Partial Gaussian splitting in quadratic Wasserstein distance

Let μ=e-V\,dx be a smooth probability measure on Rⁿ with ∇²V Iₙ. Suppose that u₁,…,uₖ are centered, variance-one Sobolev functions whose gradients are orthogonal in L²(μ), and let D=∑ᵢ₌₁ᵏ(∫|∇ uᵢ|²\,dμ-1). We prove that there is a k-dimensional subspace E for which the squared quadratic Wasserstein distance from μ to the product of a standard Gaussian on E, translated by the projected mean, and the actual marginal of μ on E^⊥ is at most $15D$. The estimate has no ambient-dimension factor and requires no smallness assumption on D. The proof combines the Caffarelli contraction and Gaussian near-affinity estimates with an aggregate selection of directions and an explicit coupling that preserves the complementary marginal. Preprint, version 1.0. This record contains the manuscript PDF and its LaTeX source with build instructions. The manuscript has not undergone journal peer review.