Modular Pole Chains and the Emergence of GUE Spacing

We investigate the spacing distribution of poles generated by modular TST chains in PSL(2,Z)PSL(2,Z)PSL(2,Z).By linking hyperbolic random walks and continued fraction dynamics, we derive subgaussian fluctuations in pole spacing without invoking circular arguments. Under four structural conditions—repulsion at short distance, Gaussian confinement at large scale, non-abelian symmetry (su(2)), and mean normalization—the spacing distribution is uniquely determined as the GUE Wigner surmise: p(s)=32π2s2e−4s2/π.p(s)=32/π^2s^2 e-4s^2/π.p(s)=π232s2e−4s2/π. This framework suggests that GUE statistics arise naturally from modular non-commutativity and global geometric constraints, providing a structural bridge between arithmetic dynamics and spectral universality.