💬 NLP

All Circuits Lead to Rome: Rethinking Functional Anisotropy in Circuit and Sheaf Discovery for LLMs

This paper challenges the assumption that LLM functions rely on unique internal mechanisms by demonstrating that multiple structurally distinct, faithful circuits can coexist, introducing an overlap-aware method to uncover these non-canonical explanations and proposing a theoretical framework based on high-dimensional superposition to explain this inherent functional anisotropy.

Original authors: Xi Chen, Mingyu Jin, Jingcheng Niu, Yutong Yin, Jinman Zhao, Bangwei Guo, Dimitris N. Metaxas, Zhaoran Wang, Yutao Yue, Gerald Penn

Published 2026-05-14

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Original authors: Xi Chen, Mingyu Jin, Jingcheng Niu, Yutong Yin, Jinman Zhao, Bangwei Guo, Dimitris N. Metaxas, Zhaoran Wang, Yutao Yue, Gerald Penn

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Idea: There Is No Single "Right" Way to Solve a Problem

Imagine you ask a massive, super-intelligent robot (a Large Language Model) to solve a puzzle. For years, scientists believed that inside the robot's brain, there was one specific, unique path of wires and gears that made the answer happen. They thought if you found that one path, you would have found the "secret sauce" of how the robot thinks.

This paper argues that this belief is wrong.

Instead of one secret path, the robot has dozens of completely different paths that all lead to the exact same correct answer. It's like asking, "How did you get to the Empire State Building?" and getting three different answers:

"I took the subway."
"I walked up the stairs."
"I took a helicopter."

All three are true. All three got the person there. But the paths share almost no common ground. The paper calls the old belief the "Functional Anisotropy Hypothesis" (a fancy way of saying "functions are located in one specific spot"). The authors prove this is false.

The Experiment: Finding the "Hidden Roads"

To prove this, the researchers used a method called Circuit and Sheaf Discovery (CSD). Think of this as a way to turn off parts of the robot's brain to see which parts are actually doing the work.

Usually, scientists run this test once, find a "circuit" (a group of active wires), and say, "Aha! This is how it works!"

The authors did something different. They ran the test 20 times on the same task, but they added a special rule: "Don't pick the same wires you picked last time." They forced the discovery process to find new ways to solve the problem.

The Results:

They found 20 different "circuits" (groups of wires) that all solved the task perfectly.
These circuits looked nothing like each other. They used different layers of the brain and different connections.
The overlap between them was tiny (less than 5%). It was as if they were using completely different maps to get to the same destination.

The "Three-Edge" Surprise

The researchers took this a step further. They tried to find the smallest possible circuit. They found a "sheaf" (a functional circuit) made of only three edges (connections).

At first, it looked like these three edges were the "essential" core—the one thing you absolutely needed. But when they tested it, they realized: Even these three edges aren't strictly necessary.

If you remove one of those three edges, the robot can still find another set of three edges to do the job. It's like a city where you think a specific bridge is the only way across the river, but if you close it, traffic instantly reroutes through a completely different set of tunnels and ferries, and nobody notices the difference.

Why Does This Happen? (The "Superposition" Analogy)

The paper offers a theory for why this happens, called the Distributive Dense Circuit Hypothesis.

Imagine the robot's brain is a giant, crowded room full of people (neurons) talking at once. This is called superposition. Because there are so many people and so many ways to combine their voices, there are millions of different "combinations" of people that can say the same sentence.

Old View: There is one specific group of people who must speak to say "Hello."
New View: There are thousands of different groups of people who can all say "Hello" at the same volume and tone. If you silence one group, another group instantly steps up to say it for you.

Because the robot is so big and complex, it has built-in redundancy. It doesn't rely on one fragile path; it relies on a dense web of competing paths.

What This Means for Scientists

The paper doesn't say the robot is broken or that we can't understand it. It says we need to change how we interpret our findings.

Don't look for "The" Circuit: When scientists find a circuit, they shouldn't say, "This is the way the model works."
Look for "A" Circuit: They should say, "This is one valid way the model works, among many."

Summary

The paper "All Circuits Lead to Rome" tells us that Large Language Models are not like a machine with a single, unique engine. They are more like a massive, flexible city with infinite routes. If you block one road, the traffic finds another. There is no single "correct" map of how the model thinks; there are many valid maps, and they all lead to the same destination.

Technical Summary: All Circuits Lead to Rome

Problem Statement

The paper challenges a central, largely implicit assumption in the field of mechanistic interpretability known as the Functional Anisotropy Hypothesis. This hypothesis posits that specific capabilities or tasks within Large Language Models (LLMs) are supported by a single, unique, or near-unique internal mechanism (a specific circuit or sheaf). Under this view, model functions are assumed to be anisotropically distributed, meaning each capability corresponds to one privileged subgraph.

The authors argue that this assumption underpins dominant evaluation paradigms in Circuit and Sheaf Discovery (CSD), where success is measured by recovering a "ground truth" circuit or finding the smallest possible mechanism that preserves task performance. The paper questions whether these methods are actually identifying a unique underlying mechanism or merely one of many possible realizations.

