Exact joint eigenvalue densities of non-Hermitian random matrices are Calogero scattering states

What the math gives to ML

The paper provides an exact spectral mechanism for non-Hermitian matrices with transposition symmetry: after factoring out a Vandermonde term, the joint eigenvalue density is a Calogero-model scattering wavefunction with inverse-square interactions. This differs from a conventional Coulomb-gas model and predicts non-Gaussian, power-law spectral correlations and spacing tails. A transferable neural-network construction is to regularize the spectrum of recurrent, state-space, or neural-ODE Jacobians with a Calogero-inspired inverse-square spectral barrier while separately enforcing a contraction boundary. The experiment should test eigenvalue-collision statistics and long-horizon stability at a predicted spectral-radius boundary, rather than merely reporting a benchmark improvement.