Four Gate Certification for a Weil Hurwitz Route to the Riemann Hypothesis

This paper advances a Weil–Hurwitz spectral program beyond the residual-to-gap closure of E1–E3 [13,14] and the subsequent two-master-theorem reduction [15]. Four gates are isolated and treated by explicit operator theorems. Gate I replaces a postulated finite-scale determinant by a non-circular construction from an ordinary boundary triple for a symmetric operator of defect one. Krein’s resolvent formula gives an exact trace identity, and a normalized characteristic function of the corresponding self-adjoint extension has only real zeros. Equality with the finite-scale Mellin function is reduced to a logarithmic-derivative identity and one base-point normalization. Gate II gives a certified form-residual capture theorem for an isolated finite-rank spectral cluster, including the reflection-even–odd doublet required by the one-boundary trial state. Gate III transfers this unweighted capture to one positive exponential weight through a conjugated, preconditioned complement equation. Gate IV iterates the one-step conjugation estimate on a scale of weighted spaces and thereby obtains every exponential weight needed for compact-uniform Mellin convergence. The four gates combine into a conditional closure theorem: if one exact finite Weil model verifies the stated boundary-triple trace identity, cluster enclosure, weighted conjugation bounds, and residual rate budget along an unbounded scale sequence, then its finite-scale functions are real-rooted and converge locally uniformly to the completed Riemann function; Hurwitz’s theorem then implies the Riemann Hypothesis. The abstract implications and operator estimates are proved in full. The model-specific verification ledger is finite and falsifiable. No unconditional proof of the Riemann Hypothesis is claimed.