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
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 GMC(n) is false for every n ≥ 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arXiv preprint, not peer-reviewed, single author.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, but no independent check is on the public record yet. 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.
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.
Nobody outside the lab has checked this yet.
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History of this entry
- CorrectedRecorded the vibemathed record for this result as a registration. No grade moved.
- AddedEntered the registry graded Author verified and AI-led.
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Cite this entry
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-momentsBibTeX
@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}
}