Solution to the Yang–Mills Mass Gap Problem Based on the Ultimate Background Wave Field Theory

Aiming at the millennium problem of Yang–Mills mass gap in non-Abelian gauge field theory, as well as inherent defects including infrared divergence, ambiguous mass generation mechanism and redundant theoretical assumptions in traditional theories, this paper takes the Ultimate Background Wave Field Unified Theory as the theoretical basis, and constructs a complete modified Yang–Mills gauge field dynamics system through strict field theory coupling and variational derivation relying on the axiom system and core field equations of the theory. Starting from the total Lagrangian density of the background wave field, this paper derives the effective Lagrangian of the coupling between gauge field and background field step by step, introduces the covariant derivative modified by background field density gradient, completes the complete mathematical proof of non-Abelian gauge invariance, gives the comparison formula of gauge field propagators before and after modification, and strictly verifies the dimensions of coupling parameters. Without introducing additional Higgs scalar field or unknown new elementary particles, it realizes the natural generation of gauge field dynamic mass, completely eliminates low-energy infrared divergence mathematically, and analyzes the physical essence of Yang–Mills mass gap. The full text has a closed logical derivation, all field equations meet four-dimensional general covariance, energy-momentum conservation and high-energy renormalization requirements, and the numerical constraints of coupling parameters are completed combined with particle physics experimental data. It has rigorous falsifiability and original academic value, provides a new systematic solution to the basic problems of gauge field theory, and further improves the framework of quantum gravity unified theory.