Paton Recursive Pressure Field Equation (PRPFE): A Minimal Structural Operator for Admissible System Evolution

This paper introduces the Paton Recursive Pressure Field Equation (PRPFE) as a minimal structural operator governing the evolution of admissible systems. The equation does not replace existing scientific theories, but provides a structural form describing how systems evolve once admissibility conditions are satisfied. The framework identifies three core elements present across domains: recursive dependence on prior states, constraint-driven modulation through a pressure coefficient, and tolerance-dependent stability. These elements appear in physical, biological, cognitive, and engineered systems as shared structural patterns. Within the Paton System, the PRPFE is positioned as a Tier-5 continuation operator. It operates only after admissibility conditions are satisfied (Tier-3) and prior to constraint field analysis (Tier-6). The equation does not introduce new physical mechanisms or unify domain-specific laws, but clarifies a common structural pattern underlying system evolution. This work establishes the PRPFE as a cross-domain structural operator suitable for further empirical and theoretical investigation.