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Gaussian Moments Conjecture

Probability & statistics · posed by Harm Derksen, Arno van den Essen, Wenhua Zhao, 2017 · disproved

2 comments

Statement

The Gaussian Moments Conjecture asks whether, for complex polynomials P,QP,Q in nn independent standard real Gaussian variables, E(Pm)=0\mathbb{E}(P^m)=0 for all m≥1m\geq 1 forces E(QPm)=0\mathbb{E}(QP^m)=0 for all large mm. Explicit counterexamples with E(Pm)=0\mathbb{E}(P^m)=0 and E(QPm)=m!≠0\mathbb{E}(QP^m)=m!\neq 0 exist in three variables (a five-term quartic PP) and four variables, so the conjecture is false in every dimension n≥3n\geq 3.

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No person has examined this. Nothing here has been checked at all. say whether it holds →

  1. construction · #1

    GPT-5.6 Sol Pro, Claude Fable 5, with Christopher D. Long

    The record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.

    AI involvement
    ai discovered
    — the result was found by a model.

    Per the paper's AI-provenance section, the four-variable construction was produced by ChatGPT 5.6 Sol Pro without human intervention after the initial prompt, which told it the Jacobian conjecture had been disproved and asked whether a small Gaussian-moments counterexample might follow; shown that example, Claude Fable 5 found the three-variable construction and supplied independent algebraic checks. The author bears responsibility for the mathematics and exposition.

  2. Recorded elsewhere on #1 · not checked here

    recorded: correctVibeMathed site check

    scope Reproduction by the VibeMathed site

    Re-derived by the site on 2026-08-02: both counterexamples were rebuilt from the paper's stated polynomials and evaluated in exact rational arithmetic against the standard Gaussian moment rules (E(W^a Z^b) = a! when a = b, else 0; E(T^c) the double factorial), independently of the paper's own algebra. For m = 1 through 10 both give E(P^m) = 0 and E(QP^m) = m! exactly, and the term counts and degrees match the paper (five terms of degree four in three variables, six of degree three in four). The surrounding exposition has had no independent review.

    Repeated from the source; nothing was checked here.

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