Article 1E — Field Equations of the Circuital Universe

Abstract We present the dynamical field equations of the Circuital Universe, a framework in which the vacuum behaves as an active dielectric medium endowed with causal memory, nonlinear response, and scale-dependent permittivity. Building on the Relative Zero Operator, the Active Vacuum formalism, and the cosmic dielectric interpretation developed in previous work, we construct nonlocal equations of motion that couple matter, geometry, and vacuum polarization via memory kernels. We show how these equations modify the standard Einstein and fluid equations, derive effective Friedmann-like relations, and identify the conditions under which the vacuum saturates into an emergent cosmological constant. We analyze the Renormalization Group (RG) stability of the vacuum–matter couplings and demonstrate how the resulting dynamics provide a theoretical mechanism that could be relevant to the H₀ and S₈ tensions through impedance-matched multiscale evolution. This article provides the phenomenological backbone of the Circuital Universe, connecting operator theory, dielectric vacuum response, and cosmological observations within a single dynamical framework. Keywords Circuital Universe; Active Vacuum; Cosmic Dielectric; Nonlocal Field Equations; Causal Memory; Relative Zero Operator; Renormalization Group; Cosmological Tensions; H_0; S_8.