The Approximation Rank of Softmax Attention: Sharp Geometric Laws and Robust Interaction Dimension

Which geometry controls the rank complexity of normalized softmax attention? We study maximum-row-$\ell_1$ approximation rank, exactly the least unrestricted rank preserving every bounded vector-valued output. Two sharp worst-case laws isolate support geometry: for fixed $d$ and error $\varepsilon$, spherical self-attention has rank $\Theta_{d,\varepsilon}(\min\{n,(1+\beta)^{(d-1)/2}\})$, while full-ball geometry adds one radial degree and, for $\beta\ge\beta_0(d,\varepsilon)$ and $n\ge C_de^{\beta/8}$, gives $\Theta_{d,\varepsilon}(\beta^{d/2})$. For a fixed head, row-softmax then quotients away row-scalar logit directions: the remaining visible query–key interaction dimension $r$ yields an $r/2$ per-instance upper law, and bounded constructions show this exponent is minimax sharp. Approximate interaction subspaces incur an explicit residual output error and yield a tolerance-indexed SVD dimension. On an 84-head BERT-base calibration set, we observe modest effective-dimension reductions across many head–temperature settings, together with positive associations with finite constructive rank upper certificates.