Boundary Inversion Under Forced Continuation

This paper formalises a structural transition occurring at admissibility boundaries within the Paton System. Under admissible propagation, systems produce stable, compressed structure at the datum, allowing continuation through constraint-consistent traversal. However, when propagation is forced beyond admissibility, the same boundary no longer permits continuation and instead produces termination or reactive resolution. This behaviour is defined as boundary inversion: a mode transition in which an identical boundary element switches function from release to refusal under excess propagation pressure. In admissible mode, controlled propagation enables traversal and stabilisation, producing coherent output and continuation. In forced mode, accelerated or unconstrained propagation attempts to cross the boundary without satisfying admissibility conditions, resulting in failure to stabilise and termination or system reaction. The transition occurs when propagation exceeds admissibility thresholds, expressed structurally as a shift from permitted traversal to blocked continuation. Once this threshold is exceeded, continuation cannot remain neutral. The system must resolve excess propagation through breakdown, overload, or reactive response. This framework introduces no new domain-specific mechanisms. It provides a structural interpretation of boundary behaviour under varying propagation regimes, applicable across physical, computational, and complex systems.