Structural Precedence under Irreversibility: A Viability-Theoretic Formulation of Safety in Non-Ideal Systems

This work formalizes the principle of structural precedence in non-ideal systems subject to irreversibility, finite resources, and bounded control authority. The central claim is that robust safety cannot be guaranteed purely through policy, monitoring, or corrective feedback, but must instead arise as a geometric property of the reachable and viable regions of the extended state space. An extended state representation is introduced, incorporating physical state, architectural mode, and information state. Within this space, effective irreversibility is defined relative to resource and control limitations. Under these assumptions, structural safety is shown to be equivalent to viability: safe operation is possible if and only if the system state lies within the viability kernel of the admissible set, or equivalently outside the adversarial backward reachable set of effectively irreversible states. The formulation is connected to Hamilton–Jacobi reachability and viscosity solution theory, enabling the definition of structural margin as the value function distance to inevitability. This margin provides an operational measure of how far a system is from regions where failure becomes unavoidable regardless of control policy. The result is domain-agnostic and applies to cyber-physical systems, autonomous systems, distributed decision systems, and learning systems operating under non-ideal constraints.