Ultrametric Asymptotics of Anomalous Chronon Decay: Spectral Analysis of p-adic Pseudodifferential Evolution Operators and Non-Archimedean Stochastic Dynamics in Totally Disconnected Local Fields

Mathematical Applications of Science Fiction We present a rigorous spectral analysis of the anomalous decay of discrete time quanta, or chronons, modeled within the non-Archimedean framework of p-adic analysis. By defining the evolution of the chronon field through a Vladimirov–Taibleson pseudodifferential operator acting on the Bruhat–Schwartz function space, we derive the exact heat kernel propagator and establish the existence of ultrametric diffusion modes that defy standard Euclidean behavior. We prove the self-adjointness of the evolution operator on square-integrable function spaces over the p-adic domain and explicitly compute its point spectrum on compact subsets, revealing a hierarchy of decay rates determined by the p-adic valuation. Furthermore, we construct the corresponding stochastic process—a random walk on the fractal tree of the p-adic numbers—and analyze its first-passage time density, showing that chronon decay follows heavy-tailed asymptotics typical of Lévy-type processes. This framework provides the mathematical foundation for quantum gravity models that postulate a discrete, totally disconnected topology of spacetime at the Planck scale.