A multimodal model writes numbers along different directions for text and images, but one rotated frame covers both
AspireAI community · arXiv
When a model adds 17 and 35 from text and 91 and 39 from a rendered image, does it use one internal concept of number? Similarity metrics cannot settle it, because they score identical and rotated directions the same way. This paper uses causal interventions instead: it learns a small subspace of Gemma4-12B's residual stream in which swapping coordinates swaps the model's answer.
Reusing the text subspace on an image recovered only 53.2 percent of its causal effect. A single rotation and rescaling lifted that to 91.4 percent, so the two modalities share one geometry expressed in different coordinate frames. Splitting each subspace into 9 directions the output layer can read and 13 it cannot, the hidden directions still carried the number while barely changing the digit being written. When two numbers share their first digit, swapping the number in those hidden directions left the visible digit intact but redirected the next digit 77.6 percent of the time.
The scope is narrow in a way that matters: this is one 12B model, arithmetic over a handful of conditions, and a subspace fitted per condition rather than a general account of how any model represents quantity. The authors write that the finding is that a model forms an abstract numerical concept upstream and unfolds it across time, which is a strong claim resting on a single system.
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