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Abstract 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> 0.1L_⊙) white dwarf. Recently, three objects that 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⁻¹withm_r< 21 corresponding to an intrinsic luminosity ofL> 0.0176L_⊙). 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 almost parallel to the line of sight.