  Following recent work on heavy-light correlators in higher-dimensional
conformal field theories (CFTs) with a large central charge $C_T$, we clarify
the properties of stress tensor composite primary operators of minimal twist,
$[T^m]$, using arguments in both CFT and gravity. We provide an efficient proof
that the three-point coupling $\langle \mathcal{O}_L\mathcal{O}_L
[T^m]\rangle$, where $\mathcal{O}_L$ is any light primary operator, is
independent of the purely gravitational action. Next, we consider corrections
to this coupling due to additional interactions in AdS effective field theory
and the corresponding dual CFT. When the CFT contains a non-zero three-point
coupling $\langle TT \mathcal{O}_L\rangle$, the three-point coupling $\langle
\mathcal{O}_L\mathcal{O}_L [T^2]\rangle$ is modified at large $C_T$ if $\langle
TT\mathcal{O}_L \rangle \sim \sqrt{C_T}$. This scaling is obeyed by the
dilaton, by Kaluza-Klein modes of prototypical supergravity compactifications,
and by scalars in stress tensor multiplets of supersymmetric CFTs. Quartic
derivative interactions involving the graviton and the light probe field dual
to $\mathcal{O}_L$ can also modify the minimal-twist couplings; these local
interactions may be generated by integrating out a spin-$\ell \geq 2$ bulk
field at tree level, or any spin $\ell$ at loop level. These results show how
the minimal-twist OPE coefficients can depend on the higher-spin gap scale,
even perturbatively.
