Arithmetic in the Wild: Llama uses Standard Addition to Reason About Cyclic Concepts

Does structure in representations imply structure in computation? We study how Llama-3.1-8B reasons over cyclic concepts (e.g., what month is three months after November?). Even though Llama-3.1-8B has circular representations for these concepts, we find that instead of directly computing modular addition in the period of the cyclic concept (e.g., 12 for months), the model re-uses a generic addition mechanism across tasks that operates independently of concept-specific geometry. First, it computes the sum of its two inputs using non-modular addition (three + November=14). Then, it maps back to cyclic concept space (14->February). We show that Llama-3.1-8B uses task-agnostic Fourier features to compute these sums—in fact, these features have periods that respect standard base-10 addition, e.g., 2, 5, and 10, rather than the cyclic concept period (e.g., 12). Furthermore, we identify a sparse, reusable set of 28 MLP neurons (approximately 0.2% of the MLP at layer 18) that can be partitioned into disjoint clusters that each compute the sum for a Fourier feature with a different period. Our work highlights how an interplay between causal abstraction and feature geometry can deepen our mechanistic understanding of LMs.