Imaginary Geometry and Multi-Carrier Phase Dynamics in the Cyclic Algebra Cn

Abstract We introduce a unified theoretical framework in which the imaginary axis of the complex field is generalized into a multi-dimensional imaginary geometry generated by the cyclic algebra Cₙ with jn=−1j^n = -1jn=−1. Each imaginary direction jkj^kjk acts as a distinct phase carrier, forming a multilayered phase manifold that extends the traditional complex plane. This structure provides a natural analytic basis for internal degrees of freedom, hidden/visible norm sectors, curvature-induced temporal dilation, and discrete phase-order transitions such as C₆ → C₃. We show that the imaginary geometry of Cₙ leaves observable signatures through projection, producing discontinuities in curvature, redshift plateaus, and multi-mode interference structures. The resulting phase–geometric model offers a compact yet powerful extension of both Dirac dynamics and Lorentz geometry.