I'm very happy with my progress here! There is currently only a single remaining open problem that I'm working on relating to sieved matrices, which I discussed in this chost. Despite its roots in the INVERT transform and linear recurrences in general, it is wholly its own beast with its own eccentricities and related problems.
It's been really surprising to find how detailed the relationship between the signature function and base sequences, with so many identities directly relying on the exactness of the signature function. The only area which is lacking is the presence and application of signature convolution, but I've come to terms with the fact that I'm mortal and thus only capable of doing so much. But this definitely isn't the end!