Cyclical time is represented as a circle, replicated and causally validated

Cyclical temporal concepts (days of the week, months) are each independently represented as points on a circle in the residual stream, linearly decodable from activations [karkada-etal-2026] Despite the "toroidal" id, the confirmed finding is a single circle per cyclic variable, not a joint torus; no cited paper shows two such circles coexisting as orthogonal factors of one representation [karkada-etal-2026] Karkada et al. independently derive the same single-circle geometry analytically from translation-symmetric co-occurrence statistics [karkada-etal-2026] Engels et al. discover the same circles in GPT-2-small (layer 7) and Mistral-7B by clustering SAE dictionary elements, and causally validate them via activation patching on Mistral-7B and Llama-3-8B, where patching to a rotated point changes a day/month arithmetic answer [engels-etal-2024] Tiblias et al. recover the same circular date/month structure via a supervised distance-fit (sMDS) across Qwen2.5-3B, Llama-3.2-3B, and Gemma-2-2B, persisting at 3B/8B/70B Llama scale [tiblias-etal-2025] The circle is task-dependent: the identical date context collapses into a non-cyclic clustered or linear manifold when the completion cue asks for season or temperature rather than recency [tiblias-etal-2025] It is causally necessary: noise confined to the located 2D subspace degrades temporal-reasoning accuracy as much as noise across the full residual stream, while an equally-sized random subspace has negligible effect [tiblias-etal-2025] Bhalla et al. reproduce the days-of-week circle in Llama-3.1-8B (layer 19) via PCA and causally steer along its principal components to shift the predicted day-of-week token smoothly [bhalla-etal-2026]