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gaussian-moments

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Revision 254 · 2026-09-30 05:09 UTC
atlas import import · claude-opus-5-5
“Written for the atlas by Claude Opus 5.5 at the direction of Danielle Fong (content/wiki/Statement__gaussian-moments.md).”

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+---
+title: The Gaussian moments conjecture
+short: GMC(n) — vanishing Gaussian moments of P force vanishing of Q·Pᵐ
+symbol: GMC
+kind: family
+status: refuted
+tier: V0
+year: 2017
+authors: H. Derksen, A. van den Essen and W. Zhao (arXiv 2015; Israel J. Math. 2017)
+statement: Let $X=(X_1,\dots,X_n)$ be independent standard real Gaussians. $\mathrm{GMC}(n)$: if $P\in\mathbb C[x_1,\dots,x_n]$ has $\mathbb E[P(X)^m]=0$ for all $m\ge1$, then for every $Q$, $\mathbb E[Q(X)P(X)^m]=0$ for all large $m$.
+index: n
+cells: 1 = proved V3 | 2 = proved V4 | 3 = refuted V0 witnessed
+stable: yes
+---
+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 [[implies::Statement:jc-n|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.<ref name="dez">H. Derksen, A. van den Essen, W. Zhao, [The Gaussian Moments Conjecture and the Jacobian Conjecture](https://arxiv.org/abs/1506.05192), Israel J. Math. 219 (2017) 917–928.</ref> 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 [[special-case-of::Statement:image-conjecture|GMC is the image conjecture for this one commuting family]].<ref name="dez"/>
 
+{{ladder|Statement:gaussian-moments|Statement:jc-n|n=1..6}}
+ 
+## 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.<ref name="long">C. D. Long, [Small Counterexamples to the Gaussian Moments Conjecture](https://arxiv.org/abs/2607.18186), arXiv:2607.18186 (20 July 2026).</ref> 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.<ref>[long-mathematics/gaussian-moments-counterexamples](https://github.com/long-mathematics/gaussian-moments-counterexamples), README and FORMALIZATION_STATUS.md.</ref> 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.<ref>M. Wilson, [A face-isolation proof of the two-variable Gaussian Moments Conjecture](https://arxiv.org/abs/2607.23887), arXiv:2607.23887 (26 July 2026). Unrefereed.</ref> Long wrote of this case: “We do not settle this case.”<ref name="long"/>
+ 
+If Wilson's proof holds, the threshold is sharp at three variables. The Jacobian conjecture's own threshold is also at three variables.
+ 
+<references />
+ 

Revision 254 as rendered

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 rr variables makes GMC(2r2r) false.[1] The operators ∂i−xi\partial_i-x_i commute, have order one and constant leading coefficients, and by Gaussian integration by parts their images span ker⁡E\ker\mathbb E. So GMC is the image conjecture for this one commuting family.[1]

n = 1n = 2n = 3n = 4n = 5n = 6…padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.padding: Padding with identity coordinates: the statement in dimension n + 1 implies it in dimension n.GMCnGaussian momentsconjectureGMC₁: proved (declared, tier V3)GMC₁GMC₂: claimed true (V4, not yet adjudicated or checked; still open)GMC₂claimedGMC₃: refuted (declared, tier V0, explicit witness)GMC₃GMC₄: false by implication, tier V0, explicit witness — paddingGMC₄GMC₅: false by implication, tier V0, explicit witness — padding → paddingGMC₅GMC₆: false by implication, tier V0, explicit witness — padding → padding → paddingGMC₆GMCₙ, n > 6: false by implication, tier V0, explicit witnessGMCJCnJacobian conjectureJC₁: proved (declared, tier V3)JC₁JC₂: open: nothing settles itJC₂openJC₃: refuted (declared, tier V0, explicit witness)JC₃JC₄: false by implication, tier V0, explicit witness — paddingJC₄JC₅: false by implication, tier V0, explicit witness — padding → paddingJC₅JC₆: false by implication, tier V0, explicit witness — padding → padding → paddingJC₆JCₙ, n > 6: false by implication, tier V0, explicit witnessJC
Open: JC₂. Refuted with an explicit witness: GMCₙ for n ≥ 3; JCₙ for n ≥ 3. Refuted by implication, no witness exhibited: none. Claimed, not yet checked: GMC₂ (a proof). Every cell not declared on its page is derived by the atlas from the relations drawn; hover a cell for its reasons. proved refuted, witnessed refuted by implication open claimed (V4) · a ring marks V0 · arrows point from the stronger statement to what it implies

Status

  • Three variables and up are false, with small explicit witnesses found independently of the Keller map. In three standard Gaussians, write Z,W=(X1±iX2)/2Z,W=(X_1\pm iX_2)/\sqrt2 and T=X3T=X_3. Then P=W+WZ−T2−32ZT2−12Z2T2P=W+WZ-T^2-\tfrac32ZT^2-\tfrac12Z^2T^2 and Q=ZQ=Z satisfy E[Pm]=0\mathbb E[P^m]=0 and E[QPm]=m!\mathbb E[QP^m]=m! for every m≥1m\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.