I present a theoretical framework for open physical systems in which persistence, nonequilibrium structure, and weak-field gravity are linked within a covariant scalar-tensor model. The construction proceeds in four steps. First, irreversible loss in open systems is represented by a positive semi-definite Universal Selection Operator, and persistence is defined by a repair inequality balancing maintenance against loss. Second, persistent organization is represented by a structural density that obeys a continuity law, so sustained structure requires ongoing energy throughput. Third, admissibility is promoted to a dimensionless spacetime scalar whose field equations recover General Relativity in a stationary limit and produce Yukawa-screened corrections in the weak-field regime. Fourth, the structural density of driven nonequilibrium systems is promoted into the covariant matter sector, generating a nonzero effective trace in the stress-energy tensor even when the underlying electromagnetic sector is traceless. This structural trace sources the admissibility field and yields a modified Poisson equation in which continuously driven, highly constrained systems can, in principle, generate measurable local deviations in effective gravitational acceleration. The paper states the action, derives the field equations, identifies the equilibrium recovery limits, and formulates experimental tests. The theory is falsifiable: if local gravimetric, interferometric, or resonant measurements near high-throughput, high-constraint systems fail to scale with the predicted structural source term after standard electromagnetic, thermal, acoustic, vibrational, and buoyancy backgrounds are removed, the framework is ruled out in its present form. I distinguish clearly between what is derived, what is inferred, and what remains conjectural.