  We consider time dependent backgrounds with a time-like linear dilaton which
leads to a weakly coupled theory at asymptotic future known as the
Supercritical Linear Dilaton (SCLD) phase. Even after projecting out the
tachyon there are naive divergences in the partition function due to the
spectrum of relevant deformations. Although the partition function seems
divergent, the background is actually well defined in the far future due to
vanishing backreactions, provided we start from an appropriate initial state
called the "capping" state. We use the technique of interpolating functions to
construct this state by embedding the SCLD phase in a strong coupling
completion of string theory. Although we are unable construct a specific
example, we do provide the general strategy in cases where S-duality may
provide the completion.
