Based on the rigorous foundation of the Pythagorean Circular Truncated Cone System V8.8, this paper presents the extended mathematical version V8.82. Within the framework of 4D closed structural mathematics, it realizes the strict unification of ring theory and number theory. This paper inherits primitive ontology, five-element closure, and 45° steady-state double-cone geometry, strictly defines the 4D structural set, operation rules and closure, and proves that the model naturally forms a self-consistent algebraic structure. It fundamentally corrects the underlying defects of traditional mathematics, such as the zero-element paradox, failure of unique factorization, and patchwork expansion of number systems. It also provides a unified structural geometric explanation for several long-standing classic problems, including Hilbert’s 8th, 9th, 10th problems, the Riemann Hypothesis, solvability of Diophantine equations, and factorization reconstruction of general rings. This system is logically closed, non-contradictory, presuppositionless, and free of negative numbers and imaginary numbers, realizing the isomorphism between mathematical self-consistency and physical reality, and providing a unified framework for pure mathematics and theoretical physics.
Zhenmin Wang (Sun,) studied this question.