  We construct an exact analytic solution of the revised small-$x$ helicity
evolution equations derived recently. The equations we solve are obtained in
the large-$N_c$ limit (with $N_c$ the number of quark colors) and are
double-logarithmic (summing powers of $\alpha_s \ln^2(1/x)$ with $\alpha_s$ the
strong coupling constant and $x$ the Bjorken $x$ variable). Our solution
provides small-$x$, large-$N_c$ expressions for the flavor-singlet quark and
gluon helicity parton distribution functions (PDFs) and for the $g_1$ structure
function, with their leading small-$x$ asymptotics given by \begin{align}
  \Delta \Sigma (x, Q^2) \sim \Delta G (x, Q^2)
  \sim g_1 (x, Q^2) \sim \left( \frac{1}{x} \right)^{\alpha_h} , \notag
\end{align} where the exact analytic expression we obtain for the intercept
$\alpha_h$ can be approximated by $\alpha_h = 3.66074 \, \sqrt{\frac{\alpha_s
\, N_c}{2 \pi}}$. Our solution also yields an all-order (in $\alpha_s$)
resummed small-$x$ anomalous dimension $\Delta \gamma_{GG} (\omega)$ which
agrees with all the existing fixed-order calculations (to three loops).
Notably, our anomalous dimension is different from that obtained in the
infrared evolution equation framework developed earlier by Bartels, Ermolaev,
and Ryskin (BER), with the disagreement starting at four loops. Despite the
previously reported agreement at two decimal points based on the numerical
solution of the same equations, the intercept of our large-$N_c$ helicity
evolution and that of BER disagree beyond that precision, with the BER
intercept at large $N_c$ given by a different analytic expression from ours
with the numerical value of $\alpha_h^{BER} = 3.66394 \, \sqrt{\frac{\alpha_s
\, N_c}{2 \pi}}$. We speculate on the origin of this disagreement.
