Capillary stability threshold for a kinetic freezing front (${\Gamma }_c=S/V$)

For $0<S<1$ and $V=1-S$, the translational branch of one-sided kinetic solidification dispersion has $\sigma =(S/V-\Gamma ){\alpha }^2+O({\alpha }^4)$. Expand $p=V+(\sigma +{\alpha }^2)/V+O({\alpha }^4)$ and substitute to prove this limit. At zero growth, $p_0=[V+\sqrt{V^2+4{\alpha }^2}]/2$, and the destabilizing right side has sign ${\alpha }^2(SV/p_0^2-\Gamma )$. For nonnegative real $\sigma$, the residual has derivative at least one, so there is one growing root precisely when $\Gamma <SV/p_0^2$. Thus instability exists exactly when $\Gamma <S/V$, with unstable band ${\alpha }^2<p_c(p_c-V)$, $p_c=\sqrt{SV/\Gamma }$ for $\Gamma >0$. At threshold the next term is $\sigma =-2S{\alpha }^4/V^3+O({\alpha }^6)$. Surface energy removes short-wave growth even when long waves remain unstable.