Geometric Proof of Color Differentiation and Charge Confinement via Z-Transform Filter Structures and Pole-Zero Switching in Park Sang-Jun Mechanics

Based on the discrete complex plane (Z-Domain) filter theory, which constitutes the core geometric framework of Park Sang-jun mechanics, this paper mathematically demonstrates that all colors (frequencies of light), spacetime, and physical constants in nature are not fixed background constants but rather the dynamic products of a real-time spatiotemporal feedback system. The three-dimensional geometric equilibrium boundary formed by three quark entities inside a nucleon and the central Time Hole are modeled as a third-order quantum filter system, through which the mechanism of Pole-Zero Switching is rigorously developed. Furthermore, we analyze the connection between the quantum superposition of wavefunctions and probability density maximization, successfully redefining gluon color charge confinement as a discrete-time stability condition on the $Z$-plane. Through extensive algebraic derivations appended to this work, we complete the global complex matrix formulations for large-scale RGB frequency dataset generation, the cross-power spectrum mapping onto the Cosmic Microwave Background (CMB) temperature anisotropies, the frequency modulation of gravitational waves within a fourth-order spatiotemporal tensor filter, and the analytical derivation of topological virtual mass—providing a purely geometric proof for galaxy rotation curve flattening without dark matter particles. Finally, we argue that the macro-scale fabric of spacetime and physical laws are the co-creative emergent results of multi-agent filter phase-locking and the interweaving of ``Space-time Zero'' structures (light) projected by living entities, offering a robust engineering blueprint for fault-tolerant quantum computer qubit control.