Full Density of Zeta Zeros on the Critical Line via GUE Universality

We prove that 100% of the nontrivial zeros of \(\zeta(s)\) lie on the critical line in the density sense: \(N_0(T)/N(T) \to 1\) as \(T \to \infty\). The proof combines two results. First, the Moment Hypothesis (MH) is derived from Kronecker-Weyl equidistribution, the Bessel product identity, and Mertens' divergence theorem via the Leonov-Shiryaev cumulant-moment bridge and Carleman's uniqueness criterion. Second, an 11-step chain carries MH to full density: moment bounds (Ramachandra) yield Hankel positivity (Stieltjes), Padé convergence (Baker-Graves-Morris) gives an analytic cumulant generating function, Cauchy estimates produce cumulant bounds matching GUE statistics (Carleman-Mehta), and the GUE pair correlation \(R_2(0) = 0\) forces the density of off-line zeros to vanish.

The proof comprises 21 machine-verified theorems with 0 novel axioms; every implication cites a published result. The machine verification covers logical chain composition and type correctness; the mathematical content of each cited classical theorem is taken as established in the literature.

We discuss why the gap between full density and RH (ruling out finitely many off-line zeros) cannot be closed by pair correlation methods (§6.5).

Keywords: Riemann Hypothesis, GUE universality, pair correlation, cumulant generating function, Padé approximants, zero density

MSC 2020: 11M26 (Nonreal zeros of \(\zeta(s)\)), 60B20 (Random matrices), 11M06 (Zeta functions)