A no-go theorem for large $N$ closed universes

Under conservative assumptions, it is established at a mathematical level of rigor that if correlation functions of single trace operators in a sequence of states of $O(N^0)$ energy of a two-sided holographic conformal field theory admit a large $N$ limit, then they must be described a pure state in the large $N$ Hilbert space of free field theory on two copies of vacuum AdS. This result clarifies recent discussions concerning the possible emergence of a semiclassical baby universe in the large $N$ limit of a low-energy partially entangled thermal state. The proof heavily relies on dominated convergence arguments from mathematical analysis. Some comments are also offered on the relationship between various recently proposed ways of restoring the emergence of the baby universe and the relaxation of different assumptions of the theorem.