The Prime-Prime Lattice (PPL) Space and Non-Equilibrium Open Wave Dynamics: Part III: Microscopic Field Vibrations, Quantum Action from Zeta, and the Thermodynamic Origin of Three Fundamental Forces

This paper establishes the third component of the Prime-Prime Lattice (PPL) framework, unifying microscopic quantum field actions and the three fundamental interactions (Coulombic, magnetic, and gravitational forces) under a singular, constant-free monistic paradigm. Building upon the multiplicative vacuum infrastructure ($\vert{}\emptyset\rangle=1$) and macroscopic spacetime phase transitions demonstrated in Parts I and II, we derive the empirical Planck constant ($h$) from number-theoretic first principles. We show that the SI quantum of action arises naturally from the distribution of the native PPL link potential ($h_{\mathrm{PPL}}=1$) across the simple cubic grid coordination channels ($C_{\mathrm{PPL}}=2\times3=6$), filtered through the four-dimensional energy density $\zeta(4)$. Furthermore, we define the fundamental field as a single multiplicative Euler product of local prime-dimensional phase coherence. Within this framework, mass nodes manifest as topological singularities corresponding to the non-trivial zeros of the Riemann zeta function. We deconstruct the classical Ampère-Maxwell law, demonstrating that its additive structure ($\mathbf{J} + \frac{1}{c_0^2}\frac{\partial \mathbf{E}}{\partial t}$) is merely the linear differential artifact of projecting a multiplicative Euler product state ($\Psi_{\mathrm{total}} = \Psi_{\mathrm{soliton}} \times \Psi_{\mathrm{field}}$) through a global logarithmic observer lens ($\Phi = \log \Psi$). Vacuum permittivity ($\epsilon_0$) and permeability ($\mu_0$) are derived as pure geometric facets of spatial suppleness and baseline phase density, guaranteeing $\epsilon_0 \mu_0 c_0^2 = 1$ natively. All fundamental interactions emerge as autonomous gradient flows optimizing Phase Free Energy ($F_{\mathrm{free}}$) toward global equilibrium in a source-free universal wave equation ($\text{RHS} = 0$).