= Projection kernel determinant integration
{title2=$\int D_n\,d\mu(x_n)=(r-n+1)D_{n-1}$}
Suppose an <integral kernel> reproduces under convolution and has integrable diagonal with integral $r$, and the displayed single-variable integrals exist. In the <determinant> $D_n=\det[K(x_i,x_j)]$, <permutations> fixing $n$ contribute $rD_{n-1}$ after integration. For every <permutation> on $n-1$ letters there are $n-1$ ways to insert $n$ into a cycle; convolution contracts the two adjacent factors, and insertion reverses the <permutation> sign. These terms contribute $-(n-1)D_{n-1}$. This proves the formula without a symmetry assumption on the kernel.
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