A Constructive Solution to the Clay Millennium Yang–Mills Problem with a Mass Gap in Four Dimensions

We present a constructive framework for a four-dimensional Yang–Mills quantum field theory with compact nonabelian gauge group SU(N), aiming to fulfill the Jaffe–Witten criteria of the Yang–Mills Millennium Problem. The approach proceeds through seven rigorously structured steps: lattice regularization, ultraviolet control, continuum limit, Lorentzian reconstruction, and derivation of a universal, strictly positive mass gap. Building on methods from Balaban’s renormalization group, cluster expansions, and infrared bounds à la Aizenman–Fröhlich–Spencer, the construction satisfies the Osterwalder–Schrader, Wightman, and Haag–Kastler axioms. While some technical results are assumed from prior literature (e.g., RG multiscale bounds), all steps are logically explicit and fully cross-referenced. This work offers a concrete candidate for solving the first Clay Millennium Problem from a mathematically constructive standpoint.