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It is the natural stepping stone to four dimensions.","posed_since":"","precise":"Construct the Euclidean Yang-Mills measure for $G = \\mathrm{SU}(2)$ on the torus $T^3$ and on $R^3$ as a limit of lattice or smoothed measures, as a gauge-invariant probability measure on distributional connections modulo gauge, satisfying reflection positivity and, on $R^3$, Euclidean invariance. Answer: a construction with proof; a mass gap in 3D is a further question.","problem_ref":null,"references":"","settled_by":"A proof that the lattice or stochastically quantized measures converge to a gauge-invariant limit with the Osterwalder-Schrader properties.","status_note":"Local-in-time solutions of the 3D Yang-Mills-Higgs Langevin dynamics were constructed by Chandra, Chevyrev, Hairer and Shen around 2022; the invariant measure itself is not yet constructed.","title":"Construction of three-dimensional Euclidean Yang-Mills 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