Phase Geometry Beyond the Complex Plane: C3 Algebra, Multi–Carrier Trigonometry, and the Foundations of Phase–Time Dynamics

Abstract This book develops a foundational extension of complex analysis and phase geometry based on the cubic algebra C₃, defined by the relation j³ = −1. Although roots of −1 are well known in algebraic contexts, their systematic use as generators of phase geometry, analytic structure, and physical interpretation has remained largely unexplored. Building on the observation that conventional analysis is intrinsically restricted to the binary phase structure of C₂, defined by i² = −1, we construct a coherent mathematical framework in which phase space is naturally decomposed into three sectors. This cubic phase geometry gives rise to multi-carrier exponential functions, generalized trigonometric systems, higher-order differential operators, and a novel interpretation of phase–time dynamics. The framework emphasizes explicit constructions and internal consistency, positioning phase geometry itself as the minimal organizing principle underlying both continuous and discrete physical behavior.