Spectral Decomposition and Real-Plane Projection in the Cyclic Phase Algebra C6

Abstract We develop a spectral decomposition framework for the cyclic phase algebra C₆, focusing on the emergence of visible and hidden phase channels and their reduction to a real-valued norm. By identifying intrinsic idempotent structures and a natural C₂ subalgebra generated by the cubic phase element, we show that the algebra admits a canonical projection onto a real two-dimensional plane in which the norm becomes strictly positive and scalar. This construction provides a geometric foundation for probability-like quantities and clarifies the role of spectral separation in multi-phase quantum models.