Random matrix theory predicts level-spacing statistics. GOE (chaotic systems) shows Wigner-Dyson repulsion P(s)∝s·e^(−πs²/4); Poisson (integrable) shows bunching P(s)=e^(−s).
GOE matrix: H = (A+Aᵀ)/√(2N), A random Gaussian. Eigenvalues unfolded to mean spacing 1. Level spacing histogram compared to Wigner-Dyson and Poisson predictions. The Σ² number variance grows ∝ (2/π²)ln(L) for GOE vs L for Poisson.