= Passing invariant for unequal point-force spheres
{title2=$r(\delta r+3D/2)\sin^2\theta=\mathrm{constant}$}
In the <leading interaction of two sedimenting spheres>, put $c=3D/4$ and let $\theta$ be the angle from the upward vertical. The equations are $\dot r=-(\delta+2c/r)\cos\theta$ and $r\dot\theta=(\delta+c/r)\sin\theta$. Their ratio separates to
$$
\frac{\delta r+c}{r(\delta r+2c)}dr=-\cot\theta\,d\theta,
$$
so $r(\delta r+2c)\sin^2\theta$ is invariant. When $\delta,c$ have the same nonzero sign, choose labels so both are positive. For a noncollinear encounter the invariant is positive, ensuring a strictly positive closest separation. Meanwhile $\dot z=-\delta-c(1+\cos^2\theta)/r<0$, so the vertical order reverses and eventually $r\geq|z|\to\infty$. Exactly collinear contact is outside this passing argument.
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