One Constant Rules All 2D Spectra: Universal Convergence to the Ginibre Kernel

Random matrix universality is the statement that local eigenvalue statistics depend only on symmetry class, not on microscopic details. For matrices with complex eigenvalues—a 2D point process in the complex plane—we show that this universality is absolute: every non-Hermitian random matrix with a genuinely two-dimensional spectrum converges to the same nearest-neighbor distance (NND) distribution, characterized by the single number ⟨s²⟩ = 1. 08746866652609, the Ginibre kernel value, computable exactly via Palm-corrected gap probability. This universal class encompasses complex Ginibre matrices, random doubly stochastic (DS) matrices, row-stochastic matrices, directed Markov chains, quantum channels (CPTP maps), truncated unitary matrices, band matrices (b/n ≥ 0. 10), sparse matrices (p ≥ 0. 05), and permutation mixtures (k ≥ 3). The onset exponent is β = 3 (cubic NND repulsion), consistent with 2D determinantal structure but not governed by a Painlevé transcendent. DS matrices converge as ⟨s²⟩ (n) = 1. 08747 + a·n^-2/5; Ginibre converges faster as ~n^-1. The Sinkhorn iteration acts as a discrete heat equation (KL gradient flow) with mixing time ~1. 1–1. 3 steps, and NND statistics saturate after ≤10 iterations regardless of matrix size. We compute 20 exact kernel moments to 14-digit precision and show their excess ratios grow linearly with slope ≈1/ (3π), suggesting Γ (3π) structure. The universality boundary is sharp: a 9× gap separates matrices with 2D spectra (universal) from symmetric/reversible matrices (non-universal), with no intermediate cases. Over 101 systematic experiments totaling >10⁷ matrix diagonalizations at sizes n = 3–500 across 15+ ensemble types support these conclusions.