Sparse Minimality of Gaussian-Moments Counterexamples in Dimension Three

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.