A simple and short mathematical proof is given for quarks confinement. The analysis is based on the exact nonperturbative ‘renormalization’ group equation for the unification inverse coupling of all fundamental interactions.

We begin with Klein’s original modular space Γ(7) for whichD=DimΓ(7)=SL(2,7)=336.Subsequent compactificationD=336 Dc=D+16k 339and conformal transformation leads toDc 339 Dcom=(Dc)(1/ ) 548.We observe that this result is identical to summing over all dimensions of the exceptional Lie symmetry groups hierarchy G2, F4, E6, E7 and E8 and adding A1, A2 and the standard model SM gauge boson to the result. That means and therefore This result is a neat confirmation of the basic group theoretical assumptions of the standard model, namely SU(3) SU(2) U(1) = 12 as well as our previous expectation number of the elementary particles in an extended standard model: particles.