In the previous chapter, we established a number of important variational properties of the subsidiary radii of a differential module over a disc or annulus. In this chapter, we continue the analysis by showing that under suitable conditions, one can separate a differential module into components of different subsidiary radii. That is, we can globalize the decompositions by spectral radius provided by the strong decomposition theorem, in case a certain numerical criterion is satisfied. As in the previous chapter, our discussion begins with some observations about power series, in this case identifying criteria for invertibility. We use these in order to set up a Hensel lifting argument to give the desired decompositions; again we must start with the visible case and then extend using Frobenius descendants. We end up with a number of distinct statements, covering open and closed discs and annuli, as well as analytic elements. As a corollary of these results, we recover an important theorem of Christol and Mebkhout. That result gives a decomposition by subsidiary radii on an annulus in a neighborhood of a boundary radius at which all of the intrinsic subsidiary radii tend to 1.