  Gravitational shockwaves are insensitive to higher-curvature corrections in
the action. Recent work found that the OPE coefficients of lowest-twist
multi-stress-tensor operators, computed holographically in a planar black hole
background, are insensitive as well. In this paper, we analyze the relation
between these two limits. We explicitly evaluate the two-point function on a
shockwave background to all orders in a large central charge expansion. In the
geodesic limit, we find that the ANEC exponentiates in the multi-stress-tensor
sector. To compare with the black hole limit, we obtain a recursion relation
for the lowest-twist products of two stress tensors in a spherical black hole
background, letting us efficiently compute their OPE coefficients and prove
their insensitivity to higher curvature terms. After resumming the lowest-twist
stress-tensors and analytically continuing their contributions to the Regge
limit, we find a perfect agreement with the shockwave computation. We also
discuss the role of double-trace operators, global degenerate states, and
multi-stress-tensor conformal blocks. These holographic results suggest the
existence of a larger universal structure in higher-dimensional CFTs.
