Proceedings of the Annual Meeting of the Cognitive Science Society

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 found 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 \emph{geometric relations} between them. Using number concepts as a target domain and language models as high-dimensional computational testbeds, 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 near-orthogonal axes. Crucially, we find that these subspaces are largely transformable into one another via linear mappings, indicating that task-specific representations, despite being located in distinct subspaces, share relational structure.