We determine the small-$x$ asymptotics of the gluon

helicity distribution in a proton at leading order in

perturbative QCD at large $N_c$. To achieve this, we begin by

evaluating the dipole gluon helicity TMD at small $x$. We then

construct and solve novel small-$x$ large-$N_c$ evolution

equations for the operator related to the dipole gluon

helicity TMD. Our main result is the small-$x$ asymptotics

for the gluon helicity distribution:

\begin{align}\label{dG_final}

\Delta G \sim \left(

\frac{1}{x} \right)^{\alpha_h^G} \ \ \ \mbox{with} \ \ \ \alpha_h^G =

\frac{13}{4 \sqrt{3}} \, \sqrt{\frac{\as \, N_c}{2 \pi}}

\approx 1.88 \, \sqrt{\frac{\as \, N_c}{2 \pi}}.

\end{align}

We note that the power $\alpha_h^G$ is approximately 20$\%$ lower than

the corresponding power $\alpha_h^q$ for the small-$x$ asymptotics of

the quark helicity distribution defined by

\begin{align}\label{dq_final}

\Delta q \sim \left(

\frac{1}{x} \right)^{\alpha_h^q} \ \ \ \mbox{with} \ \ \ \alpha_h^q =

\frac{4}{\sqrt{3}} \, \sqrt{\frac{\as \, N_c}{2 \pi}} \approx 2.31 \,

\sqrt{\frac{\as \, N_c}{2 \pi}}

\end{align}

found in our earlier work.