  Operator growth in spatially local quantum many-body systems defines a
scrambling velocity. We prove that this scrambling velocity bounds the state
dependence of the out-of-time-ordered correlator in local lattice models. We
verify this bound in simulations of the thermal mixed-field Ising spin chain.
For scrambling operators, the butterfly velocity shows a crossover from a
microscopic high temperature value to a distinct value at temperatures below
the energy gap.
