= Bosonic number-state corrections to Gaussian density correlations
{title2=$C_{12}=n_1n_2+|g_{12}|^2-\sum_\alpha N_\alpha(N_\alpha+1)|\varphi_\alpha(1)|^2|\varphi_\alpha(2)|^2$}
For a product of bosonic <number states>, direct and exchange contractions correctly count distinct modes. They overcount repeated-mode terms because $\langle a^{\dagger2}a^2\rangle=N(N-1)$, 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.
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