Synchronizability criteria based on higher-order cycles

Determine whether a synchronizability condition for higher-order group synchronization can be formulated solely in terms of the consistency of higher-order cycles, rather than the consistency of 1-cycles.

Background

The paper proves that a hyperedge potential is synchronizable if and only if all 1-cycles are consistent. It also observes that consistency of 1-cycles implies consistency of higher-order cycles, but does not establish whether consistency of higher-order cycles alone can provide an alternative characterization. Resolving this question would clarify the role of higher-order cycle structures in the mathematical foundations of synchronization.