= Solution
Let $g=Gm/r^2>0$, $H_P=-dr/d\log P>0$, $\nabla=d\log T/d\log P$, and $\nabla_\mu=d\log\mu/d\log P$. Consider a small radial <fluid displacement> $\xi$ whose sound-crossing time is short enough to keep its <pressure> equal to its surroundings. Its composition is frozen and its heat exchange negligible. This is the local <stellar convective stability> test.
For gas-pressure-dominated <ideal gas> matter, $\rho\propto\mu P/T$. The ambient gradient is therefore
$$
\frac{d\log\rho}{dr}=-\frac{1-\nabla+\nabla_\mu}{H_P}.
$$
The displaced element conserves <specific entropy> and <mean molecular weight>, giving $(d\log\rho/dr)_{\rm parcel}=-(1-\nabla_{\rm ad})/H_P$, where $\nabla_{\rm ad}=(\Gamma_2-1)/\Gamma_2$. The parcel-minus-environment <mass density> difference is
$$
\frac{\rho_{\rm parcel}-\rho_{\rm ambient}}\rho=\frac{\nabla_{\rm ad}-\nabla+\nabla_\mu}{H_P}\,\xi.
$$
Its <buoyancy> acceleration is minus $g$ times this difference. Thus
$$
\boxed{\ddot\xi+N^2\xi=0,\qquad N^2=\frac g{H_P}(\nabla_{\rm ad}-\nabla+\nabla_\mu).}
$$
Positive <stellar buoyancy frequency> squared gives a restoring force. The gas-pressure-dominated <Ledoux criterion> is therefore
$$
\boxed{\text{stable: }\nabla<\nabla_{\rm ad}+\nabla_\mu,\qquad\text{unstable: }\nabla>\nabla_{\rm ad}+\nabla_\mu.}
$$
Equality is marginal in this ideal adiabatic test. For monatomic gas $\nabla_{\rm ad}=2/5$. An inward increase of <mean molecular weight> has $\nabla_\mu>0$ and stabilizes the layer; uniform composition recovers the <Schwarzschild criterion>. For a radiative layer one substitutes the <stellar radiative temperature gradient>, $\nabla_{\rm rad}=3\kappa LP/(16\pi acGmT^4)$.
More generally define $\delta=-(\partial\log\rho/\partial\log T)_{P,\mu}$ and $\varphi=(\partial\log\rho/\partial\log\mu)_{P,T}$. The same displacement argument gives $N^2=(g/H_P)[\delta(\nabla_{\rm ad}-\nabla)+\varphi\nabla_\mu]$ and stability for $\nabla<\nabla_{\rm ad}+(\varphi/\delta)\nabla_\mu$. Both coefficients equal one in the gas-dominated ideal-gas limit used above. This is local dynamical stability against convection; it is distinct from global radial collapse and from instabilities requiring heat or composition diffusion.
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