Curvature, Phase, and Interaction: A Variational Framework from a φ-Metric Green's Function

We present a unified amplitude–phase framework in which mass, charge, and interaction laws emerge from a single variational principle defined over a prime-indexed manifold. A field-induced φ-metric generates a Laplace–Beltrami operator whose Green’s function determines the interaction structure.
In this setting, inverse-square, Yukawa, and confinement regimes arise as distinct limits of a common propagation operator, without the introduction of external gauge fields.
We further demonstrate that the energy functional acts as a Lyapunov functional under coarse-grained dynamics, yielding entropy production and an emergent arrow of time. We show that Z5 symmetry and the golden ratio φare mathematically dual through cyclotomic structure: either can be derived from the other. The framework provides a geometric origin for dynamics, interactions, and irreversibility through curvature and phase structure, recovering conformal scalar gravity (rather than full tensorial general relativity) as
a long-wavelength limit.