  In this paper, we take advantage of a reinterpretation of differential
modules admitting a flag structure as a special class of perturbations of
complexes. We are thus able to leverage the machinery of homological
perturbation theory to prove strong statements on the homological theory of
differential modules admitting additional auxiliary gradings and having
infinite homological dimension. One of the main takeaways of our results is
that the category of differential modules is much more similar than expected to
the category of chain complexes, and from the K-theoretic perspective such
objects are largely indistinguishable. This intuition is made precise through
the construction of so-called anchored resolutions, which are a distinguished
class of projective flag resolutions that possess remarkably well-behaved
uniqueness properties in the (flag-preserving) homotopy category. We apply this
theory to prove an analogue of the Total Rank Conjecture for differential
modules admitting a ZZ/2-grading in a large number of cases.
