Much like linguists seek to deduce properties of our language capacity from grammatical expressions, we believe, together with Balari et al. (2011) and Longa (2013), that a computation-oriented approach to the archaeological record open up interesting possibilities to reconstruct the nature of the capacity required to execute, use, and transmit the relevant artefacts. By focusing on the few robust differences that remain between Homo sapiens and Homo neanderthalensis artefacts (Longa 2013, Wynn et al. 2016), such as the Blombos engravings or complex projectile tools like the bow-and-arrow, we show that the interaction of at least three factors had to evolve to capture the complexity of early material cultural output of our species: self-domestication, changes in brain growth trajectory leading to the formation of certain brain circuits, and demographic expansion. None of these factors on their own are sufficient to capture the relevant artefact complexity, but their interactions can cause the sort of phase transition long suspected to underlie changes in material culture associated with our species.