Pressure-free localized stellar convective trial ($\mathbf{\xi }=f{Y}_{\ell }^j{\mathbf{e}}_r+\frac{r}{\ell (\ell +1)}\left[\frac{(r^{2}f{)}^{\prime }}{r^2}-\frac{\rho g}{\gamma p}f\right]{\nabla }_{\Omega }{Y}_{\ell }^j$)

Choose a smooth radial bump $f$ supported inside a shell where the stellar buoyancy frequency has $N^2<0$. The displayed fluid displacement has $\nabla \cdot \mathbf{\xi }=(\rho g/\gamma p){\xi }_r$, hence its Eulerian pressure perturbation vanishes identically. Its buoyancy contribution to the stellar displacement energy identity is strictly negative and self-gravity is nonpositive. The variational principle therefore proves convective instability with no unsupported assumption that incompressibility alone removes pressure work. Large spherical-harmonic degree also makes perturbed self-gravity small.