Cognitive science is confronted by several fundamental anomalies deriving from the mind–body problem. Most prominent is the problem of mental causation and the hard problem of consciousness, which can be generalized into the hard problem of agential efficacy and the hard problem of mental content. Here, it is proposed to accept these explanatory gaps at face value and to take them as positive indications of a complex relation: mind and matter are one, but they are not the same. They are related in an efficacious yet non-reducible, non-observable, and even non-intelligible manner. Natural science is well equipped to handle the effects of non-observables, and so the mind is treated as equivalent to a hidden ‘black box’ coupled to the body. Two concepts are introduced given that there are two directions of coupling influence: (1) irruption denotes the unobservable mind hiddenly making a difference to observable matter, and (2) absorption denotes observable matter hiddenly making a difference to the unobservable mind. The concepts of irruption and absorption are methodologically compatible with existing information-theoretic approaches to neuroscience, such as measuring cognitive activity and subjective qualia in terms of entropy and compression, respectively. By offering novel responses to otherwise intractable theoretical problems from first principles, and by doing so in a way that is closely connected with empirical advances, irruption theory is poised to set the agenda for the future of the mind sciences.