Bass martingales need not stretch

In classical optimal transport, the equivalence of optimality, convex gradient structure, and cyclical monotonicity is a central paradigm. Known counterexamples to this equivalence rely on carefully constructed costs. In contrast, we show that this equivalence can fail in martingale transport in dimension d ≥ 2: already for the canonical martingale Benamou-Brenier problem and irreducible marginals we have the strict inclusions \[ optimality \; \; (Bass) convex gradient structure \;\; martingale cyclical monotonicity\, . \] However, under the stronger assumption of strict irreducibility, we establish that these three notions coincide. Understanding when the gradient structure guarantees optimality is also of practical relevance, since algorithms based on the Bass gradient representation are used in financial model calibration. All our counterexamples are based on the following phenomenon: for d≥2, there exist two one-step martingales $X,Y$ with the same initial and terminal laws, and a convex function $ψ$ such that \[ 0= E[ E[ψ(Y_1) Y_0]-ψ(Y_0)] < E[ E[ψ(X_1) X_0]-ψ(X_0)] <∞. \]