Geometric Explanation of the Yang-Mills Mass Gap — A Solution from the Order Parameter Spacetime Theory

The Yang-Mills mass gap problem is one of the Millennium Prize Problems of the Clay Mathematics Institute, requiring a proof that pure quantum Yang-Mills theory possesses a strictly positive mass gap Δ > 0. This paper presents a solution based on the Order Parameter Spacetime Theory: the mass gap is not an independent mathematical proposition to be proved from axioms, but an inevitable consequence of the non-trivial geometric configuration formed by the order-parameter field near the QCD critical point. The logic is as follows: the masslessness of gluons corresponds to the vanishing of the internal space metric along the isometry direction (G_11=0); during the QCD phase transition, as the order-parameter field crosses a Type-B traversable singularity, the isometry direction is broken and G_11 changes from zero to a non-zero minimal value; substituting into the mass formula m = M · G_11 yields m_gap > 0, with its magnitude determined by the curvature scale at the singularity, naturally giving m_gap ∼ Λ_QCD. The Order Parameter Spacetime Theory does not attempt to construct a mathematically rigorous probability measure, but rather demonstrates that if the underlying physical framework is correct, the mass gap emerges as a geometric necessity. If lattice QCD calculations find a massless lowest excitation or an excitation spectrum inconsistent with the flux-tube model, this theory is falsified.