  We present a rigorous pseudo-prospective testing framework to evaluate four
generalizations of the ETAS (epidemic-type aftershock sequence) model that all
feature space-varying background events rates. We demonstrate that the model,
with (i) a magnitude-dependent Omori law exponent and (ii) a modified
Gutenberg-Richter magnitude distribution that links triggered and triggering
earthquake magnitudes, overperforms strongly. These two properties (i) and (ii)
have been previously documented empirically and derived from general physical
and scaling reasonings. One of the novel consequences of this new superior
model is that, above magnitude 6.3 (in California), earthquakes are triggered
mainly by other earthquakes of similar magnitudes. In contrast, small
non-observable events mostly trigger smaller events. This spectacular feature
is further validated by the rationalization of earthquake doublets, which are
shown to emerge from the magnitude correlations inherent in the best model, and
are not the result of some arbitrary selection bias. Our results fundamentally
question the existing paradigm that the smallest earthquakes dominate
triggering whatever the magnitude of the target events.
