zero-dual-slack edges form a critical graph, whose signed circulations generate a linear space of relative likelihood drifts. With a full-support dual optimum and a strongly connected critical graph, its rank yields a dichotomy: full relative rank permits restoring control and $O(\log b)$ excess, whereas a missing direction with nondegenerate noise forces $\Theta (\sqrt{b})$ excess. The criterion is computable by a nullspace and matrix-rank calculation on the first-order critical graph. An exact identity charges every excess sample either to terminal likelihood imbalance or to leaving the critical face. For a first-order-equivalent pair, we sharpen the two regimes to $V_{\mathrm{Flex}}^{\mathrm{cert}}(b)=rb/d+O(1),$ $V_{\mathrm{Lock}}^{\mathrm{cert}}(b)=rb/d+{\kappa }_r\sqrt{b}+o(\sqrt{b}).$ The gap persists in fixed confidence. A second pair keeps the statewise action menus and critical cycle rank fixed but changes the routing; its second-order cost changes again. Optimal frequencies and cycle count are therefore insufficient. What matters is where critical circulations point in relative likelihood space.