The Gaussian Moments Conjecture in Two Variables

This preprint proves the Gaussian Moments Conjecture for two real Gaussian variables. Using circular coordinates, Gaussian integration is decomposed into constant-term extraction in an angular variable followed by the factorial functional (U^j \mapsto j!) in a radial variable. If a polynomial with vanishing positive moments contains both positive and negative rotational weights, an exposed lower face of its Newton diagram yields a nonzero lowest radial term. The Duistermaat–van der Kallen constant-term theorem and a reduction-modulo-(p) argument then show that this term cannot cancel after applying the factorial functional. It follows that every polynomial with all positive Gaussian moments equal to zero has rotational weights of one strict sign. The mixed-moment conclusion is then immediate, proving the conjecture in dimension two. Together with Christopher Long’s recent counterexamples in every dimension at least three, this determines the exact dimensional range in which the Gaussian Moments Conjecture holds. The main proof has also been formalized in Lean.