Quantitative Ground State Closure for the Weil Operator and Diagonal Hurwitz Reduction of the Riemann Hypothesis

We prove a decisive Residual–Spectral Gap Theorem for the lowest state of a self-adjoint compact-resolvent operator and use it to complete a quantitative ground-state reduction of the Riemann hypothesis. In the terminology of the broader spectral-geometric program, the theorem compresses the tasks E1–E3–spectral isolation, trial-state residual control, and quantitative identification of the true lowest state–into the single certifiable ratio . It simultaneously controls the ground-state vector and the Ritz eigenvalue, and therefore supplies the residual-to-gap lowest-state theorem sought by the program. We then prove the weighted compactness, exact Gauss–Eisenstein Mellin identity, recursive diagonal selection, and real-root closure results needed for a Minimal Closure Theorem. If the finite-interval Weil operator has a simple even ground state, the residual-to-gap ratio tends to zero, the recentered ground states obey uniform exponential moment bounds, and the finite spectral determinants converge at each fixed scale, normalized real-rooted determinants converge locally uniformly to the Riemann -function; Hurwitz’s theorem then implies RH. This achieves the intended E1–E3 compression and the complete minimal closure of the implication chain, while retaining explicit hypotheses for the operator estimates that remain to be established in the Weil model.