= Solution
Under the Gaussian assumption, $g(\mathbf r,\mathbf r)=n(\mathbf r)$, so the coincident <normally ordered density correlation> is $(1+\epsilon)n(\mathbf r)^2$. For a homogeneous nonzero density $n$, the large-separation limit is $n^2$, giving
$$
\boxed{\text{bosons: }2,\qquad\text{fermions: }0}.
$$
These <coincident-point bunching factors> describe enhanced bosonic coincidence and excluded same-component fermionic coincidence. More generally the density-normalized ratio $C_{12}/(n_1n_2)$ tends to one at large separation. \b[The unnormalized ratio printed in the paper additionally needs matching nonzero local and remote densities]; $g_{12}\to0$ alone does not ensure this. For a bosonic Fock product state, the local normalized value instead is $2-\sum_\alpha N_\alpha(N_\alpha+1)|\varphi_\alpha(\mathbf r)|^4/n(\mathbf r)^2$, and even a large-separation factorization must be checked separately.
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