Force-free two-sphere stroke ($\Delta X=\oint K(\ell )d\ell =0$)

For spheres linked by a length $\ell (t)$, axial mobility and $F_1+F_2=0$ imply $\dot{X}=K(\ell )\dot{\ell }$ for their midpoint. The displacement over a periodic stroke is $\oint K(\ell )d\ell =0$. Unequal radii permit instantaneous midpoint motion but not net translation. This is an explicit one-shape-coordinate example of the scallop theorem.