Two-Layer C6 Geometry, Dirac Coupling, and the Emergence of Space–Time from Phase Topology

Abstract We propose a two-layer geometric framework in which the phase algebra C6 possesses a dual structural role: an internal cyclic phase topology defined by j6=−1j^6 = -1j6=−1, and an external six-dimensional Lorentz-type metric space with signature (+,+,+,−,−,−)(+,+,+,-,-,-)(+,+,+,−,−,−). These two structures, often conflated in algebraic formulations, are shown here to be categorically distinct yet dynamically coupled. The internal C6 governs phase channels, analytic continuation, and the intrinsic curvature encoded by Apéry’s constant ζ(3)\zeta(3)ζ(3), while the external C6 provides the visible/hidden decomposition that defines a Lorentzian measurement geometry. The Dirac operator is reconstructed as an intermediate layer,DC6=γk∂akjkD_{C6} = \gamma_k \partial_{a_k} j_kDC6=γk∂akjk,which couples external metric derivatives to internal phase factors without mixing the corresponding topologies. This establishes a consistent mechanism through which phase topology projects into metric geometry, leading to an emergent description of three-dimensional space, a hidden analytic time potential, and a unified norm structure∥z∥2=∥zvis∥2−∥zhid∥2\|z\|^2 = \|z_{\mathrm{vis}}\|^2 - \|z_{\mathrm{hid}}\|^2∥z∥2=∥zvis∥2−∥zhid∥2. The resulting model retains both local and global Lorentz consistency while allowing nontrivial analytic curvature effects, including a natural interpretation of redshift offsets near z≃0.19z \simeq 0.19z≃0.19 and internal phase strain governed by ζ(3)\zeta(3)ζ(3). This two-layer C6 framework thus provides a mathematically coherent and physically motivated map from cyclic phase topology to Lorentzian spacetime geometry.