Unified Placement of Lorentz and Dirac Structures in the Cn Phase Geometry

Abstract We present a unified geometric framework in which Lorentz and Dirac structures coexist without algebraic conflict by residing on distinct geometric layers of an extended phase manifold. In this construction, the Lorentz metric defines the visible spacetime base, while the Dirac operator emerges as a derived connection on an analytic cyclic fiber governed by the Cₙ algebra with generator jₙ = −1. The resulting fiber-product geometry resolves the classical incompatibility between spin structures and Lorentzian geometry by confining Dirac curvature to hidden analytic directions. This separation preserves the Lorentzian signature of the visible sector while introducing phase–time corrections that manifest as observable effects, including internal curvature, geodesic deformation, and redshift plateaus. The layered construction thus establishes the geometric core of phase-based unification within the Cₙ framework.