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Choose time $t_c$ when the passing sphere centres are at the same height, and let $S$ reflect positions in that horizontal plane. For each instantaneous configuration, Stokes flow determines particle velocities uniquely and linearly from the fixed external forces. Time reversal reverses all velocities and the dense sphere's applied force; reflecting in the horizontal plane reverses that force again. The neutral sphere remains force-free under both operations. Thus the reflected time-reversed motion obeys the same original forcing.

At $t_c$, reflection fixes both particle centres. Since the spheres have no orientational shape variables and there is no contact, uniqueness of the particle-trajectory initial-value problem makes the reflected backward motion coincide with the forward motion:This reflection symmetry of a two-sphere passing trajectory fixes the transverse coordinates and reverses the relative axial separation. Consequently, at matching large separations above and below, both spheres return to their original distances from the centreline. Their intermediate lateral excursions are undone. This uses kinematic reversibility of Stokes flow, geometric reflection symmetry and the noncontact result, not a dissipation-minimizing trajectory. It assumes the spheres pass one another; a perfectly collinear approach would not supply the side-by-side symmetry event.