  A non-perturbative and mathematically rigorous quantum Yang-Mills theory on
4-dimensional Minkowski spacetime is set up in the functional framework of a
complex nuclear Kree-Gelfand triple. It involves a symbolic calculus of
operators with variational derivatives and a new kind of infinite-dimensional
ellipticity. In the temporal gauge and Schwinger first order formalism,
Yang-Mills equations become a semilinear hyperbolic system for which the
general Cauchy problem is reduced to initial data with compact supports. For a
simple compact Yang-Mills gauge group and the anti-normal quantization of
Yang-Mills energy-mass functional of initial data in a box, the quantum
energy-mass spectrum is a sequence of non-negative eigenvalues converging to
infinity. In particular, it has a positive mass gap. Furthermore, the
energy-mass spectrum is self-similar (including the mass gap) in the inverse
proportion to an infrared cutoff of the classical energy scale.
