Multi-Carrier Phase Geometry: A Unified Algebraic and Geometric Framework Beyond Classical Analysis

Abstract This book develops a unified algebraic–geometric framework based on multi-carrier phase algebras Cₙ, with particular emphasis on the C₃, C₄, and C₆ structures. In contrast to classical analysis, which relies on a single continuous phase carrier, the framework introduces multiple phase channels that naturally combine continuous U(1) geometry with discrete Zₙ sectoral dynamics. It provides a coherent mathematical foundation for hybrid smooth–discrete geometries, phase-dependent norms, and generalized differential and integral calculus. As a result, classical incompatibilities—such as circle–polygon dualities, smooth versus cornered structures, and wave–particle separation—are resolved within a single algebraic setting. Beyond pure mathematics, the framework offers a natural language for physical phenomena involving phase transport, sectoral collapse, and multi-channel wave dynamics, with direct implications for quantum mechanics, relativistic structures, and time-potential models.