  We derive constraints on the operator product expansion of two stress tensors
in conformal field theories (CFTs), both generic and holographic. We point out
that in large $N$ CFTs with a large gap to single-trace higher spin operators,
the stress tensor sector is not only universal, but isolated: that is, $\langle
TT{\cal O}\rangle=0$, where ${\cal O}\neq T$ is a single-trace primary. We show
that this follows from a suppression of $\langle TT{\cal O}\rangle$ by powers
of the higher spin gap, $\Delta_{\rm gap}$, dual to the bulk mass scale of
higher spin particles, and explain why $\langle TT{\cal O}\rangle$ is a more
sensitive probe of $\Delta_{\rm gap}$ than $a-c$ in 4d CFTs. This result
implies that, on the level of cubic couplings, the existence of a consistent
truncation to Einstein gravity is a direct consequence of the absence of higher
spins. By proving similar behavior for other couplings $\langle T{\cal
O}_1{\cal O}_2\rangle$ where ${\cal O}_i$ have spin $s_i\leq 2$, we are led to
propose that $1/\Delta_{\rm gap}$ is the CFT "dual" of an AdS derivative in a
classical action. These results are derived by imposing unitarity on mixed
systems of spinning four-point functions in the Regge limit. Using the same
method, but without imposing a large gap, we derive new inequalities on these
three-point couplings that are valid in any CFT. These are generalizations of
the Hofman-Maldacena conformal collider bounds. By combining the collider bound
on $TT$ couplings to spin-2 operators with analyticity properties of CFT data,
we argue that all three tensor structures of $\langle TTT\rangle$ in the
free-field basis are nonzero in interacting CFTs.
