Application of Stochastic Calculus of Variations for Reflection-Positive Measure Construction in Yang-Mills Theory

In this sequential manuscript, we use the foundational concepts developed in our previous work on the spectral analysis and differential topology of 4D Yang-Mills theory, specifically the marginal stability discriminant dictating the dynamics of the theory, to address the fundamental integrability pathologies of infinite-dimensional configuration spaces. We use advanced infinite-dimensional stochastic analysis and Malliavin calculus to evaluate the continuum weak limit of the gauge measure. The functional analytic behind the concentration of the four-dimensional statistical weight path integral measure onto a one-dimensional tubular neighborhood is formalized via the Donsker delta distribution within the Watanabe-Sobolev space. The constrained measure obtained (via the Donker delta distribution) is proved to satisfy the Bouleau-Hirsch non-degeneracy condition and adherence to Haag’s theorem. Furthermore, we show that this measure concentration via the Donsker delta distribution confines the path integration to the Fundamental Modular Region, evading the Gribov ambiguity. Also, this dynamically generated topological framework resolves ultraviolet triviality through exact spatial Sobolev bounding, while infrared convergence is secured via uniformly summable multi-scale polymer cluster expansions driven by the framework’s emergent topological mass gap. We also present proof that the constructed Yang-Mills measure is reflection-positive, which represents a significant step towards a complete formulation of an interacting constructive Yang-Mills theory in four dimensions.