  Multiple channels have been proposed to produce Type Ia supernovae, with many
scenarios suggesting that the exploding white dwarf accretes from a binary
companion pre-explosion. In almost all cases, theory suggests that this
companion will survive. However, no such companion has been unambiguously
identified in ancient supernova remnants -- possibly falsifying the accretion
scenario. Existing surveys, however, have only looked for stars as faint as
$\approx 0.1 L_\odot$ and thus would have missed a surviving white dwarf
companion. In this work, we present very deep DECAM imaging $(u, g, r, z)$ of
the Type Ia supernova remnant SN 1006 specifically to search for a potential
surviving white dwarf companion. We find no object within the inner third of
the SN 1006 remnant that is consistent with a relatively young cooling white
dwarf. We find that if there is a companion white dwarf, it must have formed
long ago and cooled undisturbed for $> 10^8$ yr to be consistent with the
redder objects in our sample. We conclude that our findings are consistent with
the complete destruction of the secondary (such as in a merger) or an
unexpectedly cool and thus very dim surviving companion white dwarf.
