Constructive Proof of the 4D Quantum Yang–Mills Existence and Mass Gap A Rigorous Euclidean-to-Wightman Formulation Satisfying the Clay Millennium Criteria

This document presents a full, non-perturbative and constructive demonstration of the 4-dimensional quantum Yang–Mills existence and the positivity of the mass gap mgap>0m_{\mathrm{gap}} > 0mgap>0 for any compact simple gauge group GGG.It is built in four main parts: Classical Euclidean setup and specification of the functional analytic domain for gauge fields. Reflection positivity, lattice construction, character expansion, and exponential clustering. Tightness, Osterwalder–Schrader axioms (OS0–OS4), measure existence and continuum limit. Reconstruction of the Wightman theory, spectral representation, and a rigorous lower bound for the mass gap (Theorem E.1). In the appendices, ultraviolet stability via renormalization group flow is established, along with all technical lemmas (character expansion, compactness, OS conditions, spectral measure).This version includes corrections to Appendix E ensuring the Theorem E.1 header is intact, and a professional front cover page. Relation to previous versions: This version (v3.0, October 2025) is the polished complete manuscript ready for peer review. Prior versions (v1.x, v2.x) exist as drafts uploaded previously (if applicable). The versioning mechanism links this final version to the earlier drafts, enabling readers to track the revision history. Citations should refer to this specific version’s DOI; the “concept DOI” for all versions remains valid for referencing the evolving work. This record (v3.0, October 2025) supersedes versions 10.5281/zenodo.17261037 and 10.5281/zenodo.17260379, integrating full corrections and Appendix E reconstruction.