A Non-Perturbative Proof of the Yang–Mills Mass Gap via Null-Cone Equal-Weight Decomposition and Construction of the Hilbert–Pólya Operator

We construct quantum Yang-Mills theory on four-dimensional Minkowski spacetime within the Epstein-Glaser causal perturbation theory framework, rigorously establishing the Wightman axioms and proving the existence of a positive mass gap \Delta >0 together with asymptotic freedom. The construction proceeds from two postulates—the massless wave equation \square \phi = 0 and Poincaré invariance—through the angular momentum decomposition of the retarded Green's function on the null cone. The equal-weight condition P_{\ell}(1) = 1, a consequence of the Peter-Weyl theorem for the unit element in every irreducible representation of \mathrm{SO}(3), ensures that all angular momentum modes contribute identically at the light-cone vertex. The spectral sum \Sigma^{(4)}(t) = \sum_{\ell = 0}^{\infty}(2\ell +1)e^{- (2\ell +1)t / 2} admits the closed form \cosh (t / 2) / [2\sinh^{2}(t / 2)], whose small-t expansion encodes the Riemann zeta function at negative odd integers via \zeta (- 1) = - 1 / 12, \zeta (- 3) = 1 / 120, etc. From the constant term 1 / 12 and the group-theoretic factor C_{2}(G), we derive the one-loop \beta-function coefficient b_{1} = 11C_{2}(G) / (12\pi) analytically without Feynman diagrams, establishing asymptotic freedom as a geometric consequence of null-cone causality. The mass gap is proven through two independent arguments: a distributional proof that non-abelian vertices extend propagator support to the timelike region, and a Carleman-Fredholm determinant argument excluding zero-mass poles. We verify all Wightman axioms via the reconstruction theorem. Furthermore, the framework reveals deep structural connections to random matrix theory (Dyson's threefold classification and Migdal's large-N reduction), and to number theory through the \mathrm{SL}(2,\mathbb{C}) holonomy R(2\pi) = - \mathbb{I} and the Selberg trace formula, providing a construction of the Hilbert-Pólya operator whose spectrum corresponds to the non-trivial zeros of the Riemann zeta function.