Topological Construction of the Phase Rhythm Field ΦA

This preprint presents the phase function $\Phi_A$ (Phi-A), which defines a complex-valued field over space-time $(t, x)$ through a summation of rotating oscillatory components. Each component is characterized by amplitude $A_k$, angular frequency $\omega_k$, spatial wavenumber $k$, and initial phase $\theta_k$. Under natural conditions on $A_k$ and $\omega_k$, the function exhibits uniform convergence and smooth differentiability. Its image is interpreted as a continuous section of a fiber bundle over $\mathbb{R}^2$ with $S^1$-valued phase fibers.Within the Gyeol–Wave–Rhythm (GWR) theoretical framework, Phi-A serves as the minimal topological configuration from which subsequent rhythm structures Phi-B through Phi-J are hierarchically derived. It encodes the foundational generative structure of rhythm across time and space.Protected under Korean patent application no. 10-2025-0084208. Unauthorized reproduction or implementation is strictly prohibited.