  We study stress tensor correlation functions in four-dimensional conformal
field theories with large $N$ and a sparse spectrum. Theories in this class are
expected to have local holographic duals, so effective field theory in anti-de
Sitter suggests that the stress tensor sector should exhibit universal,
gravity-like behavior. At the linearized level, the hallmark of locality in the
emergent geometry is that stress tensor three-point functions $\langle
TTT\rangle$, normally specified by three constants, should approach a universal
structure controlled by a single parameter as the gap to higher spin operators
is increased. We demonstrate this phenomenon by a direct CFT calculation.
Stress tensor exchange, by itself, violates causality and unitarity unless the
three-point functions are carefully tuned, and the unique consistent choice
exactly matches the prediction of Einstein gravity. Under some assumptions
about the other potential contributions, we conclude that this structure is
universal, and in particular, that the anomaly coefficients satisfy $a\approx
c$ as conjectured by Camanho et al. The argument is based on causality of a
four-point function, with kinematics designed to probe bulk locality, and
invokes the chaos bound of Maldacena, Shenker, and Stanford.
