Corollary: Uniform bounded-family version

For a fixed finite $J$, suppose

Then all coordinate mean errors, variances, mean-square errors, and spectral distances to their displacement-compatible targets vanish uniformly in $j$. For commuting families the joint Wasserstein distance also vanishes.

Proof. The common upper bound is $ϵ(4n_{ϵ}+2)K^2$, which tends to zero.