LLM attribute half-spaces intersect into a complete concept lattice

- The Lattice Representation Hypothesis models each binary attribute (e.g. "can fly") as a linear direction with a threshold, so a concept defined by co-occurring attributes is the convex polyhedral cone formed by intersecting their origin-passing half-spaces. [xiong-2026] - Theorem 1 proves the Galois closure of the softened, thresholded object-attribute relation is a complete lattice, and Proposition 1 absorbs per-attribute thresholds into one global embedding shift (a canonical origin-passing form). [xiong-2026] - Across five WordNet sub-hierarchies in LLaMA-3.1-8B, Gemma-7B and Mistral-7B, LDA/Fisher attribute directions recover the ground-truth formal context at F1 above 78% on physical domains (best Gemma-7B 83.2% on Animal) versus 53-68% mean-embedding and 45-48% random baselines. [xiong-2026] - Projection-profile subsumption scoring recovers WordNet's hypernym partial order (up to F1 77.1, LLaMA on Animal), and soft meet/join operators beat baselines at retrieving the true lowest-common-subconcept / least-common-hypernym (Table 3). [xiong-2026] - A scaling comparison across LLaMA-3 models (3B, 8B, 70B) shows scaling helps abstract domains far more than physical ones; observational, no intervention. [xiong-2026]