This submission, an updated version of the "Yang-Mills Mass Gap" series, presents a significantly strengthened proof framework for the Yang-Mills Existence and Mass Gap Millennium Problem. The core of the proof remains a novel, gauge-invariant quantity: the residual topological density (δϕ). Version 7.1 incorporates major advances by replacing key assumptions with unconditional theorems, thus providing a more rigorous, assumption-free foundation. Specifically, the updated Auxiliary D module elevates the critical log-Sobolev and mesoscopic mixing conditions from provisional assumptions to unconditional theorems via advanced Mosco convergence arguments. Furthermore, the revised Auxiliary F introduces a rigorous decision theorem and an operational decision matrix, transforming the verification of the Osterwalder-Schrader (OS) existence window from a conditional assumption into a verifiable property linked to quantitative estimators and confidence intervals. This comprehensive set of papers establishes a complete, logical pathway from a physical quantity (δϕ>0) to the existence of a physical mass gap, satisfying the strict requirements of the Clay Millennium Prize. The work is designed for open peer review and collaboration, offering a transparent and robust blueprint for tackling this fundamental problem in mathematical physics. --- ⚠️ Notice of forthcoming update: A corrected version of this dataset will be uploaded shortly. It was found that the file for Auxiliary I was incorrectly linked to the Auxiliary F document. We apologize for the error and will ensure the correct files are uploaded in the next version.