Language Models Represent and Transform Concepts with Shared Geometry

How concepts are represented in neural networks is a fundamental question in machine learning. The dominant view treats concept representations as stationary geometric objects. Yet concepts appear in context, and context transforms them. Drawing from neural population geometry, we formalize concept representations as point-cloud manifolds and contextual transformations as vector fields, and instantiate this framework in large language models. Across six model families of varying scales, we find that context moves each concept differently. The variance in these displacements is semantically organized, correlating with lexical concreteness and density. Importantly, both the concepts being transformed and this variance structure are shared across models: displacement structure transported from one model predicts held-out displacements in others significantly above chance. Together, these findings show that models share a common geometry not only in how concepts are represented, but more importantly in how context transforms them, a structure with richer organization than prior work has recognized.

The Representational Geometry of Number

A central question in cognitive science is whether conceptual representations converge onto a shared manifold to support generalization, or diverge into orthogonal subspaces to minimize task interference. While prior work has discovered evidence for both, a mechanistic account of how these properties coexist and transform across tasks remains elusive. We propose that representational sharing lies not in the concepts themselves, but in the geometric relations between them. Using number concepts as a testbed and language models as high-dimensional computational substrates, we show that number representations preserve a stable relational structure across tasks. Task-specific representations are embedded in distinct subspaces, with low-level features like magnitude and parity encoded along separable linear directions. Crucially, we find that these subspaces are largely transformable into one another via linear mappings, indicating that representations share relational structure despite being located in distinct subspaces. Together, these results provide a mechanistic lens of how language models balance the shared structure of number representation with functional flexibility. It suggests that understanding arises when task-specific transformations are applied to a shared underlying relational structure of conceptual representations.