A Gauge Theory of Superposition: Toward a Sheaf-Theoretic Atlas of Neural Representations

Intelligent systems must maintain and manipulate task-relevant information online to adapt to dynamic environments. This capacity, known as working memory, is fundamental to human reasoning. Yet, human working memory is strikingly limited, maintaining only three to four items in a brain with billions of neurons. Surprisingly, large language models (LLMs), despite different substrates and direct access to prior context through attention, exhibit similar working memory limitations. Why should such different systems face analogous constraints? We propose that working memory limitations reflect a general trade-off of shared representations: representational compression and reuse support efficient learning and generalization, but also cause simultaneously active representations to interfere. We show a two-layer transformer trained on a working memory task can solve it perfectly, but diverse trained LLMs exhibit human-like limitations: performance declines with memory load, while retrieval is biased by recency and stimulus statistics. Mirroring humans, working memory performance in LLMs is also associated with broader model capability. Mechanistically, we show that LLMs encode multiple memories in entangled representations --- a condition we call in-context superposition --- and progressively suppress competing content while aligning the target with the readout. Moreover, a causal intervention that suppresses interfering information improves performance. Together, these findings suggest that working memory capacity reflects the ability to select task-relevant information under interference, a computational challenge shared by biological and artificial systems.

A defining feature of human intelligence is the ability to adapt to changing environments by inferring latent task structure from sparse observations. Neuroscientific research indicates that this capability relies on the hippocampus constructing abstract representations, expressed as low-dimensional, approximately orthogonal manifolds in neural state space. However, the internal mechanisms of large language models (LLMs) remain largely opaque, making it unclear whether they form comparable abstract representations or instead rely on task-specific statistical regularities when performing comparable reasoning tasks. Here we adapt a contextual reversal-learning paradigm to a text-based setting and compare humans and LLMs at both the Behavioural and representational levels. We report that although LLMs exhibit generalizable reasoning less frequently than humans, when such inference occurs, their internal states exhibit abstract geometric structures that resemble those reported in the hippocampus. Notably, this representational geometry is not uniformly distributed but is organized hierarchically across model depth: whereas lower layers show early, stable encoding of stimulus identity, higher layers form a hippocampal-like functional band enriched for abstract context geometry associated with inference. Furthermore, complementary intervention experiments mechanistically implicate geometry in reasoning: task-sequence language modelling induces geometric disentanglement, whereas geometric regularization of higher layers increases the emergence of generalizable inference. Together, these findings establish abstract representational geometry as a mechanistic principle supporting inference in large language models.

Existing hypotheses represent a concept in an LLM as a single point, a linear direction, or a Gaussian cluster, yet it remains unclear how and why such structures emerge. Here, we show that concept geometry can be precisely characterized via Laguerre Geometry, in which a concept is defined as a region--a Laguerre-Voronoi cell or a union of cells--allowing us to strictly define, measure, and separate concepts. Building on this formulation, we show that finer-grained concept structures, such as inclusion and hierarchy, are naturally revealed by the Laguerre weights. We then push this geometry inside the transformer. Decomposing each layer into piecewise-linear operators, we show that a token's hidden trajectory is governed by two coupled mechanisms: a static tree of self-contained piecewise-linear flow, and a dynamic transport that hops the trajectory across trees when cross-token attention fires. This decomposition yields Geometric Lens, a training-free, hyperparameter-free method for reading out the exact concept a hidden vector encodes at any layer. We also develop Laguerre Autoencoder, a 2D visualizer that renders both the decision geometry and a model's full reasoning trajectory in one view. Finally, we move beyond explanatory geometry toward actionable interpretability, showing that Geometric Lens recovers the correct factual token when a model is prompted with in-context interference. The code is available on GitHub.