4D Hierarchical Set, Five-Position Cycle and Global Coordinate System Subtitle: Unified 4D Mathematics System

Within the framework of 4D mathematics from V6.0 to V6.4, this paper establishes a global hierarchical set, an intrinsic five-position cycle, and a three-in-one global 4D coordinate system, achieving full unification and logical closure. A core principle is that dimensions are not externally imposed constructs but emerge and grow organically from existence itself, with four dimensions as the minimal complete and stable spatial structure. Evolutionary sequences are recorded via photon marks. With 1⁰ as the universal latent base set and hierarchical units ₁M, ₁P, ₁A, ₁B, ₁F, ₁D, ₁E covering microscopic, macroscopic, and all intermediate scales, the five-position cycle 1→0→1⁰→0→1 governs structural manifestation, isolation, reset, and regeneration. A torsionally coupled 4D matrix is introduced to characterize spiral extension and vertical convergence. Three mutually consistent coordinate perspectives are integrated into a unified global 4D coordinate observable through future 4D visualization tools. This work completes the logical closure of the 4D mathematics system and provides a purely spatial, intrinsic, and paradigm-free foundation for unifying cosmic structure, quantum phenomena, field dynamics, and fundamental open problems in physics.