= Capillary stability threshold for a kinetic freezing front
{title2=$\Gamma_c=S/V$}
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.
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