Zeta-Regularization of Spectral Determinants in Selberg Trace Formulae for Multiverse Topologies

Mathematical Applications of Science Fiction We present a rigorous derivation of the spectral determinant for the Laplacian operator on a disjoint union of noncompact hyperbolic manifolds, modeled as a multiverse topology represented by a countable family of quotient spaces of hyperbolic n-space by discrete groups. By extending the Selberg trace formula to include off-diagonal scattering terms arising from inter-topological tunneling, we construct a regularized zeta function that encodes the vacuum energy of the system. We prove the meromorphic continuation of the resolvent kernel across the critical line and establish a functional equation connecting the spectral determinant to the determinant of the scattering matrix. The analysis employs motivic integration and derived stack techniques to describe the moduli space of boundary components. Finally, we demonstrate that the regularized determinant remains invariant under specific deformations of the fundamental domain, providing a topological invariant that characterizes the stability of the multiverse vacuum state.