Intrinsic Interaction Geometry Controls the Low-Rank Complexity of Softmax Attention

How much matrix rank is required to preserve every bounded value output of normalized softmax attention? We study the unrestricted maximum-row-(ℓ1) approximation rank (r_varepsilon(A)), exactly the least rank achieving uniform error over all bounded vector-valued values. Row softmax exposes the intrinsic interaction (C=Pm(log A)PN), whereas invertible (Q/K) gauges leave (A) fixed while changing the Euclidean geometry of a chosen query/key factorization. We replace that coordinate-dependent description by a projective residual (q(C-T)) and an attained factor-radius size (kappa(T)). For every rank-(r) retained interaction with (τ(T)<varepsilon), we prove r_varepsilon(A)≤ minleft N,; Crleft( 1+frackappa(T) (varepsilon-τ(T))2 right)r/2 right, with the same unknown dimension constant as the underlying weighted Gibbs-row cover. The profile is gauge invariant, termwise no worse than native retained-subspace bounds at the same declared dimension, and has a worst-case sharp (r/2) size exponent at fixed (r) and (varepsilon). We then measure (r_varepsilon(A)) directly on learned attention using 9,978 certified brackets across BERT, GPT-2, Qwen2.5, and two ViT checkpoints; where certificates do not close, the optimum remains interval-valued. A pre-specified 2,302-cell held-out study further shows that the historical native-coordinate geometry block contains coarse, mostly head-level information but no detectable incremental information beyond a strong calibrated baseline. The new intrinsic descriptor is not evaluated in that study. Together, the theory and measurements distinguish an operator-intrinsic complexity control from a stronger empirical explanation that the learned-head evidence does not support.