Researchers have made a breakthrough in quantum physics, demonstrating that a spectrally isolated quartet can maintain a local two-qubit description while acquiring a loop holonomy that exceeds local subgroup limitations. This phenomenon is observed in three distinct topological settings: a BHZ ribbon, a spinful SSH chain, and a BBH corner quartet. By modifying only the loop, the transport between algebraic points is altered, showcasing the quartet's unique properties. The discovery has significant implications for quantum computing and information processing, as it reveals new possibilities for manipulating quantum states in localized systems1. This finding matters to practitioners because it opens up new avenues for exploring topological quantum computing and potentially leads to the development of more robust and efficient quantum information processing systems.