Discontinuities permeate culture, and present a formidable challenge to mathematical models of cultural evolution. Cultural discontinuities have their origin in cognitive processes that include metaphor, analogy, cross-domain transfer, and self-organized criticality. This paper shows how cultural discontinuities can be accommodated by a theory of cultural evolution using cognitive reflexively autocatalytic foodset-generated (RAF) networks. RAF networks, originally developed to model the origin of life, have been used to models the origin of cognitive structure capable of evolving culture. Mental representations of knowledge and experience play the role of catalytic molecules, interactions amongst them (e.g., associations, affordances, or concept combinations) play the role of reactions, and thought processes are modelled as chains of such interactions. The approach tags mental representations with their source, i.e., whether they were acquired through social learning, individual learning (of pre-existing information), or creative thought (resulting in new information). This makes it possible to track the emergence and transformation of cultural novelty. We illustrate how the approach accommodates discontinuities using a historical example, and show how it is amenable to modelling cultural contributions of groups. We provide a RAF interpretation of the self-made worldview, and discuss how the approach can be used to think more concretely about possible future cultural trajectories. Because cross-domain thinking produces cultural discontinuities, it is impossible to pre-specify what features or traits will be present in future iterations of a cultural output. This suggests that cultural lineages are comprised not of external outputs or ‘memes’, but the conceptual networks that generate them.