Stability and collapse phenomena appear across many scientific and computational domains, including physics, computation, organisational systems, and biological processes. These behaviours are typically studied within domain-specific frameworks, leading to the impression that each field possesses distinct mechanisms of stability and failure. This paper introduces a formal structural result within the Paton System framework: the Cross-Domain Stability Isomorphism. Systems are modelled as recursive state transitions governed by domain-specific constraint sets and admissibility conditions. Continuation occurs only when recursive updates remain compatible with governing constraints. The theorem demonstrates that when recursive update structure and admissibility boundaries are preserved under a structure-preserving mapping between domains, continuation and collapse correspond exactly across those domains. Under these conditions, the systems are stability-isomorphic. This result provides a structural explanation for why stability boundaries recur across distinct scientific domains and establishes a formal basis for interpreting stability behaviour through admissibility conditions governing recursive systems.