  In this article, we study the analytical properties of the solutions of the
complex Yang-Mills equations on a closed Riemannian four-manifold $X$ with a
Riemannian metric $g$. The main result is that if $g$ is $good$ and the
connection is an approximate ASD connection, then the extra field has a
positive lower {bound}. As an application, we obtain some gap results for
Yang-Mills connections and Kapustin-Witten equations.
