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AI-assisted small counterexamples to the Gaussian Moments Conjecture

Christopher D. Long reports explicit complex polynomials P and Q in three independent real Gaussian variables with E(P^m)=0 and E(QP^m)=m! for every positive integer m. A five-term degree-four example refutes the conjecture in every dimension at least three; a six-term cubic example works in four variables.

Sources for this record

What was already known

The conjecture asks whether vanishing moments of P force eventual vanishing of mixed moments QP^m. A route from the announced Jacobian counterexample gives a nonexplicit failure in dimension 158; these small explicit examples are separate constructions.

What changed and why it matters

Refutes the Gaussian Moments Conjecture with small explicit witnesses, beyond a nonconstructive implication in a much larger dimension.

The AI contribution

The AI-provenance section states that ChatGPT 5.6 Sol Pro produced the four-variable example after a human prompt, and Claude Fable 5 produced the three-variable example after receiving it. Long takes responsibility for the mathematics and exposition.

Limitations

Complex coefficients are essential. Dimension two remains unsettled, and dimension one was already established. These examples were not derived from the coordinates of the Jacobian counterexample. The manuscript’s algebraic checks by Claude are not independent human verification. AIProofWatch reviewed the theorem, comparison and full-text AI disclosure, not the complete proof.

Verification evidence

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Author claim

The source reports this precise contribution; statement review does not establish a proof audit.

Accessible artifact / proof

Accessible supporting proof or code; not compiled or reproduced by AIProofWatch.

Correction and publication history

2026-10-11 · publication

Public-discussion audit found this substantive scoped contribution missing. Source statement, baseline, attribution and importance reviewed; no full proof audit. Historical addition without subscriber alerts.

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