Non-Abelian Holonomy and the Poisson–GUE Transition on Modular Pole Lattices

We develop a Möbius lattice framework on Z⁴, where modular generators {T, S, ST, TS} generate a non-abelian channel algebra. The curvature of this algebra is explicitly realised through square holonomies, leading to a controlled transition between Poisson and GUE spectral statistics. A central result is the explicit expression for the effective level-repulsion exponent: beta_tilde = 2 * eta_p * ( (5/21) * kappa_phi + (16/21) * eta_P ) eta_P = 24/43 defines a Pell-induced curvature floor (~0.425) kappa_phi ∈ (0,1] measures Fibonacci (golden-ratio) coherence eta_p ∈ [0,1] represents Hecke-induced arithmetic correlation This reveals a three-layer mechanism controlling spectral statistics: Geometric layer — holonomy projectors separating Fibonacci and Pell sectors Curvature layer — eigenvalue scaling determining intrinsic repulsion strength Arithmetic layer — Hecke action modulating global correlations We show that Poisson statistics arise from abelian (flat) channel algebras GUE statistics arise from maximally curved non-abelian (su(2)-saturated) structures Intermediate regimes are determined by curvature and arithmetic decorrelation Furthermore, the Y-channel is identified as a holonomy bundle, and GUE rigidity emerges as the spectral signature of closed Y-paths under an involution.