Reinterpreting Wave Expressions through Phase Structure and Unfolding of the Complex Exponential Function

Modern physics has used the complex exponential function and the wave equation as fundamental frameworks for describing a wide range of physical phenomena. However, it remains difficult to intuitively explain what structural meanings phase and imaginary structure actually carry. This paper takes the structure of the complex exponential function as its starting point and reexamines the rotational structure contained in the wave equation. In particular, it considers whether complex numbers may be interpreted not merely as computational tools, but as structures expressing phase variation and rotational relations. To this end, the relationships among the unit circle, phase, and projection are analyzed step by step, and the rotational structure of the complex exponential function is reinterpreted through a process referred to as unfolding. As a result, this paper discusses the possibility that wave structure may be understood not as a process in which phase is constructed from space and time, but as a process in which time and wave expressions emerge from phase structure. Based on the results obtained from the minimal structure, this paper also examines the possibility that current wave expressions may represent a symmetric whole structure from a particular direction. This perspective may provide a clue for understanding the common structure of various phenomena transmitted through spherical propagation. This work is an independent theoretical interpretation and has not undergone peer review.