Sparse Minimality of Gaussian-Moments Counterexamples in Dimension Three
Year: 2026
Type: preprint
Abstract: This preprint studies minimal counterexamples to the Gaussian Moments Conjecture in three real variables, the first dimension in which the conjecture fails. Christopher Long recently constructed a counterexample of total degree four with five monomials. We prove, by an exhaustive exact computer-assisted classification, that no counterexample exists with both total degree at most four and support size at most four in Gaussian Witt coordinates. Consequently, Long’s five-term quartic is Pareto-minimal with respect to degree and support size. The calculation enumerates all 59,535 relevant supports, constructs saturated moment ideals over (\mathbb{Q}), and reduces the surviving cases to three explicit families. These families are then shown to satisfy the conjectural mixed-moment conclusion at all orders. The result does not exclude denser cubic counterexamples or four-term counterexamples of degree greater than four. Verification scripts are supplied for reproducibility.
Author: van Rijn, Roy
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7170766937
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