Cross-Domain Structural Isomorphism of Admissibility and Continuation: A Synthesis of Tier-7 Instantiations within the Paton System

A series of domain-specific papers within the Paton System demonstrate that stability, persistence, and collapse across diverse scientific disciplines share a common structural property: continuation occurs only when state updates remain compatible with governing constraints. This paper synthesises those results and shows that across mathematics, physics, computation, biological systems, and organisational systems, system evolution can be expressed as recursive state updates evaluated against admissibility conditions. When admissibility holds, continuation is possible; when it fails, instability or collapse occurs. The analysis demonstrates that these domain-specific results are structurally isomorphic instantiations of a single admissibility–continuation rule. The work clarifies the Paton System as a domain-neutral structural framework describing the minimal conditions required for system membership and continuation across systems.