Cyclic-concept arithmetic uses base-10 Fourier addition, not a modular clock

measured in 1 paper

Feucht, Haklay et al. confirm Llama-3.1-8B represents cyclic concepts (months, weekdays, hours) as circles at the input-token position [feucht-haklay-etal-2026-arithmetic-in-the-wild] Distributed alignment search causally isolates the input concept and offset (rank k=8-16) with over 95% accuracy before layer 18, but the circular representation is not recoverable at the final-token position until layers 22-25 [feucht-haklay-etal-2026-arithmetic-in-the-wild] Offset arithmetic is computed with the same base-10 Fourier periods (T in {2,5,10,20,50,100}) found for plain integer addition, not a mod-12/mod-7/mod-24 clock mechanism [feucht-haklay-etal-2026-arithmetic-in-the-wild] Cross-task activation patching between addition and cyclic tasks recovers the pre-modulo sum, and the computation localizes to a shared 28-neuron layer-18 circuit (~0.2% of the MLP) [feucht-haklay-etal-2026-arithmetic-in-the-wild] This dissociates representational geometry (a circle) from computational geometry (base-10 Fourier addition) for the same concept [feucht-haklay-etal-2026-arithmetic-in-the-wild]

Structure

Generalized Helix, Circle

Context

dissociation between representational geometry (circle, for the cyclic concept itself) and computational geometry (base-10 Fourier/helix addition, for the offset arithmetic), DAS-based causal isolation of input concept/offset (rank k=8-16) with >95% accuracy, contrasted with late re-emergence of the circular representation only at layers 22-25, extension of Kantamneni & Tegmark's Fourier-period set (T=2,5,10,100) to include T=20,50, replicated as causal for cyclic-concept arithmetic, cross-task activation patching (addition<->months/weekdays/hours) demonstrating a shared computational mechanism, a sparse, shared 28-neuron layer-18 circuit (~0.2% of the MLP) computing offsets for all four tasks via disjoint per-period clusters, explicit rejection of a modular (mod-12/mod-7/mod-24) "Clock" algorithm extension for cyclic-concept reasoning

Confirmed in models

Llama-3.1-8B · Llama

Method

Distributed Alignment Search (DAS), Activation patching, Activation Steering (Addition)

Papers

Arithmetic in the Wild: Llama Uses Base-10 Addition to Reason About Cyclic Concepts — Feucht, Sheridan, Haklay, Tal, Bhalla, Usha, Wurgaft, Daniel, Rager, Can, Sarfati, Raphaël, Merullo, Jack, McGrath, Thomas, Lewis, Owen, Lubana, Ekdeep Singh, Fel, Thomas, Geiger, Atticus2026 · arXiv:2605.01148