Decoupling and Tail Laws for Small-Order Metric-Valued Nonlocal Energies: A Structural View of the Maz'ya--Shaposhnikova Formula

We develop a measure-theoretic framework for small-order limits of nonlocal energies with metric-valued maps. For globally $L^p$ maps, we consider interaction measures whose marginals have uniformly bounded $L^\infty$ densities $a_α$ and $b_α$. If these densities converge weakly-* to $a_\infty$ and