Validity of the impulsive-kick approximation. Suppression of kick-driven dispersions when each recoil is spread over a duration τ (train of ten sub-impulses), relative to instantaneous kicks with identical realizations. The solid curve shows the predicted transfer-function suppression for the Jupiter–Saturn period-ratio random walk, weighted over the ejection-event schedule; measured values (black points; M_ej = 10⁻⁴ and 10⁻⁶ M_⊙) follow it. Dashed curves show the corresponding naive per-planet predictions for the eccentricity dispersions; the measured eccentricity dispersions (colored diamonds, τ = 30 yr; offset horizontally for visibility) instead remain at unity, because angular-momentum-deficit excitation is dominated by Uranus and Neptune—for which any plausible event duration is deeply impulsive (P ≫ τ)—and is redistributed to Jupiter and Saturn through secular coupling. Physically motivated event durations (shaded band) lie far inside the impulsive regime for all channels.