Equivalence of synchronization states in the Kuramoto flow: a unified framework

On the fiftieth anniversary of the Kuramoto model, synchronization remains a central paradigm for collective behavior in complex oscillator systems. While phase locking, frequency synchronization, and order-parameter coherence are widely used to characterize synchronization, their precise dynamical relationship has remained unclear, especially in finite-dimensional systems beyond mean-field limits. In this work, we show that, in the fully connected Kuramoto model, full phase locking, phase locking, frequency synchronization, and order-parameter synchronization are dynamically equivalent. Moreover, we demonstrate that the equivalence between phase locking and frequency synchronization is governed by a topology-independent mechanism and persists for generic symmetric coupling structures. These results provide a unified dynamical framework for synchronization and clarify the structural role of the Kuramoto order parameter.