Equations as Structural Roles: A Unifying Interpretation within the Paton System

This paper unifies a series of structural placement analyses within the Paton System by demonstrating that established physical equations occupy consistent structural roles rather than independently defining reality. By mapping equations such as the Schrödinger equation, Einstein field equations, Navier–Stokes equations, and related operators onto admissibility, recursion, continuation, projection, constraint, and stability, a coherent framework emerges. Within this interpretation, physical equations are not treated as isolated descriptions of reality, but as domain-specific expressions of underlying structural roles. This work does not modify any physical laws, but clarifies their placement within a shared admissibility-based architecture spanning Tier-3 to Tier-6 of the Paton System.