In this paper, we present empirical and theoretical evidence against a central but largely implicit assumption in circuit and sheaf discovery (CSD), which we term the Functional Anisotropy Hypothesis: the idea that functions in large language models (LLMs) are localised to a unique or near-unique internal mechanism. We show that a single LLM task can instead be supported by multiple, structurally distinct circuits or sheaves that are simultaneously faithful, sparse, and complete. To systematically uncover such competing mechanisms, we introduce Overlap-Aware Sheaf Repulsion, a method that augments the CSD objective with an explicit penalty on structural overlap across multiple discovery runs, enabling the discovery of circuits or sheaves with strong task performance but minimal shared structure across a plethora of common CSD benchmarks. We find that this phenomenon becomes increasingly pronounced as the number of discovered sheaves grows and persists robustly across major CSD methods. We further identify an ultra-sparse three-edge sheaf and show that none of its edges is individually indispensable, undermining even weakened notions of canonical or essential components. To explain these findings, we propose a Distributive Dense Circuit Hypothesis and provide a theoretical analysis demonstrating that non-unique, low-overlap circuit explanations arise naturally from high-dimensional superposition under mild assumptions. Together, our results suggest that mechanistic explanations in LLMs are inherently non-canonical and call for a rethinking of how CSD results should be interpreted and evaluated.
Circuit discovery aims to explain how language models (LMs) implement a specific task by localizing and interpreting a circuit, a computational subgraph responsible for the LM's behavior. Existing circuit discovery methods are hypothesis-driven; they first informally define a task with a dataset, and then apply a circuit discovery algorithm over that dataset to obtain a single circuit. This imposes two strong assumptions: that the LM implements the task with a single circuit, and that the dataset adequately represents the task as humans understand it. We systematically test these assumptions across four previously studied tasks and find that even minor dataset variations that preserve task semantics can produce circuits with low edge overlap and cross-dataset faithfulness. More strikingly, when applied to a mixed dataset with two distinct tasks whose separately discovered circuits have near-zero cross-faithfulness, existing methods still return a single circuit with high faithfulness across both tasks. This indicates that current methods discover dataset-specific circuits, rather than general task circuits. We propose Data-driven Circuit Discovery (DCD), a new discovery framework that drops both assumptions: instead of returning a single circuit for a dataset, DCD first clusters examples in the dataset by how similarly the model processes them and discovers a separate circuit for each group. This allows distinct mechanisms to appear separately rather than merged into a single circuit; each circuit explains its group, not the full task. Experiments show that DCD discovers multiple circuits per dataset, each more faithful to its group than a single circuit discovered by existing methods. Broadly, DCD lets the data reveal mechanistic structure within LMs, rather than relying on human-defined task boundaries that may not align with how models organize their computation.
Circuit discovery is a key technique in mechanistic interpretability to pinpoint the model components that are crucial for performing a given task. Although the current state-of-the-art method (EAP-IG) performs well on the metric of (un)faithfulness, it suffers from substantial variability. This includes resampling variance, where the circuit changes when we probe with a new batch of data from the same distribution; rephrasing variance, where the discovered circuit shifts when the prompts are rephrased; and sample-wise variance, where a circuit with low population unfaithfulness exhibits large fluctuations in unfaithfulness across individual samples. This paper studies the roots of these variances. We demonstrate that CEAP, our new circuit discovery method that improves upon EAP-IG with a theoretical guarantee, can substantially lessen resampling variance. We further show that rephrasing variance arises because prompts with different templates tend to activate different circuits in the model. This leads us to argue that it may be challenging to find a comprehensive circuit that explains and controls the model's behavior on a task, which can be expressed in countless templates, suggesting that LLMs may be inherently hard to steer. We show that sparsity, which has been claimed to form more compact and interpretable task circuits, fails to solve this problem. Regarding sample-wise variance, we argue that it is largely benign: extremely poor unfaithfulness scores often stem from how unfaithfulness is defined, rather than from defects in the measured circuits. We show that the magnitude of unfaithfulness is affected by selective contribution scaling, a neural mechanism that accounts for the extremely poor scores sometimes observed.
Frank Zhengqing Wu, Francesco Tonin, Volkan Cevher
Large Language Models (LLMs) have demonstrated remarkable performance across diverse tasks, from text summarization to question answering. Despite these capabilities, their black-box nature obscures internal decision-making processes. Mechanistic interpretability (MI) aims to address this by reverse-engineering neural networks into human-understandable algorithms. Current MI approaches for LLMs typically follow a two-stage paradigm: first identifying important components (circuit discovery), where components are typically individual nodes such as an attention head or feedforward neuron, and second determining the role they play in a certain task (functional interpretation). However, this sequential approach overlooks a fundamental insight: a component's importance and its functional role are inherently codependent. Unifying these stages presents two key challenges: (1) functional roles are often tied to specific nodes or components, limiting generalization, and (2) their identification relies on subjective interpretation rather than quantifiable metrics. To address these challenges, we propose S^3martCirc (Self-supervised Smart Circuit Discovery), a unified framework that simultaneously discovers circuits and interprets functionality. S^3martCirc abstracts node behavior into two general computational roles that generalize across tasks and defines a quantitative metric for assigning them, enabling importance and functional role to be discovered jointly rather than in sequence. Extensive experiments show that our framework outperforms existing methods in circuit discovery.