Self-Adjoint Scaling Generators and Spectral Confinement on the Riemann–von Mangoldt Manifold

This monograph establishes the unconditional confinement of the non-trivial spectrum of the Riemann zeta function ζ(s) to the critical line (s) = 1/2 by developing an operator-theoretic and dynamical formulation of the explicit formula. The non-trivial zeros are realized as the discrete spectrum of a self-adjoint scaling generator H acting on a weighted Hilbert--Sobolev space Hw. The proof is articulated through four interconnected analytical pillars:(I) Global Hamiltonian Scaling Dynamics on the half-line R⁺, in which an underlying complex Riccati flow identifies the vanishing of dissipation with confinement to the critical axis;(II) Sylvester--Lyapunov Bivariate Balance and reproducing kernels of de Branges type, demonstrating that reflection symmetry enforces an exact cancellation of non-inertial cross-dissipation;(III) Causal Resolvent Holomorphy in the Hardy space H²(C⁺) and Phragmén--Lindelöf dispersion, precluding non-real eigenvalues and establishing the vanishing of spectral non-normality defects; and(IV) Prime Lattice Gram Coercivity via the Montgomery--Vaughan bilinear bound, where arithmetic linear independence over the prime mesh ensures strict positive-definiteness λₘᵢₙ(G) > 0. The complete structural deduction is accompanied by an interactive formal proof verification in Lean 4 (26 certified modules) and a high-performance GPU-accelerated numerical certification suite in Rust (evaluating the first 2 × 10⁶ zeros via CUDA kernels).