The Geometric Cost of Normalization: Affine Bounds on the Bayesian Complexity of Neural Networks

ArXi:2603.27432v1 Announce Type: new LayerNorm and RMSNorm impose fundamentally different geometric constraints on their outputs - and this difference has a precise, quantifiable consequence for model complexity. We prove that LayerNorm's mean-centering step, by confining data to a linear hyperplane (through the origin), reduces the Local Learning Coefficient (LLC) of the subsequent weight matrix by exactly $m/2$ (where $m$ is its output dimension); RMSNorm's projection onto a sphere preserves the LLC entirely. This reduction is structurally guaranteed before any.