In this chapter, we introduce Dwork’s technique of descent along Frobenius in order to analyze the generic radius of convergence and subsidiary radii of a differential module, primarily in the range where Newton polygons do not apply. In one direction, we introduce a somewhat refined form of the Frobenius antecedents introduced by Christol and Dwork. These fail to apply in an important boundary case; we remedy this by introducing the new notion of Frobenius descendants. Using these results, we are able to improve a number of results from Chapter 6 in the special case of differential modules over ??_??. For instance, we get a full decomposition by spectral radius, extending the visible decomposition theorem and the refined visible decomposition theorem. We will use these results again to study variation of subsidiary radii, and decomposition by subsidiary radii, in the remainder of Part III.