An Analytic Scaling Law for the Mass Gap From a Nonlinear Elastic Medium to the Yang–Mills Spectrum

The Yang-Mills mass gap problem — why the lightest excitation in pure gauge theory possesses a strictly positive mass — is one of the Clay Millennium Prize Problems. Lattice QCD simulations yield a mass scale of approximately 200 MeV, but an analytic expression depending solely on calculable parameters has remained absent. This paper presents the analytic scaling law for the mass gap within the Gluon Elastic Lattice Theory (GELT) framework: the restoring frequency of the medium in the nonlinear hardening regime determines the frequency scale of the lowest excitation through the tube wall hardening mechanism. The mass gap is given by the basic parameters of the medium — restoring frequency ω₀, lattice spacing d₀, elastic wave speed c, hardening coefficient η=0. 573, and the string tension geometric factor ζT — with no free parameters. The physical origin of the mass gap lies in the nonlinear hardening coefficient η > 0 of the medium: in a purely linear medium, stiffness is uniform everywhere and the mass gap vanishes. This analytic scaling law anchors a Millennium Prize problem in the classical physics framework of nonlinear elastic medium mechanics.