Circles or Orderings? A Symmetry Test for Cyclic Concepts in Language Models

The linear representation hypothesis treats a concept as a direction. Cyclic concepts such as days of the week and months look like an exception: several studies report that language models place them on a circle (Engels et al., 2025; Karkada et al., 2026; Feucht et al., 2026). We show that the tests behind such reports cannot tell a cycle from an ordering. A fitted successor operator has its spectrum at the $K$th roots of unity for any tokens in a fixed order, random nouns included. Concentration of variation in the fundamental Fourier mode is as high for number words, ordinals, and temperature adjectives as it is for months. A short step from the last element back to the first also appears for graded scales whose two extremes lie close together. We derive a test from the symmetry itself. Advancing a concept by one step acts as an isometry on its $K$ states exactly when their Gram matrix is circulant, so we measure the departure from circulance. Across nine models from three families (Pythia 160M to 12B, Llama-3.1-8B base and instruct, Qwen2.5-7B-Instruct), days and months are more circulant than every same-length ordered control in every model, measured at the concept token. Clock hours cannot be distinguished from a temperature scale, and compass directions are organized by cardinal versus intercardinal type rather than as an eight-cycle. Behavior does not follow automatically from the representation. With the same closed representation of months, Llama shows a wrap-around penalty consistent with the base-10 mechanism reported by Feucht et al. (2026), while Qwen does not. On the compass, Llama never answers a 45-degree step correctly, which matches its two-axis representation.