  It has long been known that the coarse-grained approximation to the black
hole density of states can be computed using classical Euclidean gravity. In
this work we argue for another entry in the dictionary between Euclidean
gravity and black hole physics, namely that Euclidean wormholes describe a
coarse-grained approximation to the energy level statistics of black hole
microstates. To do so we use the method of constrained instantons to obtain an
integral representation of wormhole amplitudes in Einstein gravity and in
full-fledged AdS/CFT. These amplitudes are non-perturbative corrections to the
two-boundary problem in AdS quantum gravity. The full amplitude is likely UV
sensitive, dominated by small wormholes, but we show it admits an integral
transformation with a macroscopic, weakly curved saddle-point approximation.
The saddle is the "double cone" geometry of Saad, Shenker, and Stanford, with
fixed moduli. In the boundary description this saddle appears to dominate a
smeared version of the connected two-point function of the black hole density
of states, and suggests level repulsion in the microstate spectrum. Using these
methods we further study Euclidean wormholes in pure Einstein gravity and in
IIB supergravity on Euclidean AdS$_5\times\mathbb{S}^5$. We address the
perturbative stability of these backgrounds and study brane nucleation
instabilities in 10d supergravity. In particular, brane nucleation
instabilities of the Euclidean wormholes are lifted by the analytic
continuation required to obtain the Lorentzian spectral form factor from
gravity. Our results indicate a factorization paradox in AdS/CFT.
