Rademacher type and En o type coincide

A nonlinear analogue of the Rademacher type of a Banach space was introduced in classical work of En o. The key feature of En o type is that its definition uses only the metric structure of the Banach space, while the definition of Rademacher type relies on its linear structure. We prove that Rademacher type and En o type coincide, settling a long-standing open problem in Banach space theory. The proof is based on a novel dimension-free analogue of Pisier's inequality on the discrete cube.