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Abstract 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, TTO=0 $$ \left\langle TT\mathcal{O}\right\rangle =0 $$, where O≠T $$ \mathcal{O}\ne T $$ is a single-trace primary. We show that this follows from a suppression of TTO $$ \left\langle TT\mathcal{O}\right\rangle $$ by powers of the higher spin gap, Δgap, dual to the bulk mass scale of higher spin particles, and explain why TTO $$ \left\langle TT\mathcal{O}\right\rangle $$ is a more sensitive probe of Δgap 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 TO1O2 $$ \left\langle T{\mathcal{O}}_1{\mathcal{O}}_2\right\rangle $$ where Oi $$ {\mathcal{O}}_i $$ have spin s i ≤ 2, we are led to propose that 1/Δgap 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 〈TTT〉 in the free-field basis are nonzero in interacting CFTs.