Unistochastic reduction of multi-matrix coherent-state kernels

We revisit the finite-$N$ coherent-state overlap integrals appearing in multi-matrix BPS correlators. For commuting coherent-state parameters, these integrals take the form $I_N(C)={\int }_{U(N)}d{\mu }_H(U)\exp \text{\hspace{0.17em}⁣}\left({\sum }_{ij}{C}_{ij}\mathrm{\mid }{U}_{ij}{\mathrm{\mid }}^2\right)$. We show that $I_N(C)$ is naturally the Laplace transform of the Haar-pushforward measure on the unistochastic set. Row and column stochasticity reduce the source variables from $N^2$ to $(N-1{)}^2$. For $N=3$, using the known Haar-induced measure on the unistochastic body, we obtain a compact three-dimensional Bessel-integral representation. For general finite $N$, the same formulation gives an exact low-order moment expansion via Weingarten calculus and a direct Haar-pushforward sampling program.