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Sparse autoencoders have recently produced dictionaries of high-dimensional vectors corresponding to the universe of concepts represented by large language models. We find that this concept universe has interesting structure at three levels: (1) The “atomic” small-scale structure contains “crystals” whose faces are parallelograms or trapezoids, generalizing well-known examples such as (man:woman::king:queen). We find that the quality of such parallelograms and associated function vectors improves greatly when projecting out global distractor directions such as word length, which is efficiently performed with linear discriminant analysis. (2) The “brain” intermediate-scale structure has significant spatial modularity; for example, math and code features form a “lobe” akin to functional lobes seen in neural fMRI images. We quantify the spatial locality of these lobes with multiple metrics and find that clusters of co-occurring features, at coarse enough scale, also cluster together spatially far more than one would expect if feature geometry were random. (3) The “galaxy”-scale large-scale structure of the feature point cloud is not isotropic, but instead has a power law of eigenvalues with steepest slope in middle layers. We also quantify how the clustering entropy depends on the layer. 

This paper presents an ACT-R model designed to simulate voting behavior on full-face paper ballots. The model implements a non-standard voting strategy: the strategy votes first from left to right on a ballot and then from top to bottom. We ran this model on 6600 randomly-generated ballots governed by three different variables that affected the visual layout of the ballot. The findings suggest that our model’s error behavior is emergent and sensitive to ballot structure. These results represent an important step towards our goal of creating a software tool capable of identifying bad ballot design.