We analyze the uniqueness of extremal conformal field theories under modular invariance and sparsity conditions. While the uniqueness of the partition function can be established, the general existence remains open beyond low levels. We further study the convergence of the Poincaré series representation of the partition function via Rademacher regularization. Finally, we classify non-perturbative handlebody saddles of three-dimensional AdS gravity through modular group orbits. Notes on v23: This manuscript explicitly addresses the analytical obstruction in the large central charge regime. It includes rigorous mathematical annotations regarding the unitarity constraints of Fourier coefficients for extremal CFTs (where $k > 3$) and details the analytical continuation of the regularized Niebur-Poincaré series via Kloosterman sums and modified Bessel functions.