An AI-discovered smooth random fast dynamo on $\mathbb{T}^3$

Abstract

We construct a random, time-dependent divergence-free velocity field on $\mathbb{T}^3$---refreshing iid on finite time blocks and obeying deterministic $C^\infty_{t,x}$ bounds---that exhibits fast dynamo behavior. That is, for every fixed, sufficiently small resistivity, the almost sure exponential growth rate of the magnetic field solving the associated linear resistive induction equation is at least $1/2$; the exceptional null set may depend on the resistivity. We in fact get a time-uniform lower bound---with a random prefactor obeying a uniform-in-$κ$ inverse moment bound. The argument relies on a particular algebraic structure in Fourier space of the induction equation solution operator that allows us to propagate expected growth of the logarithmic size of three specially chosen Fourier modes. This allows us to reduce to a simple recursion, avoiding the complicated infinite-dimensional dynamics typical to the dynamo problem. The central proof idea was generated autonomously by ChatGPT 5.6 Sol Ultra; the manuscript was written (and verified) by the author.