 Bosonic number-state corrections to Gaussian density correlations

For a product of bosonic number states, direct and exchange contractions correctly count distinct modes. They overcount repeated-mode terms because $⟨a^{†2}{a}^2⟩=N(N-1)$, while Wick theorem would give $2N^2$. Subtracting the excess yields the displayed exact two-point correction. At higher order the occupation decrements carry falling factorials. This distinction is essential when comparing a fixed-number condensate with independent thermal modes.