The four-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 cubic witness in four real variables →Theorem · version 1
A cubic witness in four real variables
Statement
For j = 1,2 put and . Set and . For every m ≥ 1, and .
Assumptions
- The four real coordinates X₁,Y₁,X₂,Y₂ are independent standard Gaussians.
- Each Wⱼ is the conjugate of Zⱼ.
Scope
A different witness: degree three with four real coordinates. The lower-dimensional quartic does not replace this degree statement.
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 v2 · Selected context
Gaussian moments conjecture →
Supporting result · Proved
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v6
Gaussian moments conjecture →
Supporting result · 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-four-variable-witness
- Statement version
- 1
- SHA-256
- 3beff1d51d7f1b50f0e2ba7b26420e4c7a222d5d93792727497a7ba29705b6a4