 Reflection symmetry of a two-sphere passing trajectory

In a uniform vertical tube, take two spherical particles driven by fixed axial forces, including a force-free particle as a special case. At a side-by-side passage time $t_c$, their centres lie in one horizontal reflection plane $S$. Reflection reverses the applied axial forces, and time reversal reverses them again. Uniqueness of the particle-velocity initial-value problem therefore identifies the reflected reverse trajectory with the original forward one. Their transverse positions before and after passage are equal at matching axial separations. The argument requires passage without contact and neglects inertia or other nonreversible effects.