arXiv:2512.14729v1 Announce Type: new
Abstract: Collective synchronization in complex systems arises from the interplay between topology and dynamics, yet how to design and control such patterns in higher-order networks remains unclear. Here we show that a Dirac spectral programming framework enables programmable topological cluster synchronization on directed hypergraphs. By encoding tail-head hyperedges into a topological Dirac operator and introducing a tunable mass term, we obtain a spectrum whose isolated eigenvalues correspond to distinct synchronization clusters defined jointly on nodes and hyperedges. Selecting a target eigenvalue allows the system to self-organize toward the associated cluster state without modifying the underlying hypergraph structure. Simulations on directed-hypergraph block models and empirical systems–including higher-order contact networks and the ABIDE functional brain network–confirm that spectral selection alone determines the accessible synchronization patterns. Our results establish a general and interpretable route for controlling collective dynamics in directed higher-order systems.