Distributions of consecutive level spacings of circular unitary ensemble and their ratio: finite-size corrections and Riemann ζ zeros

We compute the joint distribution of two consecutive eigenphase spacings and their ratio for Haar-distributed U(N) matrices (the circular unitary ensemble) using our framework for Jánossy densities in random matrix theory, formulated via the Tracy-Widom system of nonlinear PDEs. Our result shows that the leading finite-N correction in the gap-ratio distribution relative to the universal sine-kernel limit is of O(N⁻⁴), reflecting a nontrivial cancellation of the O(N⁻²) part present in the joint distributions of consecutive spacings. This finding points to the possibility of extracting subtle finite-size corrections from the energy spectra of quantum-chaotic systems, and provides a potential explanation of why the deviation of the gap-ratio distribution of the Riemann zeta zeros {1/2 + iγn}, γn ≈ T ≫ 1 from the sine-kernel prediction scales as (log (T/2π))−3.