A careful analysis of the sum over topologies in 2d gravity led to an averaged form of the holographic correspondence: pure 2d gravity is dual to a random matrix theory rather than to a single Hamiltonian. In this talk, I will go one dimension higher and study the sum over topologies in 3d gravity and its relation to the statistical interpretation of the boundary theory. I will formulate a statistical version of the conformal bootstrap that organizes universal properties of CFT data, namely typicality and crossing symmetry. I will then identify a set of surgery moves on bulk manifolds that directly reflect these properties. We call the process of generating new topologies via these surgery moves the “gravitational machine.” Starting from a genus-g handlebody, I will show that the machine generates non-handlebodies, produces only hyperbolic manifolds, and does not generate all hyperbolic manifolds. These results indicate a large range of possible choices for which manifolds may be included in the gravitational path integral, reflecting a broad class of ensembles consistent with crossing symmetry and typicality. Based on work with Alexandre Belin, Scott Collier, Lorenz Eberhardt and Boris Post (arXiv:2601.07906).