When does a gravitational wave truly 'let go' of passing matter?

When a gravitational wave sweeps through space, it can permanently shift or accelerate matter — the so-called memory effect. This paper shows that requiring the wave's amplitude to simply drop to zero isn't sufficient to guarantee that particles eventually move freely afterward. By deriving sharper decay conditions, the authors classify three distinct fates for test particles and find a subtle case where residual drift affects only particles already in motion, leaving the more famous displacement memory intact.