= Solution
For fixed smooth $\xi_y,\xi_z$, the only appearances of $\xi_x$ are in $\delta\Pi=A-Td$ and the nonnegative tension contribution $Ck_y^2|\xi_x|^2$. Positivity of $T$ makes the <pressure> square nonnegative too. Impose $d=A/T$ by choosing
$$
\boxed{\xi_x=\frac1{ik_x}\left[\frac{\rho g\xi_z+Cik_y\xi_y}{T}-ik_y\xi_y-\xi_z'\right].}
$$
Then $\delta\Pi=0$ exactly and $\xi_x=O(k_x^{-1})$ as $|k_x|\to\infty$, removing both nonnegative contributions in the limit. Since the remaining terms do not involve $\xi_x$, no other choice can lower their limiting value. This is the <short transverse wavelength limit for magnetic buoyancy>.
Take the chosen vertical and along-field displacements compactly supported in the interior when constructing instability trials; the resulting $\xi_x$ also vanishes near the boundaries, satisfying the surface condition. The argument constructs an infimum, not necessarily a minimizing finite-wavelength eigenfunction. Whenever the limiting energy is strictly negative, sufficiently large finite $|k_x|$ already gives negative energy and hence the instability asserted by the supplied variational criterion.
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