Cosmic Topology: A Geometric Unification Theory Based on Noncommutative Geometry (SGCU 7.0 — Complete Rigorous Mathematical Derivation)

Starting from the spectral triple axioms of noncommutative geometry, we establish a complete and rigorous mathematical framework for SGCU (Cosmic Topology) through the non-perturbative regularization of matrix models. On the two-dimensional noncommutative torus, we numerically verify the multi-well potential structure and the existence of point-like solitons induced by noncommutativity using the eigenvalue density iteration method, which constitutes the core mathematical foundation of the theory. Based on the rigorous two-dimensional results and a reasonable generalization of noncommutative K-theory, we derive the universal mass ratio of the three families of particles in four-dimensional space as $m_0 : m_1 : m_2 = 1 : 2.9 \pm 0.4 : 5.4 \pm 0.7$, and systematically construct the complete particle periodic table based on the triple-well potential, covering all known particles (69 species) and all predicted particles (10 species). This paper honestly marks the open problems that cannot currently be closed from first principles, including systematic errors in the dimensional correction factors from two to four dimensions, UV/IR mixing effects, and the open mathematical problem of the complete non-perturbative construction of four-dimensional noncommutative space. The core value of SGCU lies in providing a unified framework where the particle spectrum, three-generation structure, and decoherence mechanism naturally emerge from the microscopic structure of spacetime, while clearly pointing the ultimate fate of the theory to the experimentally testable 47 MeV scalar boson.

 

Keywords: noncommutative geometry, matrix model, SGCU, triple-well potential, particle periodic table, 47 MeV, noncommutative soliton, falsifiability