Rigorous Construction and Mass Gap in Yang–Mills Theory from Knotified Correlators

This paper introduces a novel approach to understanding the mass gap problem in Yang-Mills theory through the lens of knot theory. By reformulating the configuration space in terms of framed knots in $\mathbb{R}^4$, we develop a rigorous mathematical framework that connects topological invariants to the spectral properties of the theory. The central innovation is the introduction of a tensorized knot invariant that measures the topological complexity of gauge field configurations, which is incorporated into the energy functional with a coupling parameter $\kappa$. We prove that this formulation naturally leads to a well-defined continuum measure that satisfies the Osterwalder-Schrader axioms, ensuring the existence of a corresponding quantum field theory in Minkowski space. Through detailed analysis of specific examples, including trivial knots, Hopf links, and pathological configurations, we demonstrate how topologically complex configurations are exponentially suppressed in the measure, providing a mechanism for the emergence of a mass gap proportional to the coupling parameter. This approach offers a new perspective on confinement in non-Abelian gauge theories and establishes a direct connection between the topological structure of the configuration space and the spectral properties of the physical theory.