Ultrametric Spectral Asymptotics of the Chronon Decay Operator: Analytic Derivations of Non-Archimedean Dissipation Rates via p-adic Pseudo-Differential Calculus

Mathematical Applications of Science Fiction We present a rigorous derivation of the spectral properties of the Chronon Decay Operator, a pseudo-differential operator acting on the Hilbert space of complex-valued functions over the field of p-adic numbers. Motivated by the hypothesis of discrete time at the Planck scale, we model the decay of temporal quanta, or chronons, as a diffusion process on an ultrametric fractal structure. We explicitly construct the heat kernel associated with a Vladimirov-type operator with a confining potential. Using Hardy–Littlewood–Tauberian methods adapted to local fields, we derive an asymptotic Weyl law for the eigenvalue distribution. We prove that the spectral dimension of the chronon field differs from its topological dimension, indicating a non-Archimedean dissipation rate that mitigates vacuum catastrophe scenarios. The paper concludes with the formulation of an Ultrametric Uncertainty Principle governing temporal localization.