The Yang-Mills Existence and Mass Gap Problem: A Critical Survey of Formulations, Obstructions, and Approaches

The Yang-Mills existence and mass gap problem asks for the construction, for every compact simple gauge group $G$, of a nontrivial quantum Yang-Mills theory on $\R^4$ satisfying axiomatic conditions at least as strong as those cited in the Clay formulation and having a finite positive mass gap. This survey gives matched Minkowski/Wightman and Euclidean/Osterwalder-Schrader formulations; states explicit working criteria for existence and nontriviality; and supplies self-contained proofs of standard implications relating spectral gaps, clustering, and relativistic mass gaps under the stated reconstruction hypotheses. We distinguish gauge-theoretic constraints—including Singer's no-global-slice theorem, the Gribov ambiguity, Elitzur's theorem, and Strocchi-type positive-metric obstructions—from the four-dimensional scalar triviality theorem, which serves only as a diagnostic comparison. We next review finite-cutoff lattice gauge theory, strong-coupling results, and the rigorously constructed two-dimensional holonomy-field theory, followed by the Balaban programme and related multiscale constructions, probabilistic lattice methods, stochastic quantization and singular stochastic partial differential equations, and Hamiltonian, semiclassical, and dual approaches. For each programme we separate complete theorems, model- or cutoff-specific results, proposed mechanisms, and remaining steps toward the four-dimensional Clay problem. We also assess numerical evidence, the distinction between mass gap and confinement, and a set of intermediate open problems designed to expose dependencies among ultraviolet construction, gauge-invariant continuum limits, and infrared spectral control. The article provides expository proofs and derivations of several standard results but claims no new research theorem.