Yang–Mills Existence and Mass Gap Continuum Limits and Osterwalder–Schrader Positivity

The construction of a mathematically rigorous four-dimensional quantum Yang–Mills theory with a strictly positive mass gap remains one of the central problems in mathematical physics. The principal mathematical requirements are the existence of a non-trivial continuum limit and the preservation of Osterwalder–Schrader reflection positivity. The former requires removal of the ultraviolet lattice cutoff without destroying gauge invariance, locality, non-triviality, or finite physical correlation functions.The latter is required for the reconstruction of a positive Hilbert space and a self-adjoint Hamiltonian from Euclidean Schwinger functions.This paper formulates the relevant structures using non-Abelian gauge fields, Wilson lattice gauge theory, renormalization-group transformations, reflection-positivemeasures, transfer operators, and spectral representations. It explains why positivityat finite lattice spacing is not by itself sufficient and why the mass-gap estimatemust remain uniform when the lattice spacing tends to zero. A constructive researcharchitecture is proposed, combining gauge-covariant renormalization, continuumcompactness, Osterwalder–Schrader reconstruction, and cutoff-independent infrared estimates. The discussion describes a mathematical framework for a possible proofbut does not claim to provide a completed proof of the Clay Mathematics Instituteproblem.