 Uniform finite convex-block criterion for superreflexivity

A Banach space $X$ is superreflexive exactly when, for every $\theta >0$, some $N$ forces every sequence $(x_i{)}_{i=1}^N\subseteq B_X$ to have two convex combinations, one before and one after a cut, at distance below $\theta$. Failure produces a nonreflexive ultrapower; finite representability transfers a violating finite sequence back from any nonreflexive space finitely representable in $X$.