(SAPZ v15.0) The Unconditional Derivation of the GUE Sine Kernel from Weighted VMO Decay: A Unified Functional Analytic Approach

This paper presents the culminating result of SAPZ Step 2 by rigorously proving that the vanishing of the weighted VMO deviation δ(T) for log |ζ(1/2 + i t)| implies convergence of empirical pair correlation operators to the GUE sine kernel in the operator-theoretic sense. By synthesizing results from recent SAPZ reinforcement papers—covering VMO–Fourier duality, spectral energy decay, Hilbert transforms, and mollifier-controlled error bounds—we establish a functional analytic bridge from local oscillation decay to GUE universality.

The proof integrates:

• Sharp VMO decay estimates derived via entropy bounds, Dirichlet LDP, and John-Nirenberg inequalities,

• Fourier tail control using a precise VMO–Fourier duality without ε-loss,

• Spectral energy localization via a weighted Parseval identity,

• Explicit formula connections between ζ′(s)/ζ(s) and zero distributions,

• and operator convergence from empirical kernel K_T to the GUE sine kernel K_GUE in both Hilbert–Schmidt and trace-class topology.

This result completes the SAPZ Step 2 program and prepares the foundation for the final inference from GUE statistics to the Riemann Hypothesis in SAPZ Step 3.