Lemma: Normal finite-mode baseline

The selected baseline has a unique normal density ${\rho }_0$ on ${\mathcal{F}}_{\mathrm{s}}({\mathbb{C}}^m)$ with

If $A=W\mathrm{diag}({\lambda }_1,\dots ,{\lambda }_m)W^{†}$, it is the passive rotation of the product densities

It commutes with total number.

Proof. The restricted covariance is finite, positive, centered, and has zero anomalous part. Construct independent circular complex Gaussian amplitudes of variances ${\lambda }_j$, rotate them by $W$, and form their coherent-state mixture. It is positive trace class of trace one and has exactly the displayed characteristic function.

For one variance $\lambda >0$, angular integration kills off-diagonal number terms and radial integration gives ${\lambda }^n/(1+\lambda {)}^{n+1}$. Zero variance means vacuum. Equality of finite-dimensional Weyl characteristic functions identifies the normal density and proves uniqueness. Passive rotation preserves total number.