The Balance Field Gleichung: Finite Dynamics and Physical Correspondence in RC Models

The Balance Field Gleichung (BFG) is examined as a specified finite recursive model and as a candidate for physical correspondence. The main mathematical result classifies every trajectory of its autonomous finite event: within an initial dimension-bounded number of attempts it terminates or reaches full persistence. Every infinite successful trajectory then has uniformly doubly exponential load collapse and a convergent formation kernel; only finitely many formation seeds occur. Predictive mediation is characterized relative to fixed observations, while the geometric construction requires an additional, nonunique metric. A grounded three-node RC model supplies explicit voltage states, energy accounting, mediator memory and parameter-identification conditions. On a fixed physical clock, a finite-order amplitude calibration cannot identify passive RC relaxation with the closed BFG load dynamics. Conversely, a specified state-dependent threshold clock exactly realizes the one-mode recursion and admits an energy readout, finite-error bounds and nonaccumulating event times. This implements an assumed recursion rather than independently validating it as a physical law. DC and oscillatory power are distinguished, and an explicit thermal-bath model gives a corrected stochastic energy residual with proved variance and coverage bounds. Analytic proofs and reproducible symbolic and numerical controls accompany the results. The contribution is a conditional mathematical classification and an operational account of possible and excluded physical correspondences; no laboratory confirmation or universal physical law is claimed. This 53-page research preprint includes six figures and an embedded computational supplement containing sources, scripts and verification records.