Bosonic number-state corrections to Gaussian density correlations ($C_{12}=n_1{n}_2+|g_{12}{|}^2-{\sum }_{\alpha }{N}_{\alpha }(N_{\alpha }+1)|{\phi }_{\alpha }(1){|}^2|{\phi }_{\alpha }(2){|}^2$)

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.