The Gaussian moments conjecture

The Gaussian moments conjecture is the probabilistic form of the Jacobian conjecture. It asks whether the kernel of Gaussian expectation is a Mathieu–Zhao subspace of the polynomial ring. Derksen, van den Essen and Zhao proved that the Gaussian moments conjecture in every dimension implies the Jacobian conjecture, and that a non-invertible cubic-homogeneous Keller map in $r$ variables makes GMC($2r$) false.^[1] The operators $\partial_i-x_i$ commute, have order one and constant leading coefficients, and by Gaussian integration by parts their images span $\ker\mathbb E$. So GMC is the image conjecture for this one commuting family.^[1]

Status

• Three variables and up are false, with small explicit witnesses found independently of the Keller map. In three standard Gaussians, write $Z,W=(X_1\pm iX_2)/\sqrt2$ and $T=X_3$. Then $P=W+WZ-T^2-\tfrac32ZT^2-\tfrac12Z^2T^2$ and $Q=Z$ satisfy $\mathbb E[P^m]=0$ and $\mathbb E[QP^m]=m!$ for every $m\ge1$. Long also gives a six-term cubic in four variables.^[2] His repository reports Lean 4 proofs not_GMC_three and not_GMC_of_three_le against an in-repository definition of GMC. They are pinned to Lean 4.34.0 and a fixed Mathlib commit, with an axiom audit.^[3] The atlas has not rebuilt the project or audited the faithfulness of that definition. The tier V0 is the author's report of a kernel check.
• One variable is true (Derksen–van den Essen–Zhao, Proposition 4.2).
• Two variables are claimed true. Wilson's face-isolation proof is an unrefereed preprint, so the cell is drawn as a claim.^[4] Long wrote of this case: “We do not settle this case.”^[2]

If Wilson's proof holds, the threshold is sharp at three variables. The Jacobian conjecture's own threshold is also at three variables.

1. ↑ a b H. Derksen, A. van den Essen, W. Zhao, The Gaussian Moments Conjecture and the Jacobian Conjecture, Israel J. Math. 219 (2017) 917–928.
2. ↑ a b C. D. Long, Small Counterexamples to the Gaussian Moments Conjecture, arXiv:2607.18186 (20 July 2026).
3. ↑ long-mathematics/gaussian-moments-counterexamples, README and FORMALIZATION_STATUS.md.
4. ↑ M. Wilson, A face-isolation proof of the two-variable Gaussian Moments Conjecture, arXiv:2607.23887 (26 July 2026). Unrefereed.