Theorem: Finite-family concentration

For each $j$, let ${\mu }_{j,ϵ}$ be the spectral law of ${\hat{O}}_{j,ϵ}$ in ${\tilde{\rho }}_{ϵ}$. Then

For $r>0$,

If $A(v_i,v_j)=0$ for every pair, the family strongly commutes and has a joint spectral law ${\mu }_{ϵ}$ on ${\mathbb{R}}^J$, with

The exact criterion for concentration of a selected coordinate is

The condition $ϵn_{ϵ}\to 0$ is sufficient for fixed bounded vector families, but is not necessary.

Proof. Equation (eq:displacement) and unitary invariance of spectral moments prove (eq:exact-mse). Lemma lem:number proves (eq:mse-bound). Cauchy–Schwarz gives the mean estimate. Variance about the actual mean is at most the mean-square error about a prescribed target.

There is only one coupling of a probability law with a point mass, so its squared Wasserstein distance is its squared distance moment. Markov's inequality gives the tail bound.

If all symplectic pairings vanish, all one-parameter Weyl groups commute by (eq:Weyl); hence the self-adjoint generators strongly commute. The joint spectral theorem applies. Its squared distance moment is the sum of the coordinate moments, proving (eq:joint-W2). The final criterion is the exact identity (eq:W2-exact).