A Completion-Locked Derivation of GUE Local Statistics for the Nontrivial Zeros of the Riemann Zeta Function

This paper derives GUE local statistics for the nontrivial zeros of the Riemann zeta function from a completion-locked Hilbert–Pólya operator. In the completion-locked setting, the Archimedean completion, determinant normalization, and analytic scale are specified at the level of the spectral realization, so no downstream spectral rescaling is available. The Hilbert–Pólya operator, the analytic scale, and the local spectral statistics are derived as compatible realizations of this same structure.

The equilibrium unitary sector carries a circle-valued phase structure. The associated Berezinskii–Kosterlitz–Thouless (BKT) critical point determines the analytic scale, and Born–Haar uniqueness identifies the canonical equilibrium reduction with Haar measure. Haar reduction yields the CUE eigenphase distribution; Weyl integration gives the eigenphase density; the correlation functions form a determinantal point process; and the bulk scaling limit gives the sine-kernel law.

Logarithmic spectral transfer carries this unitary bulk law from eigenphases on the unit circle to the real self-adjoint spectral line, producing GUE local bulk statistics after unfolding. Under the Hilbert–Pólya correspondence, the real spectral parameter determines the positive zeta-zero ordinates on the critical line in the complex s-plane. On compact bulk windows away from the origin, the square-root reparametrization preserves unfolded local statistics. Finite-equilibrium nonresonance removes block contamination for the completion-locked local observable class.

Consequently, the GUE local statistics of the nontrivial zeros are derived as a structural consequence of the same completion-locked framework that extracts the Hilbert–Pólya operator and determines the analytic scale.

License note: Distributed under CC BY-NC-ND 4.0.