Methodology

To systematically investigate the existence of competing mechanisms, the authors introduce Overlap-Aware Sheaf Repulsion (OASR).

Sheaf Discovery Framework: The work builds upon the DiscoGP framework, which formulates sheaf discovery as a differentiable edge-selection problem. A sheaf is defined as a subgraph that causally contributes to computation and can independently sustain task performance (unlike a standard circuit which may rely on patched activations).
OASR Mechanism: Standard discovery methods often converge to similar solutions due to random initialization or optimization landscapes. OASR modifies the objective function by adding an explicit overlap loss ( $L_{overlap}$ $L_{o v er l a p}$ ).
- After discovering an initial sheaf $R$ , the method penalizes the expected activation of edges already selected in $R$ .
- The combined objective is: $L = L_{fidelity} + \lambda_s L_{sparsity} + \lambda_c L_{complete} + \lambda_o L_{overlap}(R)$ .
- This "repulsion" forces the optimizer to find alternative sheaves that are faithful to the task but structurally distinct (minimizing Intersection-over-Union, or IoU) from previously discovered ones.
Experimental Scope: The authors apply OASR across multiple benchmark tasks (Indirect Object Identification [IOI], BLiMP, AGA, ANA, DNA variants, and Docstring) and models (GPT-2 Small and Pythia-160M). They also validate their findings across other CSD methods, including ACDC (heuristic), EAP (gradient-based attribution), and Edge Pruning (EP).

Key Results

1. Existence of Functionally Equivalent, Low-Overlap Sheaves

The primary empirical finding is that a single LLM task can be faithfully performed by multiple, structurally distinct sheaves.

IOI Case Study: Using OASR, the authors discovered two sheaves for the IOI task that both achieved 100% accuracy. However, their structural overlap (IoU) was only 4.1%, close to random chance.
Generalization: This phenomenon holds across all tested benchmarks. For every task, pairs of sheaves with high performance but very low IoU (typically 4–11%) were discovered.
Scaling Effect: As the number of discovered sheaves increases (up to 20 runs), the mutual intersection of edges shrinks drastically (often to <1% IoU), while the union of edges grows. This indicates that the space of valid mechanisms is vast and does not converge to a single core.

2. Persistence Across Methods

The breakdown of functional anisotropy is not an artifact of the OASR method or DiscoGP alone:

ACDC: Varying the traversal order of attention heads (a non-semantic ordering choice) resulted in significantly different circuits with low IoU, despite using the same thresholds.
EAP: Changing task-irrelevant information (e.g., swapping names in IOI prompts) led to systematically different circuits, challenging the stability of attribution-based methods.
Edge Pruning: When using task-specific loss functions, EP also exhibited high diversity in discovered circuits, similar to DiscoGP.

3. Absence of Indispensable Components

The authors tested a "weakened" version of the anisotropy hypothesis: that while multiple sheaves exist, they might share a small, essential "core" of indispensable edges.

Ultra-Sparse Sheaf: They identified an ultra-sparse three-edge sheaf for IOI that achieved 86.7% accuracy in isolation.
Non-Indispensability: However, when the task was decomposed (e.g., separating ABBA and BABA templates) or when these three edges were explicitly forbidden, the discovery process still found other sparse, high-performing sheaves.
Conclusion: No single edge (or even the three-edge core) is strictly indispensable for all functionally valid mechanisms. The apparent "essentiality" of the core arises only when the task is treated as an aggregated whole.

4. Theoretical Analysis

The paper proposes the Distributive Dense Circuit Hypothesis to explain these findings.

Theoretical Proof: Under mild assumptions (residual additivity and local linearity), the authors prove the existence of multiple structurally distinct, low-overlap circuits that are $\epsilon$ -faithful to the same task.
Mechanism: In high-dimensional spaces, the number of possible edge subsets grows combinatorially. Due to linear superposition, distinct subsets of edges can produce nearly identical readout vectors (subset-sum collisions). If the task decision is protected by a finite margin, these structural differences do not affect the final prediction, allowing for multiple valid explanations.

Significance and Claims

The paper does not claim to invalidate Circuit and Sheaf Discovery (CSD) as a field; rather, it calls for a fundamental rethinking of how CSD results should be interpreted and evaluated.

Rejection of Uniqueness: The results suggest that mechanistic explanations in LLMs are inherently non-canonical. A discovered circuit or sheaf represents one valid realization among many in a space of functionally equivalent mechanisms, not the mechanism underlying a task.
Critique of Evaluation: Current evaluation paradigms that seek a "ground truth" or a "minimal" unique circuit are based on a flawed assumption. The paper argues that the search for a single sparse, indispensable subgraph is insufficient to explain the observed empirical non-uniqueness.
Distributed View: The authors advocate for a distributed view of computation, where task behavior emerges from a dense regime of competing, partially redundant mechanisms.
Implications: This challenges the notion of "essential" components and suggests that redundancy in LLMs is not merely a latent backup system (activated only upon ablation) but a fundamental feature of normal model operation.

In summary, the paper provides both empirical evidence and theoretical justification that "All Circuits Lead to Rome"—multiple distinct structural pathways can and do support the same functional behavior in LLMs, undermining the assumption of functional anisotropy.