On the geometry and topology of representations: the manifolds of modular addition

Abstract: The Clock and Pizza interpretations, associated with architectures differing in either uniform or learnable attention, were introduced to argue that different architectural designs can yield distinct circuits for modular addition. In this work, we show that this is not the case, and that both uniform attention and trainable attention architectures implement the same algorithm via topologically and geometrically equivalent representations. Our methodology goes beyond the interpretation of individual neurons and weights. Instead, we identify all of the neurons corresponding to each learned representation and then study the collective group of neurons as one entity. This method reveals that each learned representation is a manifold that we can study utilizing tools from topology. Based on this insight, we can statistically analyze the learned representations across hundreds of circuits to demonstrate the similarity between learned modular addition circuits that arise naturally from common deep learning paradigms.

• The paper demonstrates that deep networks trained on modular addition converge to either a disc or torus manifold, confirming a universal representation structure.
• It uses closed-form representation theorems, PCA visualizations, and persistent homology to rigorously characterize the geometric and topological properties of neural representations.
• The study challenges prior interpretability metrics by showing that only MLP-Concat exhibits distinct circuit behavior, while other architectures share equivalent manifold structures.

The Geometry and Topology of Neural Representations in Modular Addition

Introduction

This paper investigates the internal representations learned by deep networks performing modular addition, a canonical task in mechanistic interpretability. The authors challenge previous claims that different architectures (particularly, transformers with uniform vs. learned attention) yield qualitatively distinct “circuits” for modular addition. Instead, they formally and empirically demonstrate that the representations underlying these architectures are geometrically and topologically equivalent, resolving apparent contradictions to “universality” in circuit formation. Their analysis leverages closed-form representation theorems, large-scale empirical surveys over hundreds of circuits, and tools from topological data analysis (TDA), particularly Betti numbers and persistent homology.

Theoretical Analysis of Representation Manifolds

The core theoretical contribution is the establishment of an equivalence class for the learned representations in models trained on modular addition. Leveraging the “simple neuron” model—supported by earlier mechanistic interpretability research—the authors prove that, for all relevant architectures (MLP with summed or concatenated embeddings, transformers with uniform or trainable attention), the first layer representations can be classified into one of two low-dimensional manifolds: a vector-addition disc (“pizza”) or a torus.

Under the commutativity and symmetry inherent in modular addition, the learned phase parameters in neural activations force the representation to collapse onto a disc or torus, which are linearly or nonlinearly projected depending on total phase decorrelation. The closed-form structure of these manifolds is precisely characterized (Theorem 1), and crucially, the clock solution (a pure angle-sum) cannot arise generically under these conditions.

Geometric Visualization and Manifold Structure

Principal component analysis of neuron clusters substantiates the above theory. For MLP-Add, Pizza, and Clock networks, a 2D disc is recovered; for MLP-Concat, a 4D torus emerges. All transformer variants cluster with MLP-Add and Pizza, not with MLP-Concat, confirming that “Clock” and “Pizza” are not genuinely disparate solutions in terms of learned representation geometry.

Figure 1: PCA of the first layer neuron pre-activations in single-frequency clusters—MLP-Add, Pizza, and Clock form nearly identical discs, while MLP-Concat exhibits a toroidal geometry.

Cluster-averaged post-activation heatmaps locate the strongest neuron responses along the $a=b$ diagonal for MLP-Add, Pizza, and Clock. This is invariant across network initializations and architectures, again separating MLP-Concat from the rest.

Figure 2: Normalized sum of post-activations in clusters—“Pizza” and “Clock” have activations concentrated along $a=b$, while MLP-Concat’s activations are uniformly distributed on the torus.

The mapping from the torus (or vector-addition disc) to the output logit annulus is elaborated, demonstrating that intermediate and final representations are consistent with the algebraic structure of cyclic groups. The disc and torus are the only robust attractors in representation space, and all architectures project toward the correct circle topology at the output layer.

Figure 3: Different factorizations of the torus-to-circle map, showing the collapse to a circle (logit manifold) in later layers.

Statistical and Topological Analysis

The analysis is backed by exhaustive empirical sampling—703 networks per architecture, over all $(a,b)$ input pairs. The “Phase Alignment Distribution” (PAD), capturing the $(a,b)$ pairs where a cluster’s neurons fire maximally, exhibits strong diagonal concentration for the transformer and MLP-Add/Pizza families, while MLP-Concat shows widespread toroidal coverage.

Figure 4: Log-density heatmaps of the PAD by architecture—Attention 0.0 and 1.0 (Pizza and Clock) are almost indistinguishable and aligned with MLP-Add.

Histogram analysis with a torus-distance metric quantifies this separation. Attention 0.0 and 1.0 neurons display tight alignment to the diagonal ($a=b$), in stark contrast to MLP-Concat.

Figure 5: Histograms of torus distance of neuron phase from the $a=b$ diagonal; MLP-Add is perfectly aligned, attention models are close, MLP-Concat is widely distributed.

Furthermore, persistent homology and Betti number tracking across layers reveal disc topologies in the first layer for MLP-Add, Attention 0.0, and 1.0. These converge to circle topology at the logits in deeper networks, while MLP-Concat transitions directly from torus to circle, bypassing the disc intermediary.

Figure 6: Betti number distributions over layers and seeds, showing MLP-Add, Pizza, and Clock architectures are topologically equivalent.

Evaluation of Prior Interpretability Metrics

The authors provide a thorough critique of previous metrics—gradient symmetricity and distance irrelevance—introduced to distinguish “Clock” from “Pizza” circuits. They confirm that these metrics do not reliably separate transformers with trainable versus uniform attention; in particular, neither metric consistently distinguishes Attention 1.0 from 0.0. Instead, only MLP-Concat stands apart under all metrics, echoing the findings from geometric and topological evaluations.

Figure 7: Gradient symmetricity and distance irrelevance for all architectures; only MLP-Concat consistently separates, confirming limited utility of prior heuristics.

Implications and Future Directions

This work rigorously resolves previous contradictions to the universality hypothesis—specifically, that similar architectures trained on identical data learn similar circuits. Here, “similarity” is formalized as topological and geometric equivalence of the representation manifold, not the superficial identity of parameter values or neuron-level motifs. The manifold hypothesis (that learned representations lie on low-dimensional, structured manifolds) is thus substantiated in this algebraic task.

The findings are consequential for mechanistic interpretability, suggesting that for well-structured tasks, neural networks reliably recover universal or linearly-projected manifolds encoding the algorithmic structure. It remains to be seen if such global structure persists in more complex, structured, or high-dimensional AI tasks, but the topological data analysis pipeline introduced here forms a quantitative foundation for future work.

Further, the authors hypothesize that the universality observed is a consequence of DNNs finding a “universal” representation manifold—typically a torus or its projection—in the algebraic latent space. Extrapolating this insight may inform hypotheses about manifold alignment, invariance, and representation engineering in both algorithmic and real-world domains.

Conclusion

By combining theoretical analysis, large-scale empirical evaluation, and topological tools, this paper convincingly demonstrates that transformer and MLP architectures trained on modular addition converge to geometrically and topologically equivalent internal representations. The supposed dichotomy between “Pizza” and “Clock” circuits is resolved—they are distinct projections of the same underlying torus structure. This result reinforces the plausibility of universality in representation learning and highlights the power of topological methods for mechanistic interpretability. Future research can now build on these quantitative techniques to assess representation geometry in less algebraic, more realistic neural network applications.