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 →Theorem · version 1
A quartic witness in three real variables
Statement
Set , , and . For every integer m ≥ 1, and .
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
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.
Source dated Jul 20, 2026 · Observed Oct 13, 2026
Connections
In this problem · publication v1 · Selected context
Gaussian moments conjecture →
Counterexample · Proved
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v6
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