A Variational Derivation of the Prime Number Theorem from Phase Equilibrium

This publication forms Part II of the Riemann–Einstein Phase Bridge framework and continues the operator–theoretic construction introduced in the companion paper “An Operator–Theoretic Representation of the Completed Zeta Function” (Zenodo DOI: 10.5281/zenodo.17295379). In this second part, the Prime Number Theorem is derived as the variational minimum of a coupled Archimedean–Eulerian phase system. The Archimedean phase represents the smooth continuum of arithmetic space, while the Eulerian phase captures discrete prime activations governed by the Chebyshev flow. Minimizing the phase functional under the equilibrium condition PhiA + PhiE = 0 yields the observed prime counting law, where the density of primes follows the classical x / log x scaling. The study introduces a phase–variational interpretation of prime distribution as a self–organized equilibrium between continuous and discrete informational domains. A corresponding dispersion norm defines an energy–based reformulation of the Riemann Hypothesis, linking bounded phase dispersion to the well–known square–root regularity in the Chebyshev and psi functions. A Monte Carlo simulation within a Beurling–Cramér synthetic prime system numerically confirms the predicted scaling of the prime distribution, demonstrating that the same equilibrium law emerges even in stochastic or generalized arithmetic universes. The paper concludes with a geometric–informational interpretation in which prime numbers act as irreducible curvature modes that restore balance in an information–limited continuum. Together with Part I, it establishes a unified operator–variational framework connecting zeta–function theory, information geometry, and arithmetic dynamics.