We resolve the Clay Millennium Prize problem on Yang-Mills existence and mass gap. Quantum Cell Theory (QCT) is a constructive framework derived from three postulates: discrete energy cells, energy-exchange resizing, and unit-speed light propagation. The fundamental particle is a Hopf soliton on the type-IV5 bounded symmetric domain D5 = SO0(5, 2)/(SO(5) × SO(2)). We prove that the continuum limit of the QCT path integral on D5 is four-dimensional SO(5) Yang-Mills theory with derived coupling g2 = α/14, satisfying all Wightman axioms with strictly positive mass gap Δ = me. The gap equals the electron mass by structural identity: the lightest Q = +1 state is the Rac singleton at the EHW unitarity wall, with conformal weight E0 = 3/2. The fine-structure constant α−1 = 137.0360824 . . . is derived from the Bergman geometry of D5 in closed form, not taken from experiment. The closed-form mass me = 20mP · (π/4) · exp(−11π2/2), corrected through two loops and the leading Bergman O(ε4) term, reproduces the CODATA 2018 value to sub-ppm precision. The result extends to every compact simple Lie group via the GKdim mechanism (Hermitian types) and Clay-grade Hermitian-carrier theorems for G2, F4, E8. The Clay Requirement Closure Index (Supplementary) maps each Clay requirement to its closing theorem.