Separation vs. difference in total proper motion (in standard deviations) for stars near Kepler-1651 (top) and KOI-2705 (bottom). The red star indicates the target we identify as a true companion, while black points are unassociated stars. For relatively high-proper-motion targets like these, the probability that an unassociated star has comparable proper motion and is nearby on the sky is extremely small, which is evident by the large region of unfilled parameter space.