This paper introduces a novel minimal operator framework designed to model and sustain long-lived, regenerating structure in non-equilibrium media. Rather than relying on conservation laws, Hamiltonian dynamics, or variational principles, the framework leverages a pair of interacting scalar potentials—denoted P(x,t) and E(x,t)—each with internal tri-modal structure. The resulting system is explicitly non-Hamiltonian and exhibits persistent organization through irreversible operator coupling, weak detune asymmetry, and nonlinear feedback. The governing equations combine cyclic generators, asymmetric detune operators, saturating nonlinear exchange terms, and anisotropic transport guided by an emergent structure tensor. Together, these components produce a robust, self-sustaining dynamical regime termed structural persistence—a non-equilibrium steady state (NESS) in which organized patterns endure indefinitely despite diffusive decay and stochastic perturbations. A key result is the identification of the Field Equilibrium Mode (FEM)—a distinguished detune ratio at which persistence metrics are maximized and structural decay is minimized. Both perturbative analysis and numerical simulations show that this ratio aligns with the Golden Ratio φ≈1.618, indicating a deep recursive balance between internal cycling and detune-induced phase precession. The paper formalizes the renuclearization mechanism as a variance-triggered reseeding rule that enables recovery from collapse events, thus preserving the system’s long-term dynamics. Detailed implementation guidance is provided, including the evaluation of structure tensors, transport tensors, and numerical schemes used to simulate the system. Connections are drawn to well-known families of non-equilibrium systems, including complex Ginzburg–Landau models, reaction–diffusion systems, Kuramoto-type oscillator networks, active nematics, and nonlinear optical media. Despite its simplicity, the SPR framework captures a wide range of emergent behaviors—such as tri-axial defect formation, multi-scale self-similarity, and persistent regeneration—within a deterministic, algebraically minimal formulation. This work offers both a rigorous mathematical foundation and a concrete computational model for exploring persistent structures in driven-dissipative systems. It is positioned as a theoretical benchmark and conceptual tool for researchers studying non-equilibrium pattern formation, defect dynamics, and operator-based field architectures. 10.6084/m9.figshare.30627266