{
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  "problem_id": "vibemathed:smooth-random-fast-dynamo-on-the-three-torus",
  "title": "Smooth Random Fast Dynamo on the Three-Torus",
  "canonical_statement": "Arnold's fast-dynamo problem asks for a smooth divergence-free velocity field on $\\mathbb{T}^3$, chosen independently of the magnetic diffusivity, that drives exponential growth of the magnetic field at every sufficiently small diffusivity. This constructs a genuinely $C^\\infty$ 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 $1/2$ 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 $\\mathbb{T}^3$ remains open.",
  "plain_summary": "VibeMathed reports this item as open. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.",
  "current_status": "open",
  "solution_events": [
    {
      "id": "vibemathed:smooth-random-fast-dynamo-on-the-three-torus:event",
      "problem_id": "vibemathed:smooth-random-fast-dynamo-on-the-three-torus",
      "type": "proved",
      "status": "open",
      "occurred_at": "2026-08-20T00:00:00.000Z",
      "title": "arXiv:2608.20105 - An AI-discovered smooth random fast dynamo on the three-torus",
      "summary": "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. This is one variant case of Arnold's 1994 fast-dynamo problem, not the problem itself: Arnold asks for a field that is smooth, autonomous and deterministic all at once, and this one keeps the smoothness while giving up the other two. The sibling entry on this site relaxes the opposite hypothesis, keeping an autonomous deterministic field at Lipschitz regularity. Neither settles Arnold's problem as posed, which remains open.",
      "ai_contribution": "ai_discovered",
      "attempt_ids": [
        "vibemathed:smooth-random-fast-dynamo-on-the-three-torus:attempt"
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      "source_assertion_ids": [
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      "id": "vibemathed:smooth-random-fast-dynamo-on-the-three-torus:attempt",
      "problem_id": "vibemathed:smooth-random-fast-dynamo-on-the-three-torus",
      "model": "ChatGPT 5.6 Sol Ultra",
      "provider": "OpenAI",
      "attempted_at": "2026-08-20T00:00:00.000Z",
      "human_collaborators": [
        "Keefer Rowan"
      ],
      "exposure": "unknown",
      "prompt_public": null,
      "outcome": "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.",
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      "sources": [
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          "label": "arXiv:2608.20105 - An AI-discovered smooth random fast dynamo on the three-torus",
          "url": "https://arxiv.org/abs/2608.20105",
          "kind": "primary-mathematical-source",
          "license": null
        },
        {
          "label": "VibeMathed: Smooth Random Fast Dynamo on the Three-Torus",
          "url": "https://vibemathed.com/problem/smooth-random-fast-dynamo-on-the-three-torus",
          "kind": "source-assertion",
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      "independence": "unknown"
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    {
      "id": "vibemathed:smooth-random-fast-dynamo-on-the-three-torus:verification",
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      "solution_event_id": "vibemathed:smooth-random-fast-dynamo-on-the-three-torus:event",
      "level": "unreviewed",
      "mathematical_correctness": "unknown",
      "statement_fidelity": "unaudited",
      "peer_review": "none",
      "verifier": "VibeMathed (source-reported label)",
      "verified_at": null,
      "note": "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.",
      "sources": [
        {
          "label": "VibeMathed: Smooth Random Fast Dynamo on the Three-Torus",
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          "kind": "source-assertion",
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          "kind": "primary-mathematical-source",
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  "sources": [
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      "license": null
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    {
      "label": "VibeMathed: Smooth Random Fast Dynamo on the Three-Torus",
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      "generated_at": "2026-09-13T16:28:40.178Z"
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  "uncertainties": [
    "VibeMath has not independently verified the mathematical claim.",
    "AI-attempt independence and training-data exposure are unknown unless explicitly documented.",
    "VibeMath has not independently audited the mathematical statement, proof, or novelty claim."
  ],
  "generated_at": "2026-09-13T16:28:40.178Z"
}