Lemma: Orientation-sensitive blocks and moments

For the one-particle block,

Extend compressed annihilators by zero on the flag: $b_{i,N}=P_N{b}_i{P}_N\oplus 0$. If

then the truncated number covariance is

It need not equal $K_{ji}$; the discarded moments are absent.

Proof. Equation (eq:permanent) is the matrix element of the symmetric tensor power in (eq:density); symmetrization sums over permutations and occupation normalization supplies the factorials. Its one-particle specialization is immediate.

The cutoff is invariant under passive mode rotation. In the normal-mode basis the retained, unnormalized state is diagonal, and its mean in mode $a$ is ${{\eta }_a}^{(N)}$. Rotating its one-particle matrix back and applying (eq:K) proves (eq:truncatedmoments).