Cosine similarity approximates geodesic, not Euclidean, distance on feature manifolds

Modell et al. model a feature as a metric space whose representation is a homeomorphic image on a manifold in the unit sphere, and prove that under a continuous-correspondence hypothesis cosine similarity approximates a monotone function of on-manifold geodesic distance rather than ambient Euclidean distance [modell-etal-2025] The theorem is validated on known manifold features via Kendall's tau and Chatterjee's xi between predicted and true geodesic distance [modell-etal-2025] The hue circle and day-of-year circle are found in OpenAI text-embedding-3-large raw embeddings, not SAE features [modell-etal-2025] The months circle is an SAE feature in Mistral-7B and the year axis is an SAE feature in GPT-2-small layer 7 (following Engels et al. 2025), isometric only on a log scale log(2019 - year) [modell-etal-2025]