Author verified AI-led

Counterexamples to the Gaussian moments conjecture

Explicit low-degree polynomials disproving a 2017 conjecture that had been proposed as a route to proving the Jacobian conjecture, found days after that conjecture was itself disproved.

Lab: Independent
Model: GPT-5.6 Sol Pro + Claude Fable 5
Field: Mathematics
Date: 2026-07-20
Human collaborators: Christopher D. Long
Problem posed: 2017 · open 9 yrs

Sources

Original work

arXivSmall Counterexamples to the Gaussian Moments Conjecture↗arXivDerksen, van den Essen, Zhao: The Gaussian Moments Conjecture and the Jacobian Conjecture↗

What was found

Derksen, van den Essen and Zhao proposed the Gaussian moments conjecture and showed that the Jacobian conjecture would follow from it. Long reports that a cubic counterexample in four variables was produced by GPT-5.6 Sol Pro with no human intervention after the initial prompt, which simply noted that the Jacobian conjecture had just been disproved and asked whether a small counterexample to GMC might therefore exist; Claude Fable 5 then found a quartic counterexample in three variables and supplied independent algebraic checks. Together these show $\mathrm{GMC}(n)$ is false for every $n \ge 3$ .

Novelty check

The Gaussian moments conjecture is due to Derksen, van den Essen and Zhao (Israel J. Math., 2017), building on their earlier moment-vanishing and integral conjectures. No prior counterexample appears in the literature; the conjecture's interest was precisely that it implied the Jacobian conjecture, which was disproved on 2026-07-19. The two-variable case GMC(2) is explicitly left open by this paper.

Caveats and known objections

arXiv preprint, not peer-reviewed, single author. The constructions are explicit polynomials and checkable by computer algebra, which lowers the cost of verification, and vibemathed has since re-derived both counterexamples in exact arithmetic for the first ten moments, which confirms the objects but not the argument that the identity holds for every m. The result is a corollary of the Jacobian collapse rather than an independent line of attack: the prompt that produced it began by telling the model that the Jacobian conjecture had fallen.

Independent checks

vibemathed.com re-derivation: Rebuilt both counterexamples from the paper's stated polynomials and evaluated them in exact rational arithmetic against the Gaussian moment rules, independently of the paper's algebra: for $m = 1$ to $10$ both give $E(P^m) = 0$ and $E(QP^m) = m!$ exactly. That checks instances, not the all- $m$ argument. · link ↗

Also recorded at

vibemathedgaussian-moments-conjecture

Nothing mechanically. It is a parallel listing of the same result, carrying its own verification label rather than an independent check.

Disagree with these grades?

Bring a citation: a grade moves on evidence, not on argument.

Challenge this grade Submit a check

Or on GitHub: submit a check challenge the grade send a correction or send a pull request

History of this entry

2026-09-24CheckedRecorded vibemathed's exact-arithmetic re-derivation of both counterexamples for the first ten moments. The all-m argument is unchecked, so no grade moved. source ↗
2026-08-21CorrectedRecorded the vibemathed record for this result as a registration. No grade moved.
2026-07-27AddedEntered the registry graded Author verified and AI-led.

Entries are never deleted. A grade that does not hold up is downgraded on the record, with the reason beside it.

Graded author verified for verification and ai-led for autonomy. What these mean.

Cite this entry

Plain text

whataifound.org. (2026). Counterexamples to the Gaussian moments conjecture. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2026-07-20-gaussian-moments

BibTeX

@misc{whataifound-independent-2026-moments,
  title        = {Counterexamples to the Gaussian moments conjecture},
  author       = {{whataifound.org}},
  year         = {2026},
  howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
  note         = {Result by Independent. Verification: Author verified. Autonomy: AI-led.},
  url          = {https://whataifound.org/finding/2026-07-20-gaussian-moments}
}

Related findings

Formally verifiedLean disproof of Krempa's matrix form of the Koethe conjectureOpenAI / Epoch AI · 2026-09-03 Formally verifiedCounterexample to the Jacobian conjecture in dimension threeAnthropic · 2026-07-19 ClaimedCounterexample to Talagrand's operator cotype problemIndependent · 2026-09-17

algebra polynomial-maps counterexample jacobian-conjecture

← All mathematics findings in the registry