The ‘causal-constitution’ fallacy, the ‘cognitive bloat’ worry, and the persisting theoretical confusion about the fundamental difference between the hypotheses of embedded (HEMC) and extended (HEC) cognition are three interrelated worries, whose common point—and the problem they accentuate—is the lack of a principled criterion of constitution. Attempting to address the ‘causal-constitution’ fallacy, mathematically oriented philosophers of mind have previously suggested that the presence of non-linear relations between the inner and the outer contributions is sufficient for cognitive extension. The abstract idea of (...) non-linearity, however, can be easily misunderstood and has, in the past, led to incorrect and counterintuitive conclusions about what may count as part of one’s overall cognitive system. In order to prevent any further mistakes I revisit dynamical systems theory to study the nature of the continuous mutual interactions that give rise to the aforementioned non-linear relations. Moreover, focusing on these interactions will allow us to provide two distinct arguments in support of the ontological postulation of extended cognitive systems, as well as an objective criterion of constitution. Accordingly, I put forward a version of HEC that treats continuous mutual interactions (and the resultant non-linear relations) not just as sufficient but also as necessary for cognitive extension. Such a qualified version of HEC may exclude certain alleged cases of cognitive extension where the agent does not mutually interact with his artifacts (e.g., shopping lists and directory services), but it is immune both to the ‘causal-constitution’ fallacy and the ‘cognitive bloat’ worry, and it can be sharply distinguished from HEMC. (shrink)