Phase Rhythm Function Group for Generative Existence: Topological Structure and Self-Referential Recurrence of ΦA, ΦB, and ΦC

This work introduces the phase rhythm function group PhiA(t, x), PhiB(t, x), and PhiC(t, x) as a mathematical model for generative existence. PhiA is formulated as a finite summation of rhythmic oscillators, PhiB as an integral responsive field, and PhiC as a coupled structure that integrates temporal, spatial, and amplitude–phase functions. We establish their convergence, differentiability, and topological closure, demonstrating that the group forms a self-referential recurrence within the phase cycle. Potential applications range from biological rhythms and molecular binding to gravitational wave interference, while also offering a formal model of existence in mathematics, physics, and philosophy.