= Positive-root criterion for dust self-gravity with drag
{title2=$P(0)=\epsilon\Omega(\omega^2-\Omega^2)<0$}
For any nonzero positive drag, the cubic <dust gravitational dispersion relation with gas drag> is negative at $s=0$ if $\omega^2<\Omega^2$ and positive at sufficiently large positive $s$. The <intermediate value theorem> therefore guarantees a growing real root. This existence argument is independent of the weak-drag expansion and includes the dynamically unstable $\omega^2<0$ regime.
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