This work introduces a minimal structural–dynamical mechanism governing transition occurrence in complex systems, formalized as the principle of transition accessibility. The central result demonstrates that transitions are not determined by system dynamics or state-space structure independently, but arise only through their interaction. Specifically, transitions occur only when two conditions are simultaneously satisfied: (i) geometric proximity to regions of elevated instability in reconstructed state space, and (ii) sufficient dynamical activation exceeding a critical threshold. Using embedding-based state-space reconstruction and data-driven analysis, the study shows that each condition alone yields transition probabilities close to baseline levels, whereas their conjunction produces a strong and statistically robust amplification (approximately 10×–15×). This establishes a necessary condition for transition realizability. The framework is validated through multiple independent procedures, including permutation testing, out-of-sample evaluation, embedding invariance, and cross-system validation (Lorenz system, logistic map). Results demonstrate that the identified mechanism is robust, system-independent, and not an artifact of specific modeling choices. Conceptually, the work introduces a constraint-based perspective on complex systems, in which transition realizability is governed by the geometry of accessible regions in state space and their dynamical accessibility. Rather than predicting when transitions will occur, the framework identifies when transitions become possible. This approach extends existing theories of dynamical transitions and early-warning signals by explicitly separating structural accessibility from dynamical realization, and by establishing transition accessibility as a fundamental property of state space.
Josef Piskač (Fri,) studied this question.