Transitions Are Structurally Constrained: A Unified Framework of Admissibility, Activation, and Transition Regimes in Complex Systems

This work introduces a unified structural framework for transitions in complex systems, demonstrating that transitions are not purely dynamical events, but constrained processes governed by the geometry of admissible states. The framework integrates three distinct but interacting components: (1) structural admissibility, quantified by the admissibility functional Phi, which defines the subset of state space in which transitions are possible, (2) dynamical activation, capturing fluctuation-driven processes that drive the system toward instability, and (3) structural transition regimes (SRR), representing finite-time intervals of coordinated system-wide reorganization during which transitions are physically realized. Within this formulation, transitions occur if and only if all three conditions are simultaneously satisfied: the system resides in an admissible region (Phi >= Phic), undergoes dynamical activation, and enters a structurally coherent transition regime. This establishes a strict separation between three fundamental aspects of transitions: - possibility (structure), - approach (dynamics), - realization (temporal regime). The admissibility functional Phi acts as a state-space filter, empirically separating configurations in which transitions are possible from those in which transitions are structurally forbidden. This leads to the interpretation that dynamical systems evolve within a constrained subset of state space defined by structural compatibility. The framework is validated using multiple independent approaches, including surrogate testing, Monte Carlo simulations across distinct classes of dynamical systems, clustering-based detection of transition regimes, and statistical separation analysis. Results demonstrate strong separation between admissible and non-admissible states, robustness of the admissibility threshold, and the emergence of SRR only in systems exhibiting structural organization and collective dynamics. In contrast, random and weakly structured systems show no admissibility structure and no transition regimes. These findings establish a constraint-based perspective on transitions, in which dynamics alone is insufficient to explain transition occurrence. Instead, transitions are governed by the interaction between structural constraints, dynamical preparation, and temporally extended realization regimes. The proposed framework provides an operational and empirically testable method for identifying where transitions are possible, when systems approach transition conditions, and when transitions are being realized. This work contributes to the broader understanding of complex systems by introducing the concept of a geometry of admissible states, in which the space of possible transitions is not continuous, but structured, constrained, and measurable.