Arithmetic Holonomy and Non-Integrable Accumulation as a Structural Origin of GUE Statistics in the Zeros of the Riemann Zeta Function

Abstract

The non-trivial zeros of the Riemann zeta function exhibit universal statistical properties consistent with the Gaussian Unitary Ensemble (GUE), including level repulsion and spectral rigidity. While this behavior is often interpreted through the conjectural existence of a chaotic Hamiltonian (Hilbert–Pólya framework), such an approach introduces an implicit dynamical structure that is not intrinsic to arithmetic objects. In this exploratory work, we propose an alternative structural viewpoint: the observed GUE statistics arise from a non-integrability of arithmetic accumulation itself, formalizable as an arithmetic holonomy. This holonomy does not describe an evolution in time, but a path dependence in admissible accumulation procedures (such as truncation, smoothing, or dual summation between primes and zeros). We argue that a non-vanishing but constrained holonomy naturally produces local desynchronization (level repulsion) together with global rigidity, without invoking an underlying Hamiltonian dynamics. The proposal is conceptual and structural, aiming to clarify the geometric origin of spectral rigidity in the zeta zeros rather than to construct an explicit operator.