= Solution
The <correlation functions of a point process> use factorial normalization:
$$
R_3^{(k)}(x_1,\ldots,x_k)
=\frac{3!}{(3-k)!}\int_{\mathbb R^{3-k}}
\rho_3(x_1,x_2,x_3)\,dx_{k+1}\cdots dx_3.
$$
Start with $\rho_3=D_3/6$. By <Gaudin's lemma>, integrating $D_3$ once gives $D_2$, and integrating $D_2$ once gives $2D_1$. The factorial factors cancel, so the <eigenvalues> form a <determinantal point process> with
$$
\boxed{R_3^{(1)}(x)=\widetilde K_3(x,x),}
$$
and
$$
\boxed{R_3^{(2)}(x,y)
=\det\begin{pmatrix}
\widetilde K_3(x,x)&\widetilde K_3(x,y)\\
\widetilde K_3(y,x)&\widetilde K_3(y,y)
\end{pmatrix}
=\widetilde K_3(x,x)\widetilde K_3(y,y)-\widetilde K_3(x,y)^2.}
$$
The last equality uses the real symmetry of this kernel. In particular, $\int R_3^{(1)}=3$ and $\iint R_3^{(2)}=6$, as required for three points and six ordered pairs of distinct points.
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