The Paton System — Canonical Equations I–X

This paper presents the canonical set of equations defining the Paton System as a minimal structural framework for admissibility, continuation, and system persistence under constraint. The equations formalise the core conditions governing system evolution, including recursive state generation, admissibility filtering, constraint compatibility, and boundary-defined collapse. The Paton Recursive Pressure Field Equation (PRPFE) defines the generative mechanism for candidate structural states, while admissibility conditions determine whether those states can persist. The Constraint-Compatibility Persistence Law (CCSL) and Constraint-Incompatibility Collapse Law (CICL) define the conditions for continuation and termination, respectively. The Unified Datum Line formalises observation as a projection of admissible structure, while the admissibility field and viability gradient define the geometric structure of valid system states and their distance to failure boundaries. Constraint conservation establishes the invariant transfer condition across system boundaries. Together, these equations provide a complete, domain-independent formalisation of system behaviour across physical, computational, biological, cognitive, and organisational domains. This work functions as the canonical equation set of the Paton System, unifying previously distributed formulations into a single, structurally consistent framework.