Smooth Random Fast Dynamo on the Three-Torus
Arnold's fast-dynamo problem asks for a smooth divergence-free velocity field on , chosen independently of the magnetic diffusivity, that drives exponential growth of the magnetic field at every sufficiently small diffusivity. This constructs a genuinely field with that behaviour: random and time-dependent, refreshing iid on finite time blocks, for which the almost sure exponential growth rate is at least at each fixed small enough resistivity, with a time-uniform lower bound whose random prefactor has a resistivity-uniform inverse-moment bound. The field is neither autonomous nor deterministic, so Arnold's smooth autonomous problem on remains open.
- Result
- Proved(see note)
- Status
- Variant only
- AI contribution
- AI-discovered
- Method
- Construction
- Field
- Dynamo theory
- Posed by
- Arnold's fast-dynamo problem (1994); the random formulation has no single named proposer
- Year posed
- —
- Years open
- —
- Solved
- 2026-08-20
- Model
- ChatGPT 5.6 Sol Ultra
- Vendor
- OpenAI
- Collaborators
- Keefer Rowan
- Verification
- Unreviewed
- Publication
- Preprint
- Significance
- 30 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Constructed an explicit class of smooth random, time-dependent incompressible velocity fields on T^3, obtained by alternating smooth shear flows with iid random phases on finite time blocks. For every fixed sufficiently small resistivity, the magnetic field has an almost-sure exponential growth rate at least 1/2, together with a time-uniform lower bound whose random prefactor has a resistivity-uniform inverse-moment estimate.
What the AI did
Rowan states that the original proof idea was generated essentially autonomously by ChatGPT 5.6 Sol Ultra from a prompt asking for a proof of the random fast dynamo conjecture. ChatGPT produced an initial manuscript and identified the key Fourier-space mechanism: carefully chosen shears act so that selected Fourier modes evolve through an effectively tridiagonal recursion, reducing the infinite-dimensional induction dynamics to a simple growth process. During later revisions, ChatGPT also produced the exponential-martingale argument that became Lemma 2.4. Rowan subsequently simplified the velocity fields, replaced the backward-time argument with a cleaner forward-time proof, rewrote the manuscript from scratch, and hand-checked the mathematics.
Verification
Checked by this site on 22 August 2026 against the paper (arXiv:2608.20105, Keefer Rowan, 12pp): the abstract is as submitted, the paper is titled "An AI-discovered smooth random fast dynamo on T^3" and its comments field reads "AI discovered; human written", so the contribution tier is the author's own framing rather than an inference. The mathematics was not checked here. Days-old preprint, no independent review, and the result is recorded as a variant because the velocity field is random and time-dependent where Arnold's problem asks for an autonomous deterministic one.
Source
Related entries
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