We introduce a framework for quantifying random matrix behavior of 2d CFTs and AdS_3 quantum gravity, that is manifestly compatible with the constraints of conformal and modular invariance. We explain what it means to isolate the chaotic part of the Virasoro primary spectrum of a given theory. This leads to a 2d CFT trace formula, analogous to the Gutzwiller trace formula for chaotic quantum systems, which allows us to identify conditions for random matrix universality in 2d CFT. We use this to understand the off-shell Cotler-Jensen torus wormhole of AdS_3 pure gravity from a purely microscopic CFT point of view. This wormhole is shown to be extremal, a minimal completion of the random matrix theory two-point correlator compatible with the symmetries. By factorizing this wormhole, we determine a new piece of the AdS_3 pure gravity path integral with a single torus boundary; this contains fine-grained spectral data of the black hole microstates of pure gravity.