A quartic witness in three real variables

Statement

Set $Z=(X_1+iX_2)/\sqrt2$ , $W=(X_1-iX_2)/\sqrt2$ , $P_3=(1+Z)\left(W-\frac{2+Z}{2}T^2\right)$ and $Q_3=Z$ . For every integer m ≥ 1, $\mathbb E[P_3^m]=0$ and $\mathbb E[Q_3P_3^m]=m!$ .

Assumptions

X₁, X₂ and T are independent standard real Gaussians.
Z and W are conjugate coordinates, not independent complex Gaussian variables.

Scope

Total degree four in three real coordinates. Ignoring extra coordinates gives failure for every n ≥ 3.

Explanation

Constructions · 1

The three-variable polynomial witness

The identities hold for every positive integer order in the cited proof. A finite table of moments illustrates them; it does not prove the all-orders claim.

A quartic witness in three real variables →

Small Counterexamples to the Gaussian Moments Conjecture →

Evidence

Source reports

Proved

Moment identities proved in the cited paper ↗

Exact coefficient identities for the three- and four-variable polynomials, not a finite numerical moment test.

Small Counterexamples to the Gaussian Moments Conjecture · Proposition 4.1, Theorem 5.1, Corollary 5.2, Section 7 and AI disclosure →

Source dated Jul 20, 2026 · Observed Oct 13, 2026

Connections

In this problem · publication v6 · Selected context

Gaussian moments conjecture →

Counterexample · Proved

Relationship not independently reviewed. This concerns its use in this problem.

Sources

Small Counterexamples to the Gaussian Moments Conjecture →

Proposition 4.1, Theorem 5.1, Corollary 5.2, Section 7 and AI disclosure → · Original passage ↗
Recorded wording
Gives a cubic four-variable witness and a quartic three-variable witness, both with zero pure moments and nonzero mixed moments of every order.
paraphrase · OpenMath supplies its own mathematical paraphrases and links; proof text is not republished.

Christopher D. Long · Preprint · Publication date Jul 20, 2026

Revision & attribution

Source revision: 2026-07-20. Observed Oct 13, 2026.

Long credits ChatGPT 5.6 Sol Pro with the four-variable polynomial, Claude Fable 5 with the three-variable reduction and checks, and ChatGPT with editing. Long is the paper’s author and responsible for its mathematical claims.

History

Statement versions

Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.

Assessment history · 1

Assessments retained in these publication records. Earlier statement versions retain their own source histories.

Proved

Moment identities proved in the cited paper ↗

Exact coefficient identities for the three- and four-variable polynomials, not a finite numerical moment test.

Source dated Jul 20, 2026 · Observed Oct 13, 2026 · Source passage →

Record details

Claim ID: claim-gaussian-moments-three-variable-witness
Statement version: 1
SHA-256: 7d5dcab94824b7c58d5abf67690eb8691d7e076bdd14abf4b69046a634e22647

Publication records

Gaussian moments conjecture · publication v6 · JSON →