False complex saddle in dissipative cubic dispersion ($\alpha =-1,\beta =1,\gamma =1/10$)

For these constants, every real-wavenumber growth rate is $-k^2-1/10<0$. The real-contour impulse integral decays with envelope $e^{-t/10}{t}^{-1/2}$ at fixed position. Yet the stationary candidate $k=-2i/3$ has $\mathrm{Im}\omega =4/27-1/10>0$. It cannot be a contributing spatial pinch point; otherwise it would contradict the real-contour growth bound. This explicit example shows why solving the algebraic saddle equations is insufficient for absolute wave-packet instability.