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 single new closure hypothesis is that the leading nonanalytic contribution to the collective Yang-Mills energy density inherits the squared sector-relative Hermitian-bilinear scale, κ̂², with a positive matching coefficient. Near the lemniscatic point, the Weierstrass invariants then imply κ̂² ∝ ρ⁻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.