Unified Four-Dimensional Mathematics (V6 Series) Foundational Mathematical System for Cosmic Closure

This paper establishes the complete, unified, and logically closed four-dimensional mathematical system (V6 Series) on the foundational basis of the critique and reconstruction of zero. It strictly follows four foundational axioms: Fundamental Unity, Four-Dimensional Closure, Algebraic Closure, and Unique Valuation. Primordial 1 generates four spiral motions: left rotation, right rotation, expansion, contraction, which correspond to addition, subtraction, multiplication, and division. Left-right spirals engage in occlusion, self-rotation, and resultant force; up-down spirals govern expansion and contraction to stabilize the four-dimensional unit ₄F. The system satisfies commutativity, associativity, distributivity, identity elements, and null elements. This system unifies definitions, operations, sets, structures, phases, coupling, cycles, and geometric expressions into one self-consistent framework. It is a pure mathematical system that does not depend on physical hypotheses and serves as the necessary and sufficient mathematical foundation for the Pythagorean Primary Substance System V14.0.