Constructive Proof of Yang-Mills Mass Gap and Existence via YuanXian Theory YD-T64 Framework

The Yang-Mills mass gap and existence problem is one of the seven Clay Mathematics Institute Millennium Prize Problems. Traditional approaches face severe ultraviolet and infrared divergences in four-dimensional Euclidean space. This work presents a novel constructive proof based on the YuanXian Theory (YXT) YD-T64 framework. By embedding the physical four-dimensional spacetime into a compact 64-dimensional torus \ (T^64\), natural regularization is achieved. Combined with the TCSC involution, which imposes odd parity on gauge fields and freezes zero modes, we rigorously derive a strictly positive mass gap \ (= 1/R > 0\). Numerical simulations of Wilson loops on the four-dimensional projection confirm the area law, verifying quark confinement. Furthermore, the path integral measure is constructed and the Osterwalder-Schrader axioms are formally verified in Lean 4. This approach demonstrates that the mass gap and existence of quantum Yang-Mills theory are topological consequences of \ (T^64\) compactification and TCSC symmetry.