= Solution
Substitution of a <normal mode> gives the <dispersion relation>
$$
D(k,\omega)=-i\omega-\alpha k^2+\beta k^4+\gamma=0,\qquad
\omega=i\sigma(k),\quad \sigma(k)=\alpha k^2-\beta k^4-\gamma.
$$
For real $k$, $\sigma$ is the temporal growth rate. Since $\beta\ne0$, its supremum is finite exactly when
$$
\boxed{\beta>0}.
$$
For $\beta<0$, arbitrarily short wavelengths grow arbitrarily fast. For $\beta>0$, the <finite maximum temporal growth rate> is
$$
\sigma_{\max}=\begin{cases}\alpha^2/(4\beta)-\gamma,&\alpha>0,\\-\gamma,&\alpha<0.\end{cases}
$$
The <Briggs-Bers criterion> starts the inverse temporal <Laplace transform> above all temporal singularities and then deforms its contour downward while following the spatial roots. A finite growth bound supplies such an initial contour and a causal, high-frequency-controlled <Green function>. Unbounded temporal growth prevents that standard construction.
For <absolute wave-packet instability>, a candidate <spatial pinch point> must satisfy $D=0$, $D_k=0$, and $\operatorname{Im}\omega>0$. Here
$$
D_k=-2\alpha k+4\beta k^3=0
$$
gives
$$
k_0=0,\quad \omega_0=-i\gamma;\qquad
k_\pm=\pm\sqrt{\frac\alpha{2\beta}},\quad
\omega_\pm=i\left(\frac{\alpha^2}{4\beta}-\gamma\right).
$$
Thus candidate growing saddles require $\gamma<0$ at $k_0$, or $\gamma<\alpha^2/(4\beta)$ at $k_\pm$. Together with $\beta>0$, existence of at least one such candidate requires $\gamma<\alpha^2/(4\beta)$.
\b[A growing double root is not sufficient: the roots must pinch the spatial inversion contour from opposite sides.] Collisions of branches originating in the same spatial half-plane do not obstruct the relevant deformation. This distinction is part of the <Briggs-Bers criterion>; it is stated, for example, in the primary study https://doi.org/10.1017/jfm.2016.195.
An explicit <false spatial saddle in quartic dispersion> is $\alpha=-2$, $\beta=1$, $\gamma=1/2$. Its candidates $k=\pm i$ have $\omega=i/2$, but all real modes have $\sigma=-k^4-2k^2-1/2<0$. Each imaginary collision joins two branches in the same half-plane; neither is a relevant pinch. For $\omega=i\Omega$, the spatial roots obey $k^2=-1\pm\sqrt{1/2-\Omega}$, making those same-half-plane collisions transparent as $\Omega\downarrow1/2$.
For this particular real, even dispersion relation one can also establish the actual threshold directly. At the origin its impulse <Green function> is
$$
G(0,t)=\frac1{2\pi}\int_{-\infty}^{\infty}e^{t\sigma(k)}dk.
$$
<Laplace method> selects the real maximum, and gives a positive prefactor times $t^{-1/2}e^{\sigma_{\max}t}$ because $\alpha\ne0$. Therefore the actual <absolute wave-packet instability> condition is
$$
\boxed{\beta>0,\quad\begin{cases}\gamma<\alpha^2/(4\beta),&\alpha>0,\\\gamma<0,&\alpha<0.\end{cases}}
$$
These are sufficient for this model as well as necessary. They follow after identifying relevant real saddles; the earlier algebraic double-root test alone lacks the pinch information. Equality is marginal, not exponential absolute growth.
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