Divine Benevolence is an $x^2$: GLUs scale asymptotically faster than MLPs

Scaling laws can be understood from ground-up numerical analysis, where traditional function approximation theory can explain shifts in model architecture choices. GLU variants now dominate frontier LLMs and similar outer-product architectures are prevalent in ranking models. The success of these architectures has mostly been left as an empirical discovery. In this paper, we apply the tools of numerical analysis to expose a key factor: these models have an $x^2$ which enables \emph{asymptotically} faster scaling than MLPs. GLUs have piecewise quadratic functional forms that are sufficient to exhibit quadratic order of approximation. Our key contribution is to demonstrate that the $L(P)$ scaling slope is $L(P)\propto P^{-3}$ for GLUs but only $L(P)=P^{-2}$ for MLPs on function reconstruction problems. We provide a parameter construction and empirical verification of these slopes for 1D function approximation. From the first principles we discover, we make one stride and propose the ``Gated Quadratic Unit'' which has an even steeper $L(P)$ slope than the GLU and MLP. This opens the possibility of architecture design from first principles numerical theory to unlock superior scaling in large models. Replication code is available at https://github.com/afqueiruga/divine_scaling.