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.

翻译：认知科学的一个核心问题是：概念表征究竟是收敛到一个共享流形以支持泛化，还是发散到正交子空间以最小化任务干扰。虽然先前的研究发现了支持这两种情况的证据，但对于这些特性如何共存并在不同任务间转换的机制性解释仍然难以捉摸。我们提出，表征共享并非存在于概念本身，而是存在于概念之间的几何关系中。以数字概念为测试平台，以语言模型作为高维计算基底，我们证明了数字表征在不同任务中保持稳定的关系结构。任务特定的表征嵌入在截然不同的子空间中，其中诸如大小和奇偶性等低级特征沿着可分离的线性方向进行编码。关键的是，我们发现这些子空间在很大程度上可以通过线性映射相互转换，这表明尽管表征位于不同的子空间中，它们仍共享关系结构。总之，这些结果为语言模型如何平衡数字表征的共享结构与功能灵活性提供了一个机制性的视角。它表明，当任务特定的转换应用于概念表征的共享底层关系结构时，理解便产生了。