  We study the stress tensor multiplet four-point function in the 6d maximally
supersymmetric $(2,0)$ $A_{N-1}$ and $D_N$ theories, which have no Lagrangian
description, but in the large $N$ limit are holographically dual to weakly
coupled M-theory on $AdS_7\times S^4$ and $AdS_7\times S^4/\mathbb{Z}_2$,
respectively. We use the analytic bootstrap to compute the 1-loop correction to
this holographic correlator coming from Witten diagrams with supergravity $R$
and the first higher derivative correction $R^4$ vertices, which is the first
1-loop correction computed for a non-Lagrangian theory. We then take the flat
space limit and find precise agreement with the corresponding terms in the 11d
M-theory S-matrix, some of which we compute for the first time using
two-particle unitarity cuts.
