Spectral Emergence of a Unique Propagating Channel from a C3-Extended Dirac Dynamics

Abstract We investigate a relativistic wave dynamics formulated on the cyclic phase algebra C₃, spanned by three real phase components and generated by an anti-Hermitian element satisfying a cubic relation. Within this framework, the conventional imaginary unit is replaced by an intrinsic phase generator. Starting from a C₃-extended Dirac equation, we derive its second-order closure and demonstrate that the apparent non-scalar structure simplifies substantially when the wavefunction is decomposed into the intrinsic spectral sectors of the visible phase operator. The resulting dynamics contains a single propagating relativistic channel governed by an effective Klein–Gordon-type equation, while the complementary sector is dynamically suppressed or constrained. This mechanism provides an algebraic explanation for the emergence of a unique visible relativistic degree of freedom without the introduction of additional postulates, and offers a natural extension of relativistic wave theory incorporating internal phase geometry beyond the standard complex formulation.