Operational Exclusion of Local Second-Order Singularities in a Prime-Induced Distribution

We introduce a prime-induced distribution arising from the logarithmic prime-power spectrum appearing in the lifted Weil explicit formula and investigate its local distributional structure through an operational framework based on translated Hermite–Gaussian probe families. A prime-admissible test space is constructed on which the arithmetic distribution is shown to be a well-defined continuous linear functional. For distributions admitting finite local jet expansions, we establish a forward detection theorem showing that every nonzero local second-order singularity necessarily generates an absolutely continuous component in the probe-center Fourier spectrum. Motivated by this observation, we define the class of prime-type distributions, characterized by purely atomic operational spectra supported on discrete logarithmic frequencies. The principal result proves that prime-type distributions admit no nonzero local secondorder singularity. In particular, the prime-induced distribution possesses no local second-order jet at any finite point. The proof combines the operational detection theorem with the Lebesgue decomposition of finite Radon measures to show that an absolutely continuous spectral contribution cannot be hidden within a purely atomic operational spectrum. The results are formulated entirely within the framework of distribution theory and harmonic analysis. Although motivated by arithmetic distributions arising from the explicit formula, the operational exclusion theorem is independent of any particular application and establishes a rigorous analytical foundation for subsequent investigations of prime-induced distributions.