Pythagorean Round Platform System: Paradigm Innovation and Closed-Form Solution Theory for Jacobi and Gauss-Seidel Iteration Methods

The traditional Jacobi iteration method and Gauss-Seidel iteration method are mainstream numerical methods for solving linear systems. Their core relies on matrix splitting and successive recursive approximation, which inherently suffer from strict convergence conditions, accumulated iteration errors, low efficiency in high-dimensional computation, and divergence-prone behavior for ill-conditioned matrices. Based on the primitive geometric framework of the Pythagorean Round Platform System, this paper takes the principle that uncertainty guarantees certainty as its core axiom, introduces the global normalization operator Π₁ and the certainty-uncertainty unified eigenoperator β₁, establishes a strict symbolic convention and operator operation system, and constructs a dedicated closed-form solution model for linear systems. The results show that the Pythagorean Round Platform System requires no iterative loops, is not restricted by spectral radius or diagonal dominance, and directly yields an exact analytical solution through operator paired coupling.