Phase–Riemann Geometry: A Phase-Fiber Extension of General Relativity

Abstract We introduce Phase–Riemann Geometry, a unified framework in which a cyclic analytic phase fiber governed by the algebra Cₙ is coupled to a curved Riemannian (Lorentzian) spacetime. The construction is organized in seven steps. (1) Minkowski spacetime is promoted to a general Lorentzian metric gμν(x)g_{\mu\nu}(x)gμν(x), providing the standard geometric arena of general relativity. (2) The internal phase-time variable τC\tau_CτC, built from the multi-carrier field components on the Cₙ fiber, is reinterpreted as a scalar field on curved spacetime. (3) The induced metric gμν(C)=gμν+ZC∂μτC∂ντCg^{(C)}_{\mu\nu} = g_{\mu\nu} + Z_C \partial_\mu \tau_C \partial_\nu \tau_Cgμν(C)=gμν+ZC∂μτC∂ντC is introduced, showing that the phase fiber deforms the time direction via a rank-one correction that preserves Lorentzian signature. (4) From this deformation we derive an additional, explicitly computable energy–momentum tensor Tμν(C)T^{(C)}_{\mu\nu}Tμν(C), which can be viewed as a phase-induced curvature source analogous to dark energy or scalar–tensor contributions. (5) The Dirac–Cₙ operator is lifted to curved spacetime using tetrads and a spin connection, while keeping the analytic Cₙ connection Γj\Gamma_jΓj intact on the phase fiber. (6) The Einstein equations are then extended to Gμν=8π(Tμν+Tμν(C))G_{\mu\nu} = 8\pi (T_{\mu\nu} + T^{(C)}_{\mu\nu})Gμν=8π(Tμν+Tμν(C)), establishing a consistent coupling between Riemann curvature and phase-induced curvature. (7) Finally, we analyze the C₆ → C₃ collapse and the C→∞C \to \inftyC→∞ limit in curved backgrounds, demonstrating how phase–curvature effects produce redshift plateaus, effective acceleration, and a controlled return to classical general relativity. Together, these steps define Phase–Riemann Geometry as a concrete and testable phase-fiber extension of Einstein gravity.