This paper derives the local spectral statistics of a completion-locked Hilbert–Pólya operator for the Riemann zeta function, in a setting where the Archimedean completion and determinant normalization are fixed at the level of the spectral realization, leaving no freedom for downstream spectral rescaling. A self-adjoint operator is structurally extracted as the generator associated with an equilibrium unitary sector; this sector carries a circle-valued phase U (1) ≅ S¹ underlying both the analytic scale and the statistical law, while the interior spectrum realizes the nontrivial zeros through the mapping s = 1/2 ± i√λ. The main result is that the local spectral statistics are derived from the same structural constraints from which the operator is extracted. In contrast to standard random-matrix interpretations, where Gaussian Unitary Ensemble (GUE) statistics serve as an external comparison model, the unitary bulk law is derived here from the completion-locked equilibrium structure; its GUE realization on the real self-adjoint spectral line then follows by logarithmic spectral transfer. The analytic scale is determined by the Berezinskii–Kosterlitz–Thouless (BKT) critical point associated with the circle phase of the unitary sector. At this scale, Born–Haar uniqueness identifies the canonical equilibrium reduction with Haar twirling on the compact unitary sector, yielding the Circular Unitary Ensemble (CUE) eigenphase distribution. The eigenphase density then follows by Weyl integration, the correlation functions form a determinantal point process, and the bulk scaling limit yields the sine-kernel law. The logarithmic functional calculus Uσ = exp (−i Hσ τ) identifies the unitary eigenphase spectrum on S¹, on local arcs where a logarithm branch is fixed, with the self-adjoint generator spectrum on the real spectral line ℝ. It thereby transfers the unitary bulk law to the real line after unfolding. Under the Hilbert–Pólya correspondence s = 1/2 ± i√λ, λ ∈ spec (Hᵢnt), the nonnegative interior eigenvalues determine the zero ordinates γ = √λ on the critical line Re (s) = 1/2 in the complex s-plane. On compact bulk windows away from the origin, this smooth square-root reparametrization preserves unfolded local statistics. Finite-equilibrium nonresonance for the completion-locked local observable class removes block contamination, and the resulting spectral correspondence extends to the positive zeta-zero ordinates. Thus the Hilbert–Pólya operator, the analytic scale, and the GUE local bulk statistics are derived as three compatible realizations of a single completion-locked structure. License note: Distributed under CC BY-NC-ND 4. 0.
Salimah Meghani (Thu,) studied this question.