A CONSTRUCTIVE QUANTUM FIELD THEORY APPROACH TO YANG–MILLS EXISTENCE AND MASS GAP VIA UNIFORM OPERATOR BOUNDS AND SCALE-INVARIANT EUCLIDEAN MEASURES

Abstract

Abstract. ThispaperpresentsamathematicallyrigorousformulationaddressingtheYang– Mills Existence and Mass Gap problem, one of the Clay Mathematics Institute’s Millennium Prize Problems. Moving beyond fixed-parameter cutoff approximations, we establish a con- tinuous dilation framework on Banach spaces where the scaling parameter λ ∈ [1, ∞) acts as a transient regularizer. By utilizing non-perturbative Wilson loop area laws and transfer matrix spectral bounds, we prove the existence of a non-trivial Mass Gap ∆ > 0 without assuming dimensional transmutation a priori. Furthermore, through uniform cluster expan- sion bounds, we verify the infinite volume limit (Λ → R4), demonstrating that the resulting Schwinger functions satisfy all four Osterwalder–Schrader (OS) axioms (OS1–OS4). : Declaration of Generative AI and AI-Assisted Technologies (including Claude and Gemini) : AI Use Declaration: Generative AI tools were used during manuscript preparation exclusively for language polishing, LaTeX syntax formatting, and assisting in the structuring of Lean 4 tactic workflows. All mathematical proofs, derivations, and underlying theoretical concepts were formulated, checked, and validated independently by the author, who holds sole responsibility for the work.