There is much to be said for a diachronic or interpersonal individuation of singular modes of presentation (MOPs) in terms of a criterion of epistemic transparency between thought tokens. This way of individuatingMOPs has been discussed recently within themental filesframework, though the issues discussed here arise for all theories that individuateMOPs in terms of relations among tokens. All such theories face objections concerning apparent failures of the transitivity of the ‘sameMOP’ relation. For mental files, these transitivity failures most obviously occur because mental files can merge or undergo fission. In this paper I argue that this problem is easily resolved once mental files are properly construed ascontinuants, whose metaphysics is analogous to that of persons or physical objects. All continuants can undergo fission or fusion, leading to similar transitivity problems, but there are well‐established theories of persistence that accommodate this. I suggest that, in particular, the stage theory best suits the purposes of a continuant theory ofMOPs.