Non-Archimedean Entropy and the Asymptotic Equidistribution of Prime-Valued Orbits in Weakly Interacting Dark Matter Phase Spaces: An Arithmetic-Dynamical Approach

Mathematical Applications of Science Fiction We present a rigorous derivation of the thermodynamic properties of Weakly Interacting Dark Matter(WIDM) modeled on rigid analytic spaces over the field of p-adic complex numbers Cp. By constructing anon-Archimedean Hamiltonian system on the Berkovich spectrum of the Tate algebra, we define a topologicalentropy hp(Φ) compatible with the ultrametric topology. The central contribution of this manuscript isthe proof of the Prime-Orbit Equidistribution Theorem, which asserts that the orbital points of the WIDMphase flow, when mapped to the ring of integers, exhibit asymptotic density in the set of rational primes Pproportional to the inverse of the non-Archimedean entropy. We utilize techniques from p-adic Hodge theory,Coleman integration, and spectral theory of unitary operators on p-adic Hilbert spaces. This frameworkresolves the "Ultrametric Catastrophe" in low-temperature dark matter models and suggests a discrete,number-theoretic substructure to the multiverse vacuum energy.