Gaussian Moments Conjecture
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Statement
The Gaussian Moments Conjecture asks whether, for complex polynomials in independent standard real Gaussian variables, for all forces for all large . Explicit counterexamples with and exist in three variables (a five-term quartic ) and four variables, so the conjecture is false in every dimension .
Context
Explicit counterexamples in dimensions 3 and 4, so GMC(n) fails for every n >= 3; GMC(1) was already known, and a separate human-authored preprint claims the remaining n = 2 case affirmatively
A named conjecture in the Jacobian-conjecture circle, equivalent in part to the Image Conjecture: a real if specialist target, and its failure follows the Jacobian disproof it was prompted by.
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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.
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Machine check · not human verification
machine: correctscope 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.
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