Constructive Yang-Mills Mass Gap | UIDT v3.7.1: Existence and Uniqueness via Osterwalder-Schrader Axioms, BRST Analysis, and Extended Functional Renormalization Group

We present a constructive proof of the existence and uniqueness of a positive mass gap in quantum Yang–Mills theory for the gauge group SU(3) on four-dimensional Euclidean space ℝ⁴. The proof extends pure Yang Mills theory by coupling to a fundamental information-density scalar field S(x), which transforms as a gauge singlet. Using the Extended Functional Renormalization Group (FRG) and the Banach Fixed-Point Theorem, we establish the existence of a unique mass gap Δ = 1.710 ± 0.015 GeV with Lipschitz constant L = 3.749 × 10⁻⁵ < 1. The theory satisfies all Osterwalder–Schrader axioms (OS0–OS4), enabling Wightman reconstruction to Minkowski signature. BRST cohomology defines the physical Hilbert space ℋphys = ker Q / im Q with positive-definite inner product via the Kugo–Ojima mechanism. Gauge independence follows from Nielsen identities, and renormalization group invariance from the Callan–Symanzik equation at the UV fixed point (5κ² − 3λS ≈ 0). The scalar field is auxiliary and can be integrated out, yielding pure Yang–Mills with preserved mass gap via continuous deformation. Homotopy theorems (Kato–Rellich perturbation stability) establish spectral continuity under the transformation from scalar-extended to pure gauge theory, with both formulations lying in the same infrared universality class. The predicted mass gap agrees with lattice QCD determinations: combined z-score 0.37 (p = 0.75). This work is part of the UIDT Framework and addresses the Yang–Mills mass gap problem within a constructive quantum field theory setting. Full framework documentation: DOI 10.5281/zenodo.17835200.