Two Decisive Closure Theorems for a Weil Hurwitz Approach to the Riemann Hypothesis

This paper contributes two master theorems. The first, the Weighted Isolated Spectral Branch Theorem, combines form-domain admissibility, spectral isolation, reduced-resolvent control, and exponential localization into one quantitative statement. It extends the previously proved residual-to-gap lowest-state estimate from a simple eigenvalue and an operator-domain residual to a finite-rank isolated spectral cluster and a quadratic-form residual. This extension is forced by a symmetry obstruction: the natural one-boundary Gauss–Eisenstein trial state cannot converge in the finite-interval Hilbert space to a reflection-even simple ground state. The correct object is a localized vector in an isolated even–odd spectral doublet, followed by a norm-preserving boundary unfolding. The second, the Self-Adjoint Determinant Representation Theorem, promotes the Fredholm determinant from a technical endpoint to the decisive arithmetic statement. We prove that an entire function admits a regularized determinant representation with a compact self-adjoint Schatten-class operator if and only if its nonzero zeros are real and their reciprocals have the corresponding Schatten summability. Consequently, merely postulating such a representation is equivalent to assuming the desired real-zero conclusion. A non-circular proof must instead construct the operator independently and verify an explicit logarithmic-derivative or trace identity. Finite-section convergence then supplies real-rooted approximants. Combining the two theorems gives a precise conditional closure: a verified weighted isolated branch produces locally uniform convergence to the completed Riemann function, while an independently proved self-adjoint determinant identity forces the finite-scale functions to have only real zeros; Hurwitz’s theorem then implies the Riemann Hypothesis. All abstract implications are proved here. The two remaining exact-model statements are displayed as explicit, falsifiable verification problems. No unconditional proof of the Riemann Hypothesis is claimed.