Microscopic Parametric Correlations and Spectral Rigidity in the Non-Hermitian Threefold Way

We study microscopic spectral correlations between two nearby parameter values in the three non-Hermitian Gaussian bulk classes: A (unconstrained), AI (complex symmetric), and AII (complex self-dual, with each degenerate doublet counted once). Parameter dependence is modelled by stationary matrix Ornstein--Uhlenbeck evolution, whose microscopic time scale is N-1. For every positive integer replica number n, we derive exact finite- N auxiliary-field integral representations for joint characteristic-polynomial moments and obtain their bulk asymptotics as N to infinity with n fixed. With the spectrum normalized to the unit disk, correlations near a bulk point z 0, between spectral positions z 1,z 2 at times t 1,t 2, depend only on s=Nz 1-z 2 2+N(1-z 0 2)t 1-t 2. Adopting the static Hermitian/non-Hermitian replica continuation yields explicit two-time kernels; for AI and AII these are replica conjectures and, to our knowledge, the first microscopic two-time kernels proposed for these classes. They determine density correlations, cross-time eigenvalue-count covariances, and equal-time number variances. For a disk of mean eigenvalue count y, the latter obey Var N X(D y)=κ X sqrt y+ β X/sqrt y+o(y-1/2), with explicitly given constants for each class. Exact perturbation theory further identifies eigenvector nonorthogonality as the mechanism of short-time spectral diffusion and yields a basis-independent condition number for a two-dimensional Kramers eigenspace in class AII. Direct simulations agree quantitatively with its conjectured inverse-gamma law.