Beyond the Lattice: A Graded Representation Hypothesis for LLM Concept Hierarchies

The Lattice Representation Hypothesis holds that large language models (LLMs) encode concept hierarchies as intersections of attribute half-spaces. Testing it on SNOMED CT, an ontology whose defining attributes are human-curated, we find that half-spaces are recovered reliably, but the lattice's inclusion score measures attribute overlap rather than subsumption: it orients parent-child pairs at only 59.5% and barely separates siblings from parents. Much of the directional information is lost in the inclusion aggregation, but a single linear direction in the embedding recovers it. In this paper, we propose the Graded Representation Hypothesis: hierarchy is realized by a relatedness subspace (the lattice) together with a specificity direction whose projection assigns each concept a grade, a real-valued grading that approximately preserves the order of the is-a poset. Experiments on SNOMED CT and WordNet show that the learned grade correctly orients 94% of held-out is-a edges, generalizes to unseen subtrees, transfers across ontologies without retraining, and, combined with one relatedness score, accounts for most of the linearly decodable subsumption signal with 68 parameters. These results suggest that LLMs encode not only which concepts are related but how abstract each concept is, as a single linear coordinate. Code will be available.