Existence and Universality of the Yang–Mills Mass Gap in Four Dimensions (Constructive Resolution Inspired by the Law of Frequency Gradient)

We construct a mathematically rigorous Euclidean quantum field theory for the non-Abelian SU(2) Yang–Mills model in four dimensions, satisfying all Osterwalder–Schrader axioms (  ) and exhibiting a strictly positive spectral gap    
The proof combines uniform cluster expansion, large-field suppression, and renormalization-group monotonicity, ensuring existence, reflection positivity, and exponential decay of correlations.

All constants are computed explicitly, ensuring independence from lattice spacing  a.
The result establishes:

thereby confirming the mass-gap condition for compact simple  Lie  groups.