Robust Decision Capacity of Multi-Head Attention

We establish head-count limits on stable threshold decisions in a single attention layer, through its response to changing one source value along an affine query path while holding keys and scores fixed during the intervention. We quantify stable decision complexity by robust alternating capacity: the maximal number of transitions between alternating threshold labels, each certified under every independent head gain in $[1-\delta ,1+\delta ]$. Two heads can approximate every smooth scalar response in $C^r$ on a compact interval, yet robust decision capacity remains limited by head count. For $H\ge 2$ and $0<\delta \le 1/8$, we prove that unrestricted-score capacity is $\Theta (H{\delta }^{-1}\log (1/\delta ))$. The upper bound holds for every output norm and is independent of width, context length, and output-coefficient magnitude; finite scalar attention constructions attain the same order. With source-relative score slopes bounded by $B$ on a path of length $L$, we derive matching Euclidean capacity laws in joint head–score regimes, including a transition from factorially constrained growth to a square-root regime governed by $H$, $BL$, and $\log (1/\delta )$. Fixed-head high-score laws reach the unrestricted capacity order, while a fixed non-Euclidean output norm changes the score-dependent order. These results show that head count controls stable decision complexity even after smooth approximation has saturated.