Technical Summary: Arithmetic in the Wild

Problem Statement

The paper investigates the algorithmic role of representation geometry in large language models (LLMs), specifically focusing on how models reason about cyclic concepts (e.g., months, weekdays, hours). While prior work has established that LLMs represent these concepts using circular geometry (Engels et al., 2025) and that small transformers trained on modular arithmetic operate over periodic bases (Nanda et al., 2023), a critical question remains: Does the circular structure of the input representation imply a circular computation?

The authors hypothesize that Llama-3.1-8B might compute answers to questions like "What month is six months after August?" by rotating along a circular manifold. However, they find that the model does not perform direct modular arithmetic within the concept's specific period (e.g., modulo 12 for months). Instead, the model utilizes a generic, task-agnostic base-10 addition mechanism before mapping the result back to the cyclic space.

Methodology

The study employs a combination of causal analysis and geometric probing to dissect the internal mechanisms of Llama-3.1-8B.

1. Causal Abstraction via Distributed Alignment Search (DAS):

   • The authors use DAS (Geiger et al., 2023) to localize low-rank subspaces in the residual stream that encode causal variables: the input concept, the offset, and the output concept.
   • They measure Interchange Intervention Accuracy (IIA) to determine when and where these variables are combined. This involves patching hidden states from counterfactual prompts into original prompts to see if the model's output changes predictably.
   • This analysis isolates the specific layer and token position where the computation occurs.

2. Cross-Task Patching:

   • To test the hypothesis of a shared mechanism, the authors perform "patching" experiments between standard arithmetic tasks ($a+b=$) and cyclic tasks (e.g., months).
   • They patch the residual stream from an addition prompt into a cyclic prompt (and vice versa) within the shared DAS subspaces to observe if the model outputs the sum or the cyclic concept, respectively.

3. Fourier Probing and Steering:

   • The authors train linear probes to detect Fourier features (sinusoidal representations) with specific periods $T \in \{2, 5, 10, 20, 50, 100\}$, known to represent base-10 numbers (Kantamneni & Tegmark, 2025).
   • They verify if these features, trained on standard addition, are present and causal in cyclic tasks.
   • They use these probes to steer the model's internal state, forcing the "pre-modulo sum" to a counterfactual value to see if the output shifts accordingly.

4. Neuron Identification and Ablation:

   • The authors identify a sparse set of MLP neurons at Layer 18 that write to the DAS output subspaces.
   • They analyze the activation patterns of these neurons, partitioning them into clusters based on their periodicity.
   • They perform ablation studies (zeroing out specific neurons) to quantify their causal contribution to the model's performance.

Key Results

1. A Shared Base-10 Addition Mechanism

Contrary to the hypothesis of circular computation, the model computes cyclic tasks using a base-10 addition mechanism.

• Two-Step Process: The model first computes the sum of the inputs in numerical space (e.g., $6 + 8 = 14$) and only in later layers maps this sum back to the cyclic concept space (e.g., $14 \to \text{February}$).
• Layer 18 Localization: Causal analysis reveals that input concepts and offsets are combined at the Layer 18 MLP at the last token position. Crucially, the circular geometry of the output concept (e.g., the "weekdays circle") is not present at Layer 18; it only emerges in layers 22–25.
• Cross-Task Patching: Patching from an addition prompt ($6+8=$) into a cyclic prompt causes the model to output the cyclic concept corresponding to the sum (e.g., February). Conversely, patching from a cyclic prompt into an addition prompt causes the model to output the pre-modulo sum (e.g., 14), even if the model fails to produce the correct cyclic answer in a clean run.

2. Fourier Features and Task Agnosticism

The model uses task-agnostic Fourier features to perform the addition.

• The internal representations respect standard base-10 periodicities ($T \in \{2, 5, 10, 20, 50, 100\}$) rather than the specific period of the cyclic concept (e.g., 12 for months or 7 for weekdays).
• Fourier probes trained on standard addition successfully steer cyclic tasks, confirming that the same geometric features drive the computation across domains.

3. Sparse Neuron Circuit

The authors identify a sparse set of 28 MLP neurons at Layer 18 (approximately 0.2% of the layer's MLP) that are responsible for this addition.

• Clustered by Period: These neurons can be partitioned into disjoint clusters, where each cluster computes the sum for a specific Fourier period (e.g., a cluster for $T=2$, another for $T=5$, etc.).
• Mechanism: The neurons operate in a distributed manner. Some neurons exhibit "split" activation patterns (reading the input concept via the gate vector and the offset via the up vector), while others are "mixed." Their down-projection vectors sum to a vector pointing to the correct angle on the Fourier plane, encoding the sum modulo $T$.
• Ablation Impact: Zero-ablating these 28 neurons causes a drastic drop in accuracy (e.g., from 95% to 24% for addition), while ablating all other Layer 18 neurons retains most performance.

Significance and Claims

The paper claims to provide a mechanistic understanding of how large language models perform arithmetic over cyclic concepts, bridging the gap between representation geometry and causal computation.

• Geometry vs. Computation: The work demonstrates that while representations may be circular, the underlying computation is not necessarily modular arithmetic on that circle. Instead, the model re-uses a generic base-10 addition circuit, mapping the result to the target domain only after the calculation is complete.
• Interplay of Methods: The study highlights the power of combining causal abstraction (to locate where information is processed) with geometric analysis (to understand how it is represented). Causal analysis localized the computation to Layer 18, while geometric analysis revealed that the computation relies on base-10 Fourier features rather than task-specific circular structures.
• Sparse Circuits: The identification of a small, reusable set of neurons (28 neurons) that perform addition across diverse tasks (months, days, hours, numbers) suggests that LLMs develop highly efficient, shared subcircuits for fundamental arithmetic operations, regardless of the semantic domain of the input.

The authors remain modest regarding generalization, noting that these specific mechanisms were identified in Llama-3.1-8B and may not generalize to all models or all types of cyclic reasoning (e.g., backward offsets or distance queries were outside the scope). However, the findings offer a concrete example of how "arithmetic in the wild" is implemented via the interplay of sparse neurons and Fourier geometry.