The Kuramoto model describes phase oscillators on the unit circle whose interactions are encoded by a graph. Each edge acts like a spring that pulls the two adjacent oscillators toward each other whenever their phases differ. A central question is to determine which graphs are globally synchronizing, meaning that trajectories of the Kuramoto dynamics converge to the fully synchronized state from almost all initial conditions. This property is tightly linked to the benign nonconvexity of the model's energy landscape. Existing guarantees for global synchronization rely on minimum-degree thresholds, which require the graph to be highly dense. In this work, we show that connected threshold graphs, whose density may vary from $2/n$ to $1$, are globally synchronizing. Our proof relies on a phasor-geometric analysis of the stationary points of the associated energy landscape.
