  Quenched central galaxies tend to reside in a preferentially quenched
large-scale environment, a phenomenon that has been dubbed galactic conformity.
Remarkably, this tendency persists out to scales far larger than the virial
radius of the halo hosting the central. Therefore, conformity manifestly
violates the widely adopted assumption that the dark matter halo mass Mvir
exclusively governs galaxy occupation statistics. This paper is the first in a
series studying the implications of the observed conformity signal for the
galaxy-dark matter connection. We show that recent measurements of conformity
on scales R ~1-5 Mpc imply that central galaxy quenching statistics cannot be
correctly predicted with the knowledge of Mvir alone. We also demonstrate that
ejected (or `backsplash') satellites cannot give rise to the signal. We then
invoke the age matching model, which is predicated on the co-evolution of
galaxies and halos. We find that this model produces a strong signal, and that
central galaxies are solely responsible. We conclude that large-scale `2-halo'
conformity represents a smoking gun of central galaxy assembly bias, and
indicates that contemporary models of satellite quenching have systematically
over-estimated the influence of post-infall processes.
