Structural Non-Implication, Threshold Separation, and Forward Invariance in Hybrid Dynamical Systems

This paper develops a hybrid dynamical framework for distinguishing structural non-implication, threshold separation, and forward invariance in constrained systems. First, constructive examples show that constraint–projection data and topological redundancy do not uniquely determine each other in either direction, and that the dimension of an allowed perturbation space does not determine its orientation, when no additional coupling axioms are imposed. Second, explicitly constructed hybrid systems demonstrate that discrete structural events and continuous safety failures can occur in different orders, and that compensation can change which safety margin is the first to reach zero within a specified model. Third, the paper formulates a conditional forward-invariance criterion based on continuous-flow viability, reset-map safety, and the treatment of finite-time event accumulation, under stated regularity and existence assumptions. These results establish logical distinctions and constructive possibilities within the specified mathematical models. They do not establish universal independence of structural quantities, empirical predictions, or forward invariance for arbitrary hybrid systems. Application to a concrete system requires operational definitions of its state variables, safety sets, event guards, reset maps, and admissible control paths. The framework does not provide empirical calibration or historical judgments.