= Solution
For $k_y=0$, the minimized energy <density> from part (c) is
$$
\left[-\frac{\rho^2g^2}{T}-g\rho'\right]|\xi_z|^2
=\rho g\left[-\frac{d\ln\rho}{dz}-\frac{\rho g}{T}\right]|\xi_z|^2.
$$
If the bracket is nonnegative throughout, this <density> and the omitted <pressure> square are nonnegative for every displacement, so there is no negative-energy trial. Conversely, a strict negative value at a point of a smooth equilibrium persists in a small interval. Choose a nonzero smooth $\xi_z$ supported there, set $\xi_y=0$, and choose $\xi_x$ as in part (c). This makes the <pressure> square zero and gives a strictly negative integral. Therefore the <localized interchange criterion for a magnetized atmosphere> is
$$
\boxed{\text{instability for }k_y=0\quad\Longleftrightarrow\quad
-\frac{d\ln\rho}{dz}<\frac{\rho g}{\gamma p+B^2/\mu_0}\text{ somewhere}.}
$$
These disturbances interchange flux tubes without along-field variation; both gas and <magnetic pressure> contribute to their effective compressional stiffness. Equality alone is marginal and does not give a negative energy integral.
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