  It is believed that Euclidean Yang-Mills theories behave like the massless
Gaussian free field (GFF) at short distances. This makes it impossible to
define the main observables for these theories - the Wilson loop observables -
in dimensions greater than two, because line integrals of the GFF do not exist
in such dimensions. Taking forward a proposal of Charalambous and Gross, this
article shows that it is possible to define Euclidean Yang-Mills theories on
the 3D unit torus as "random distributional gauge orbits", provided that they
indeed behave like the GFF in a certain sense. One of the main technical tools
is the existence of the Yang-Mills heat flow on the 3D torus starting from
GFF-like initial data, which is established in a companion paper. A key
consequence of this construction is that under the GFF assumption, one can
define a notion of "regularized Wilson loop observables" for Euclidean
Yang-Mills theories on the 3D unit torus.
