Crossing symmetry of OPE statistics

We study the crossing symmetry of the ensemble of large-$c$ 2D CFTs defined through 3D gravity. A central observation is that statistical moments of OPE coefficients are not independent; rather, lower and higher moments are strongly correlated. Using Virasoro TQFT, we clarify how these correlations arise and how they guarantee consistency across OPE channels. Our analysis introduces the new notion of pseudo-hyperbolic manifolds, which are a certain class of non-hyperbolic manifolds whose partition functions are nevertheless related to those of hyperbolic ones. These manifolds serve as bridges that help manifest the crossing symmetry of the CFT ensemble.