  We study four-point functions of scalars, conserved currents, and stress
tensors in a conformal field theory, generated by a local contact term in the
bulk dual description, in two different causal configurations. The first of
these is the standard Regge configuration in which the chaos bound applies. The
second is the `causally scattering configuration' in which the correlator
develops a bulk point singularity. We find an expression for the coefficient of
the bulk point singularity in terms of the bulk S matrix of the bulk dual
metric, gauge fields and scalars, and use it to determine the Regge scaling of
the correlator on the causally scattering sheet in terms of the Regge growth of
this S matrix. We then demonstrate that the Regge scaling on this sheet is
governed by the same power as in the standard Regge configuration, and so is
constrained by the chaos bound, which turns out to be violated unless the bulk
flat space S matrix grows no faster than $s^2$ in the Regge limit. It follows
that in the context of the AdS/CFT correspondence, the chaos bound applied to
the boundary field theory implies that the S matrices of the dual bulk scalars,
gauge fields, and gravitons obey the Classical Regge Growth (CRG) conjecture.
