Spatial pinch point ($D(k_{\ast },{\omega }_{\ast })=D_k(k_{\ast },{\omega }_{\ast })=0$)

A collision of spatial branches which pinches the spatial inversion contour as the temporal contour is lowered from above all singularities. The branches must originate on opposite sides of the spatial contour. A collision of two upper-half-plane branches or two lower-half-plane branches can solve the double-root equations without causing absolute wave-packet instability.

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.