Pure Analytic Calculations of the Mass Gap and Glueball Spectrum in Four-Dimensional Yang-Mills Theory

We prove the existence of a positive mass gap \Delta >0 for quantum Yang- Mills theory on four- dimensional Minkowski spacetime within the Epstein- Glaser causal perturbation theory framework, and derive analytically the glueball mass spectrum. 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 direct consequence of the Peter- Weyl theorem, ensures that all angular momentum modes contribute identically at the causal vertex. The spectral sum \Sigma^{(4)}(t) = \cosh (t / 2) / [2\sinh^2 (t / 2)] encodes the Riemann zeta values \zeta (- 1) = - 1 / 12 , \zeta (- 3) = 1 / 120 , ... in its small- t expansion; from the constant term 1 / 12 we derive the one- loop \beta - function coefficient b_{1} = 11C_{2}(G) / (12\pi) without Feynman diagrams. The mass gap is proven through two independent arguments: off- cone propagation and Carleman- Fredholm determinant estimates. All Wightman axioms are verified. Applying Boltzmann's 1877 statistical method with Yang- Mills self- interaction playing the role of Newtonian mechanics, and fixing the inverse temperature via Jacobson's thermodynamic relation \delta Q = T dS , we derive the analytic glueball mass spectrum M_{n} = \frac{j_{2,n}}{2}\Lambda ,\qquad n = 1,2,3,\ldots , where j_{2,n} are the zeros of the Bessel function J_{2} and \Lambda is the dynamical scale. The mass ratios M_{n} / M_{0} = j_{2,n} / j_{2,1} = 1:1.638:2.260:\dots agree with lattice QCD to within the expected 1 / N^{2} corrections. The framework connects to Migdal's large- N reduction, Unsal- Yaffe volume independence, and Verlinde's entropic gravity.