  We introduce a framework for quantifying random matrix behavior of 2d CFTs
and AdS$_3$ quantum gravity. We present a 2d CFT trace formula, precisely
analogous to the Gutzwiller trace formula for chaotic quantum systems, which
originates from the $SL(2,\mathbb{Z})$ spectral decomposition of the Virasoro
primary density of states. An analogy to Berry's diagonal approximation allows
us to extract spectral statistics of individual 2d CFTs by coarse-graining, and
to identify signatures of chaos and random matrix universality. This leads to a
necessary and sufficient condition for a 2d CFT to display a linear ramp in its
coarse-grained spectral form factor. Turning to gravity, AdS$_3$ torus
wormholes are cleanly interpreted as diagonal projections of squared partition
functions of microscopic 2d CFTs. The projection makes use of Hecke operators.
The Cotler-Jensen wormhole of AdS$_3$ pure gravity is shown to be extremal
among wormhole amplitudes: it is the minimal completion of the random matrix
theory correlator compatible with Virasoro symmetry and
$SL(2,\mathbb{Z})$-invariance. We call this MaxRMT: the maximal realization of
random matrix universality consistent with the necessary symmetries.
Completeness of the $SL(2,\mathbb{Z})$ spectral decomposition as a trace
formula allows us to factorize the Cotler-Jensen wormhole, extracting the
microscopic object $Z_{\rm RMT}(\tau)$ from the coarse-grained product. This
captures details of the spectrum of BTZ black hole microstates. $Z_{\rm
RMT}(\tau)$ may be interpreted as an AdS$_3$ half-wormhole. We discuss its
implications for the dual CFT and modular bootstrap at large central charge.
