  We discuss spectral correlations in coarse-grained chaotic two-dimensional
CFTs with large central charge. We study a partition function describing the
dense part of the spectrum of primary states in a way that disentangles the
chaotic properties of the spectrum from those which are a consequence of
Virasoro symmetry and modular invariance. We argue that random matrix
universality in the near-extremal limit is an independent feature of each spin
sector separately; this is a non-trivial statement because the exact spectrum
is fully determined by only the spectrum of spin zero primaries and those of a
single non-zero spin ("spectral determinacy"). We then describe an argument
analogous to the one leading to Cardy's formula for the averaged density of
states, but in our case applying it to spectral correlations: assuming
statistical universalities in the near-extremal spectrum in all spin sectors,
we find similar random matrix universality in a large spin regime far from
extremality.
