= Large-wavenumber branches of kinetic solidification
{title2=$p^3+(S-V)p^2+[(\Gamma-1)\alpha^2-2SV]p-S\alpha^2+SV^2=0$}
The cubic follows by eliminating $\sigma=p^2-Vp-\alpha^2$ from <one-sided kinetic solidification dispersion>, with $V=1-S$. For $\Gamma=0$, the interface branch tends to $\sigma=SV-SV/\alpha+O(\alpha^{-2})$. For fixed $0<\Gamma<1$, substituting $p=\sqrt{1-\Gamma}\alpha+d$ gives $2d-V=S\Gamma/(1-\Gamma)$, hence $\sigma=-\Gamma\alpha^2+[S\Gamma/\sqrt{1-\Gamma}]\alpha+O(1)$. At $\Gamma=1$ the balance is $p^3\sim S\alpha^2$, giving $\sigma=-\alpha^2+S^{2/3}\alpha^{4/3}+o(\alpha^{4/3})$. For $1<\Gamma<1+2S/V$, $p\to S/(\Gamma-1)>V/2$ and $\sigma=-\alpha^2+p^2-Vp+o(1)$. Stronger capillarity can remove the real isolated interface mode from the chosen thermal branch; it does not create growth. These distinctions explain why replacing $p$ by $\alpha$ is not a controlled fixed-capillarity limit.
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