Geometric Mass Gap in Pure Yang-Mills Theory from the Zhang Effective Metric

The Yang–Mills mass gap is one of the Clay Mathematics Institute Millennium Prize Problems, requiring proof that four-dimensional pure non-Abelian gauge theory has a strictly positive spectral gap. Within the rotational-order geometric unification programme built upon the Zhang Effective Metric, four-dimensional Yang–Mills fields arise after dimensional reduction from a 15D geometry containing a compact internal S³ fibre. The Laplace–Beltrami spectrum on S³ supplies a discrete tower of fluctuation modes. The constant n=0 zero-mode is projected out by global framing constraints of the orientation field n̂ᵃ, leaving a lowest admissible non-zero mode that defines a geometric mass gap m_gap ~ Λ_geo. The scale Λ_geo runs from the GUT scale (fixed in Paper 64) down to the infrared QCD confinement scale, matching lattice determinations of the lightest glueball mass. Gauge invariance remains unbroken, and no fundamental Higgs scalar is introduced. This construction provides a geometric pathway toward the spectral gap of pure Yang–Mills theory.