Pythagorean Unified System: Axiomatic Closure, Four-Dimensional Operations, and the Interrelation of Geometry, Algebra, and Physics

This paper establishes a self-contained, self-driven, and self-conserving axiomatic system with the primitive unit 1 as the sole ontology. Based on the origin of logic, the system regards logic as the existential law and first principle of the cosmos, rather than a mere tool. The concept of 0 is redefined as a coupling interface, structural gap, and reset node. With β₁ as the global endogenous dynamics, a unified four-dimensional operation system and core formula set are constructed, providing reasonable explanations for relevant problems in geometry, algebra, and physics. The universe originates from the four-dimensional synchronous structural growth of the singularity, identified as the primitive unit 1. Explosions are local cosmic phenomena and do not create the universe or its fundamental origin. The four-dimensional meta-mother universe possesses a near-45° steady-state structure and is highly stable against explosion, while daughter universes may explode and contain singularity fragments. All symbols, operators, and formulas are uniquely fixed. The system is self-consistent, complete, and falsifiable.