Proved
Witness extends to every higher dimension ↗
Use the same polynomial in its first three coordinates, independent of all extra Gaussian coordinates.
Source dated Jul 20, 2026 · Observed Oct 13, 2026
Theorem · version 1
The same witness works in every higher dimension
Statement
If complex polynomials P,Q in three independent standard real Gaussian variables satisfy E[P^m] = 0 and E[QP^m] = m! for every positive integer m, then for every n ≥ 3 these polynomials, independent of the extra coordinates, refute the Gaussian Moments Conjecture in n variables.
Assumptions
- The exact three-variable identities hold for every positive integer m.
- Additional coordinates are independent standard real Gaussians.
Scope
Additional independent coordinates do not change these expectations. This extension leaves the two-variable case untouched and by itself gives no conclusion about the Jacobian conjecture.
Evidence
Source reports
Connections
In this problem · publication v4 · Selected context
Gaussian moments conjecture →
Conditional implication · Proved
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v6
Gaussian moments conjecture →
Conditional implication · 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
Witness extends to every higher dimension ↗
Use the same polynomial in its first three coordinates, independent of all extra Gaussian coordinates.
Source dated Jul 20, 2026 · Observed Oct 13, 2026 · Source passage →
Record details
- Claim ID
- claim-gaussian-moments-higher-dimensions
- Statement version
- 1
- SHA-256
- 0e85973d4fb9c6acc526bd7a8691547ca9ff9b08b54c6da67d75c03bbba83911