Gives a cubic four-variable witness and a quartic three-variable witness, both with zero pure moments and nonzero mixed moments of every order.
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Gives a cubic four-variable witness and a quartic three-variable witness, both with zero pure moments and nonzero mixed moments of every order.
Passage ID: gaussian-moments:entry:long:v1
Source item: long
Source record: gaussian-moments:source:long:v1
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Problem publication v6
Source assessment
Proved
Witness extends to every higher dimension
Use the same polynomial in its first three coordinates, independent of all extra Gaussian coordinates.
Source date Jul 20, 2026 · Observed Oct 13, 2026
Source assessment
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 date Jul 20, 2026 · Observed Oct 13, 2026
Construction
The four-variable polynomial witness
Source assessment
Disproved
Universal conjecture refuted by an explicit polynomial
Three independent real Gaussian coordinates suffice; the construction extends to every higher dimension.
Source date Jul 20, 2026 · Observed Oct 13, 2026
Source assessment
Open
Two-variable case remains open in the cited survey
The three-variable counterexample does not settle dimension two.
Source date Jul 20, 2026 · Observed Oct 13, 2026
Source assessment
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 date Jul 20, 2026 · Observed Oct 13, 2026
Construction
The three-variable polynomial witness
Research activity
Explore the remaining dimension
The source leaves GMC(2) as the nearest unresolved restriction.
Higher-dimensional counterexamples cannot be projected down while preserving their moment identities.
History
Two explicit polynomial witnesses
Question
Gaussian moments conjecture
Further reading
Read the mathematical argument; inspect its Claim for assessment and scope.
Recorded authorship
Christopher D. Long
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.
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Small Counterexamples to the Gaussian Moments Conjecture
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