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A simple probe of chaos and operator growth in many-body quantum systems is the out of time ordered four point function. In a large class of local systems, the effects of chaos in this correlator build up exponentially fast inside the so called butterfly cone. It has been previously observed that the growth of these effects is organized along rays and can be characterized by a velocity dependent Lyapunov exponent. We prove a bound on this exponent that generalizes the chaos bound of Maldacena, Shenker and Stanford. We observe that many systems saturate this bound in a finite size region near the edge of the butterfly cone and the size of this region grows with the coupling. We discuss the connection to conformal Regge theory, where the velocity dependent exponent controls the four point function in an interpolating regime between the Regge and the light cone limit, and relate the aforementioned saturation of our bound to an exchange of dominance between the stress tensor and the pomeron.