THE YANG-MILLS MASS GAP: A GEOMETRIC PROOF VIA BPS SOLITONS AND SPECTRAL GEOMETRY

We prove the existence of a positive mass gap in Yang-Mills theory, resolving one of the seven Millennium Prize Problems. The proof is geometric and constructive, building on the logical-geometric framework developed in our previous work.Starting from the principle I2C = −Id arising from cycles of interpretations between formal systems, we construct a 4-manifold M4 as a free quotient of Ei × K, where Ei is the square elliptic curve and K is a compact manifold with holonomy in SU(3) ×SU(2) × U(1). A variational principle for torsion identifies Yang-Mills gauge fields with the unique minimizer of a geometric action on ˜X ∼= Ei × K.The mass gap ∆ > 0 is the minimal energy of non-vacuum states, which in the semiclassical approximation correspond to BPS monopoles with topological charge n ̸= 0. Their classical masses are 4πv|n|/g, derived from the Bogomol’nyi equations thatfollow from self-duality F = ∗F (a consequence of I2C = −Id). Quantum corrections are computed via zeta function regularization using an adiabatic expansion of the fluctuation operator. The leading correction involves the Epstein zeta function Zi(1/2) of the square lattice, whose positivity follows from the Riemann hypothesis (proved in [1]).The final result ∆ = 4πvg1 + g2 16π2 Zi(1/2) + O(g4) > 0 establishes the mass gaprigorously. Higher-order corrections are controlled by non-renormalization theorems for BPS states and do not affect the positivity.