Modular Closure of Yang–Mills Dynamics: A Falsifiable UV–IR Framework for the Mass Gap, Glueball Correlators, Lattice Benchmarks, and Experimental Continuation

We formulate a compact and falsifiable ultraviolet-to-infrared effective closure for four-dimensional pure Yang–Mills theory. The construction uses a gauge-invariant collective modular coordinate reconstructed from gradient-flowed scalar and topological densities and a sector-relative Modular-Weierstrass scale. The core constitutive hypothesis is that the leading nonanalytic contribution to the collective Yang–Mills energy density inherits the squared sector-relative Hermitian-bilinear scale, \({\widehat{\kappa}}^{2}\) with a positive matching coefficient. Near the lemniscatic point, the Weierstrass invariants then imply \({\widehat{\kappa}}^{2} \propto \rho^{- 2}\). Combined with the leading regular analytic restoring term, this produces a unique finite radial minimum and a positive scalar curvature; a CP-even periodic angular term gives a separate positive pseudoscalar curvature. The distinctive inverse-square relation is formulated as a configuration-level lattice test rather than inferred from glueball masses. Published continuum pure \(SU(3)\) spectra provide a quantitative mass-ratio benchmark, while multi-\(N\) spectra test regular large-\(N\) behavior. The BESIII \(X(2370)\) result is used only as a full-QCD continuation benchmark after explicit gluonic–quark mixing. A genuine out-of-sample pseudoscalar prediction requires all modular and mixing inputs to be fixed independently of the target resonance mass. The framework therefore yields a positive infrared quadratic spectral scale within the stated effective closure and a concrete program of falsification; it is not presented as a constructive proof of the Clay Yang–Mills existence-and-mass-gap problem.