Semantic hierarchy is encoded as orthogonality between concept levels

Park et al. prove (Theorem 8) that sibling-contrast directions at one hierarchy level are orthogonal to the parent's contrast direction and to any ancestor level [park-etal-2024] Combined with per-level categorical polytopes, this gives a direct-sum-of-polytopes structure where each level occupies its own orthogonal subspace, confirmed in Gemma-2B and Llama-3-8B (cosine near zero versus a shuffled-labels control) [park-etal-2024] The authors argue ordinary high-dimensional Euclidean space suffices, explicitly contrasting with hyperbolic-embedding accounts of hierarchy [park-etal-2024] Marinov et al. confirm the same cross-level orthogonality for language family versus individual language across Qwen3-4B, Mistral-7B-v0.3, and Llama-3-8B via a causal inner product [marinov-etal-2026]