Sharp Wasserstein stability for bounded squared laws

For a real law with X² supported in [3−√6,3+√6], set δ=3(E[X²])²−E[X⁴]. How close is the squared law to the zero-deficit Gaussian quadrature law in ordinary W₁ distance?

Why it matters and what counts as progress

Why it matters: An optimal transport bound quantifies stability throughout the feasible deficit range, even for nonsymmetric original laws.

What counts as progress: A sharp attainable bound W₁(L(X²),L(X₀²))≤√(δ/3) for every feasible 0≤δ≤30+12√6, with exact extremizers.

How the problem developed

15 Oct 2026 · Stability bound [1][2]
JD Jones, with GPT-6 Pro and Codex assistance Proved the sharp global W₁ bound between the squared law and the critical quadrature law at every feasible fourth-cumulant deficit, with equality cases and no symmetry requirement on the original variable.

AI contribution

AI-led proof gives a sharp global squared-law transport bound

For any real law with X² in [A,B], A=3−√6 and B=3+√6, let δ=3(E[X²])²−E[X⁴] and let X₀ be the four-node Gaussian quadrature law identified by zero deficit. For every feasible 0≤δ≤30+12√6, W1(L(X²),L(X₀²))≤√(δ/3), using ordinary infimum-over-couplings Wasserstein distance. The bound is attained at every feasible deficit and does not require symmetry of X.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean 4/Mathlib, Comparator, NanoDa and con-ron with exact rational certificate checking)

Human understandingFew