The number condition is not necessary

For any fixed finite family $v_1,\dots ,v_J$, choose a unit vector $e_{\perp }$ orthogonal to their finite complex span. Put $\lfloor {ϵ}^{-2}\rfloor$ particles in this mode and vacuum in its orthogonal complement.

The residual number expectation is $n_{ϵ}=\lfloor {ϵ}^{-2}\rfloor$, so $ϵn_{ϵ}\to \infty$. Nevertheless, every $\Phi (v_j)$, and every ordered Weyl product in these directions, acts on the vacuum factor independently of the occupied mode. All conclusions for the displaced vacuum in this finite family hold exactly. This proves that total-number control is not necessary for observable-specific concentration.