Pythagorean Unified System V9.21: Primitive Reconstruction of Number Theory and Ring Theory

Based on the pure mathematical axiomatic framework of the Pythagorean Unified System, this paper carries out primitive, closed and unified reconstruction of traditional number theory and ring theory based on Primitive Element 1, Background Element 1⁰, Coupling Element 0 and primitive operations. The research shows that this system can construct self-consistent, closed and conserved primitive number theory and primitive ring structure without relying on Peano axioms, infinity axioms and traditional abstract algebra presuppositions, and realize the global unification of number theory and ring theory. This paper systematically clarifies the logical limitations of traditional number theory and ring theory, gives primitive number system generation, arithmetic rules, algebraic structure and closure proof, improves the pure mathematical foundation of the Pythagorean Unified System, and provides an original path for the unification of mathematical foundations.