Neuroscientific theories aim to explain paradigm cases of consciousness such as masking, binocular rivalry or the transition from dreamless sleep to wakefulness. The most popular theories are based on computational principles. Recurrent processing is a key feature of many of these theories, such as Information Integration Theory (Tononi, 2004), Lamme’s (2006) Theory of Recurrent Processing, and Grossberg’s Adaptive Resonnance Theory (2017). Here, we point to a mathematical result proving that recurrent processing in fact cannot explain paradigm cases of consciousness.
 It is well established that both recurrent and feedforward networks can implement any input-output mapping. For example, if a recurrent network can explain visual masking, then there exists a feedforward network that explains masking as well. Hence, recurrent processing is not necessary to explain paradigm cases. Recurrent processing is not sufficient either since, for example, the kneejerk reflex is unconsciously processed in a mono-synaptic recurrent loop. There is a double dissociation between paradigm cases of consciousness and recurrent processing. For example, in IIT, consciousness is quantified by a number, φ. Feedforward systems have φ=0 (they are unconscious) and recurrent systems have φ>0 (they are conscious). Although the human brain indeed has very high φ, there is a feedforward (φ=0) network, which shows the same input-output characteristics in the paradigm cases. Hence, φ is not necessary to explain the paradigm cases. Conversely, although unconscious processes such as regulating blood pressure can indeed be implemented in feedforward (φ=0) systems, they can also be implemented recurrently (φ>0). Hence, φ is not sufficient for consciousness either.
 We propose that many computational theories are too under-constrained to explain consciousness. We suggest that the best way to proceed is to philosophically and scientifically unearth the commonalities between paradigm cases.