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The material-boundary and vorticity conditions give $\alpha =S/[r(r-1)]$ and $\beta =Sr/(r-1)$. A pressureless particle subject to linear drag, Coriolis acceleration and tidal gravity has the drag-polynomial stability for all positive stopping rates with $K=2\Omega (2\Omega -S)$, $L=\Omega (\alpha +\beta -S)$ and $P=\alpha \beta$. In Keplerian shear, $K/{\Omega }^2=1$, $L/{\Omega }^2=3(r+1)/[2r(r-1)]$ and $P/{\Omega }^2=9/[4(r-1{)}^2]$. The exact criterion is $r\ge 3$, including the boundary $K=L>P$ at $r=3$. Attraction is local to the centre under the prescribed interior velocity field; a trajectory leaving the patch cannot be followed using that field alone.

Take every finite positive drag coefficient $\gamma >0$ and defineThe coefficients of the characteristic polynomial are $c_3=2\gamma$, $c_2=K+{\gamma }^2$, $c_1=2L\gamma$ and $c_0=P{\gamma }^2$. The nontrivial Routh-Hurwitz criterion reduces exactly toThe printed strict chain $K>L>P>0$ makes every coefficient positive and both bracket terms positive, so it proves attraction for every positive drag coefficient.

There is, however, a boundary error in the asserted necessity. To have the bracket positive for every $\gamma >0$, its constant and slope as a function of ${\gamma }^2$ must be nonnegative, and cannot both vanish. Together with coefficient positivity this gives the exact drag-polynomial stability for all positive stopping rates:Necessity follows by taking arbitrarily small and arbitrarily large positive $\gamma$; if one bracket coefficient is negative, the inequality fails in the corresponding limit. Sufficiency follows because at least one coefficient is strictly positive. Thus $K=L>P$ and $K>L=P$ still give strict decay at each finite positive $\gamma$. They need not give a uniform stability margin in the zero- or infinite-drag limit.

In a Keplerian shearing sheet, $S=3\Omega /2$ andThe differences areHence $r>3$ establishes the requested strict chain and trapping conclusion. At $r=3$, however, $K=L={\Omega }^2$ and $P=9{\Omega }^2/16$, soAll coefficients remain positive. This is an explicit counterexample within the stated vortex model: $r=3$ also attracts particles for every finite positive drag coefficient, so the literal strict “if and only if” is false. The exact Keplerian all-positive-drag condition is $r\ge 3$. If $\gamma =0$ is included, the polynomial has zero and purely imaginary roots and there is no asymptotic attraction; “all drag coefficients” must mean positive drag. The local attraction claim presumes the particle remains in the patch, as is assured for sufficiently small perturbations of its centre.