We derive a new general jump condition on a broken Weierstrass–Erdmann extremal of a vectorial variational problem. Such extremals, containing surfaces of gradient discontinuity, are ubiquitous in shape optimization and in the theory of elastic phase transformations. The new condition, which does not have a one dimensional analog, reflects the stationarity of the singular surface with respect to two-scale variations that are nontrivial generalizations of Weierstrass needles. The over-determinacy of the ensuing free boundary problem suggests that typical stable solutions must involve microstructures or chattering controls.