= Solution
Put $A=(Gm/h^3)F(hk)$. The two amplitude equations have coefficient matrix
$$
\begin{pmatrix}s^2-3\Omega^2+A&-2\Omega s\\2\Omega s&s^2-2A\end{pmatrix}.
$$
Its determinant gives the <dispersion relation>
$$
\boxed{s^4+(\Omega^2-A)s^2+2A(3\Omega^2-A)=0.}
$$
Writing $q=A/\Omega^2\ge0$, its squared growth rates are
$$
\boxed{\frac{s_\pm^2}{\Omega^2}=\frac{q-1\pm\sqrt{9q^2-26q+1}}2,}
$$
and the four growth rates are both <square roots> of each displayed value. Define
$$
q_- =\frac{13-4\sqrt{10}}9,\qquad q_+ =\frac{13+4\sqrt{10}}9.
$$
For $0<q<q_-$, the <discriminant> is positive, the sum of the squared rates is $q-1<0$, and their product is $2q(3-q)>0$; both squared rates are negative. These modes have purely imaginary $s$. For $q_-<q<q_+$, the squared rates form a nonreal <complex conjugate> pair; their <square roots> include roots with positive real part. This is <overstability>, meaning growing oscillations. For $q\ge q_+$, at least the plus squared rate is positive because $q>1$, so a real growing root exists. Consequently every $q>q_-$ is exponentially unstable.
Take the maximizing mode $hk=\pi$, so $q=(Gm/(h^3\Omega^2))F(\pi)$. It is unstable whenever
$$
\boxed{\frac{Gm}{h^3\Omega^2}>\frac{13-4\sqrt{10}}{9F(\pi)}.}
$$
This is also the threshold for the onset of exponential growth among all lattice modes, since $0\le F\le F(\pi)$. At equality the maximizing mode has repeated imaginary roots and can have algebraic growth; it is a marginal collision of oscillatory modes, not a strict-decay assertion. The $k=0$ mode has $F=0$, an <epicyclic motion> sector and a zero-frequency <epicyclic guiding center> sector with possible secular shear drift. These neutral or algebraic behaviors are distinct from the strict exponential instability established by the requested inequality.
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