= Solution
The <pressure>-square term in the <Cowling energy principle for a rotating barotropic star> is nonnegative, and $\mathcal N^2>0$ makes its stratification term nonnegative. Write $j=R^2\Omega$ for <specific angular momentum>. Then
$$
\kappa^2=\frac1{R^3}\frac{d(j^2)}{dR}=\frac{2j}{R^3}\frac{dj}{dR}.
$$
For a regular star reaching the rotation axis, $j(0)=0$. The given $dj/dR>0$ therefore makes $j>0$ for $R>0$, and consequently $\kappa^2>0$. Equivalently, with the usual nonnegative angular-<velocity> convention the sign follows directly. All three energy terms are nonnegative, so
$$
\boxed{Q[\xi]\ge0\quad\Longrightarrow\quad\text{no exponentially growing axisymmetric adiabatic mode}.}
$$
This is stability within the <Cowling approximation> used throughout. The orientation-independent rotational condition is $d(j^2)/dR\ge0$, the <Rayleigh discriminant> criterion. If a fluid region excludes the axis and negative <angular velocity> is allowed, $dj/dR>0$ alone needs the additional sign of $j$; the squared-angular-momentum criterion avoids that ambiguity.
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