This deposit contains two interconnected mathematical treatises that establishing a complete, non-perturbative constructive framework to solve the Yang-Mills existence and mass gap problem on four-dimensional Euclidean space (R4) for any compact simple gauge group G. "Non-Abelian Wave Interference Geometry" (Core Theoretical Framework) This work introduces a novel paradigm shift, designating the interaction of gauge fields as the geometric interference of non-Abelian matrix-valued wavefunctions. By invoking the de Broglie relation as an intrinsic spatial boundary condition, it establishes a dynamic self-quantization mechanism. It demonstrates that the local energy density functional is strictly semi-positive definite, thereby automatically satisfying the Osterwalder-Schrader reflection positivity axiom without ad-hoc assumptions. Furthermore, it proves that the non-linear master equation prevents ultraviolet divergences by introducing a dynamical geometric cutoff via Apollonian basin dynamics. "A Axiomatic Proof of Yang-Mills and Mass Gap" (Rigorous Spectral and Topological Proof) This work provides the explicit mathematical derivation and spectral analysis finalizing the millennium prize problem. By mapping the continuous spatial deformation to the topological winding number (n) via the third homotopy group pi3 (G), it derives the non-linear quadratic energy scaling law (E = E0 * n²). It evaluates the asymptotic behavior of connected two-point Schwinger functions, demonstrating a Gaussian-type suppression at long distances. This rigorously proves a strict, non-zero lower bound for the first excited state (Delta > 0), establishing the formal mathematical existence of the mass gap. Together, these two papers bridge the historic chasm between axiomatic mathematical physics and the empirical reality of the subatomic world, providing a rigid, unyielding foundation for the continuum limit of gauge field theories.
Ren Matsuoka (Mon,) studied this question.