Ultrametric Analytic Continuation of Chronon Decay Rates: Spectral Asymptotics of p-adic Pseudo-Differential Operators in Non-Archimedean Temporal Domains

Mathematical Applications of Science Fiction We present a rigorous derivation of the spectral asymptotics for a class of pseudo-differential operators acting on the Hilbert space of complex-valued functions over the field of p-adic numbers, modeling the decay of quantized time units, or chronons, within a non-Archimedean temporal manifold. By replacing the continuum hypothesis of classical mechanics with a p-adic topology, we resolve the singularities that arise at the Planck-scale limit of chronon interactions. We construct the Vladimirov–Beleren operator and prove its self-adjointness and discrete spectrum on compact subsets of the p-adic field. Furthermore, we establish the analytic continuation of the associated spectral zeta function and derive an explicit formula for the chronon decay rate in terms of the residues of this function at its poles. Our results show that the stability of the temporal lattice depends on the prime characteristic of the underlying field, providing a novel resolution to the vacuum catastrophe through ultrametric renormalization.