  In the single-degenerate (SD) channel of a Type Ia supernovae (SN Ia)
explosion, a main-sequence (MS) donor star survives the explosion but it is
stripped of mass and shock heated. An essentially unavoidable consequence of
mass loss during the explosion is that the companion must have an overextended
envelope after the explosion. While this has been noted previously, it has not
been strongly emphasized as an inevitable consequence. We calculate the future
evolution of the companion by injecting 2-6 10^47 ergs into the stellar
evolution model of a 1 Msun donor star based on the post-explosion progenitors
seen in simulations. We find that, due to the Kelvin-Helmholtz collapse of the
envelope, the companion must become significantly more luminous (10 - 10^3
Lsun) for a long period of time (10^3 - 10^4 years). The lack of such a
luminous "leftover" star in the LMC supernova remnant SNR 0609-67.5 provides
another piece of evidence against the SD scenario. We also show that none of
the stars proposed as the survivors of the Tycho supernova, including Tycho G,
could plausibly be the donor star. Additionally, luminous donors closer than
~10 Mpc should be observable with the Hubble Space Telescope starting ~2 years
post-peak. Such systems include SN 1937C, SN 1972E, SN 1986G, and SN 2011fe.
Thus, the SD channel is already ruled out for at least two nearby SNe Ia and
can be easily tested for a number of additional ones. We also discuss similar
implications for the companions of core-collapse SNe.
