Existence of a Mass Gap in SU(N) Yang–Mills Theory via Recursive Symbolic Fields

This paper presents a constructive resolution of the Yang–Mills Mass Gap Problem using a symbolic framework known as the Pi-Knot Machine. The theory is formulated on four-dimensional Euclidean space ( \mathbb{R}^4 ) with gauge group ( SU(N) ), and defines a recursive field operator whose spectrum is discrete and bounded below by a strictly positive energy unit ( \epsilon ). The symbolic recursion replaces conventional functional integration and yields a naturally renormalizable structure. The theory satisfies the Osterwalder–Schrader axioms and demonstrates the existence of a mass gap ( \Delta = \epsilon > 0 ). This approach is independent, non-perturbative, and structurally distinct from lattice gauge theory or constructive QFT, offering a new pathway toward rigorous quantum field theory.