= False spatial saddle in quartic dispersion
{title2=$\sigma(k)=-k^4-2k^2-1/2$}
The real dispersion above is temporally stable for every real $k$, but its analytic continuation has stationary points at $k=\pm i$ with $\omega=i/2$. Writing $\omega=i\Omega$ gives $k^2=-1\pm\sqrt{1/2-\Omega}$; as $\Omega$ decreases to $1/2$, both branches at $+i$ approach from the upper half-plane and both at $-i$ from the lower. These are not <spatial pinch points>. This provides a concrete counterexample to treating growing algebraic double roots as sufficient for <absolute wave-packet instability>.
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