C6 Phase Geometry and the (3,3) Metric Signature: A Unified Framework for Phase, Curvature, and Lorentz Structure

Abstract We develop a complete algebraic and geometric formalism for the six-phase domain C6, defined by the closure condition j6=−1j^6 = -1j6=−1. The intermediate phase element satisfies (j3)2=−1(j^3)^2 = -1(j3)2=−1, creating a natural internal imaginary axis and yielding a (3,3) signature upon conjugation. This structure decomposes phase space into visible channels (1,j,j2)(1, j, j^2)(1,j,j2) and hidden channels (j3,j4,j5)(j^3, j^4, j^5)(j3,j4,j5), allowing both measurable and latent degrees of freedom to coexist within a unified framework. We introduce the phase-metric tensorgμν=ημν+κ (∂μϕ ∂νϕ)(3,3)g_{\mu\nu} = \eta_{\mu\nu} + \kappa\, (\partial_\mu \phi\, \partial_\nu \phi)_{(3,3)}gμν=ημν+κ(∂μϕ∂νϕ)(3,3),which naturally preserves Lorentz covariance and encodes curvature as a phase-gradient effect. This leads to observable consequences, including an analytic correction to the expansion rate and a predicted redshift feature near z≈0.19z \approx 0.19z≈0.19. The formalism provides a new bridge between Hilbert-space quantum mechanics and Riemannian relativity.