For a specified set of available feed streams and a specified network of chemical reactions, the attainable region is the full set of output states that can be attained by means of all possible reactor designs (within a generally broad class). The boundary of the attainable region is of great interest, for it represents the outer limit of what is actually achievable. In consideration of isothermal designs involving only reaction and mixing, it was shown in Feinberg & Hildebrandt (1997. Chemical Engineering Science, 52, 1637–1665), how, independent of the reaction network or its kinetics, the attainable region's boundary is invariably shaped by classical plug flow reactors, ideal CFSTRs and differential sidestream reactors (DSRs), each playing special roles. Differential sidestream reactors generally serve as crucial precursors to plug flow reactors, which, in turn, provide final access to the attainable region's extreme points. Here we examine properties of critical DSRs — that is, DSRs whose products lie entirely on the attainable region's boundary. We show that critical DSRs must conform to highly detailed design equations, equations that govern in a precise way the strategic addition of sidestream along the reactor's extent. In particular, the equations indicate how the sidestream addition rate, the sidestream composition, and the reaction rates must be orchestrated if the DSR is to give rise to products at the outer limit of what is achievable.