  We prove a certain transcendence property of the unipotent Albanese map of a
smooth variety, conditional on the Ax-Schanuel conjecture for variations of
mixed Hodge structure. We show that this property allows the Chabauty-Kim
method to be generalized to higher-dimensional varieties. In particular, we
conditionally generalize several of the main Diophantine finiteness results in
Chabauty-Kim theory to arbitrary number fields.
