  We construct a numerical solution of the recently-derived large-$N_c \& N_f$
small-$x$ helicity evolution equations with the aim to establish the small-$x$
asymptotics of the quark helicity distribution beyond the large-$N_c$ limit
explored previously in the same framework. (Here $N_c$ and $N_f$ are the
numbers of quark colors and flavors.) While the large-$N_c$ helicity evolution
involves gluons only, the large-$N_c \& N_f$ evolution includes contributions
from quarks as well. We find that adding quarks to the evolution makes quark
helicity distribution oscillate as a function of $x$. Our numerical results in
the large-$N_c \& N_f$ limit lead to the $x$-dependence of the flavor-singlet
quark helicity distribution which is well-approximated by \begin{align} \Delta
\Sigma (x, Q^2)\bigg|_{\mbox{large-}N_c \& N_f} \sim \left( \frac{1}{x}
\right)^{\alpha_h^q} \, \cos \left[ \omega_q \, \ln \left( \frac{1}{x} \right)
+ \varphi_q \right]. \end{align} The power $\alpha_h^q$ exhibits a weak
$N_f$-dependence, and, for all $N_f$ values considered, remains very close to
$\alpha_h^q (N_f=0) = (4/\sqrt{3}) \sqrt{\alpha_s N_c/(2 \pi)}$ obtained
earlier in the large-$N_c$ limit. The novel oscillation frequency $\omega_q$
and phase shift $\varphi_q$ depend more strongly on the number of flavors $N_f$
(with $\omega_q =0$ in the pure-glue large-$N_c$ limit). The typical period of
oscillations for $\Delta \Sigma$ is rather long, spanning many units of
rapidity. We speculate whether the oscillations we find are related to the sign
variation with $x$ seen in the strange quark helicity distribution extracted
from the data.
