Unified Phase–Riemann Dynamics: A Combined Framework for Spacetime Curvature, Phase-Time Geometry, and Cn-Extended Dirac Fields

Abstract We introduce a unified geometric framework in which spacetime curvature, phase–time dynamics, and Cₙ-extended Dirac fields coexist within a single coupled system of equations. The construction is based on a phase-extended metric that augments general relativity with a scalar phase–time field encoding the internal analytic structure of Cₙ carriers. The resulting phase-induced energy–momentum tensor acts as a geometric source of curvature without introducing new degrees of freedom into the spacetime metric. We derive the full set of governing equations, including Einstein equations with phase-curvature sources, the dynamical equation for the phase–time field, the curved-space Dirac–Cₙ equation, a multi-layer energy spectrum incorporating carrier structure, and the associated phase-induced redshift relation. Together, these results define the core of Unified Phase–Riemann Dynamics, a phase-based extension of relativistic field theory with direct observational implications, such as redshift plateaus, spectral carrier loss, and characteristic C₆→C₃ collapse signatures.