  We study the interplay between wall-crossing in four-dimensional gauge theory
and instanton contributions to the moduli space metric of the same theory on
$\mathbb{R}^{3}\times S^{1}$. We consider $\mathcal{N}=2$ SUSY Yang--Mills with
gauge group SU(n) and focus on walls of marginal stability which extend to weak
coupling. By comparison with explicit field theory results we verify the
Kontsevich--Soibelman formula for the change in the BPS spectrum at these walls
and check the smoothness of the metric in the corresponding compactified
theory. We also verify in detail the predictions for the one instanton
contribution to the metric coming from the non-linear integral equations of
Gaiotto, Moore and Nietzke.
