We introduce the neighbourhood lattice decomposition of a distribution, which is a compact, non-graphical representation of conditional independence that is valid in the absence of a faithful graphical representation. The idea is to view the set of neighbourhoods of a variable as a subset lattice, and partition this lattice into convex sublattices, each of which directly encodes a collection of conditional independence relations. We show that this decomposition exists in any compositional graphoid and can be computed efficiently and consistently in high-dimensions. In particular, this gives a way to encode all of independence relations implied by a distribution that satisfies the composition axiom, which is strictly weaker than the faithfulness assumption. Time permitting, we will discuss applications to interpreting large language models via partial orthogonality in embedding space.