ChatGPT 5.6 Sol Ultra Generates a Smooth Random Fast Dynamo on the Three-Torus

In a manuscript dated August 20, 2026, Keefer Rowan constructs a random, time-dependent, divergence-free velocity field on $\mathbb{T}^3$ with deterministic smoothness bounds and proves that it exhibits fast dynamo behavior. For every sufficiently small fixed resistivity $\kappa$, the associated magnetic field has an almost-sure, all-time lower bound of the form

$$\|B_t^\kappa\|_{L^2}\geq (L^\kappa)^{-1}e^{t/2},$$

where the random prefactor has an inverse-moment bound uniform in $\kappa$.

The velocity field refreshes independently on finite time blocks. A special Fourier-space structure reduces the induction equation to growth estimates for three selected modes, avoiding control of the full infinite-dimensional dynamics. The exceptional probability-zero set may depend on the fixed resistivity, so the theorem does not assert one common full-probability event valid simultaneously for every $\kappa$.

Rowan states that ChatGPT 5.6 Sol Ultra generated the original proof idea essentially autonomously and produced draft manuscripts after further prompting. He reports that the generated versions contained minor errors and poor exposition; after rederiving the argument and discussing it with William Cooperman, Rowan rewrote the proof from scratch in its simpler forward-time form.