  Type Ia Supernovae (SNe Ia) are securely understood to come from the
thermonuclear explosion of a white dwarf as a result of binary interaction, but
the nature of that binary interaction and the secondary object is uncertain.
Recently, a double white dwarf model known as the dynamically driven
double-degenerate double-detonation (D6) model has become a promising
explanation for these events. One realization of this scenario predicts that
the companion may survive the explosion and reside within the remnant as a fast
moving ($V_{peculiar} >1000$ km s$^{-1}$), overluminous ($L > 0.1 L_\odot$)
white dwarf. Recently, three objects which appear to have these unusual
properties have been discovered in the Gaia survey. We obtained photometric
observations of the SN Ia remnant SN 1006 with the Dark Energy Camera over four
years to attempt to discover a similar star. We present a deep, high precision
astrometric proper motion survey of the interior stellar population of the
remnant. We rule out the existence of a high proper motion object consistent
with our tested realization of the D6 scenario ($V_{transverse} > 600$ km
s$^{-1}$ with $m_r < 21$ corresponding to an intrinsic luminosity of $L >
0.0176 L_\odot$). We conclude that such a star does not exist within the
remnant, or is hidden from detection by either strong localized dust or the
unlikely possibility of ejection from the binary system near parallel to the
line of sight.
