SEERAVERSE RH Program: A Boundary-First Pick Approach to the Riemann Hypothesis

Abstract We present a boundary-first Pick formulation for the Riemann Hypothesis. The argument proceeds in two stages. First, a finite/effective analysis establishes strict positivity of a right-half-plane Pick matrix \(K_{\partial}^{(N)}\) for every depth \(N\). The proof combines a rigorous finite prefix, effective centrality estimates, a variable reflection-cap argument, an analytic product closure of the remaining finite gap, and an effective large-depth restart. Second, all-depth Pick positivity is converted, through classical positive-real interpolation and a Montel normal-family limit, into a holomorphic positive-real interpolant \(H\) on the right half-plane. On a translated half-plane, the difference \(H-R_{\partial}\) belongs to the Nevanlinna class and vanishes on a non-Blaschke lattice. Nevanlinna-class uniqueness then yields a meromorphic identity between \(H\) and \(R_{\partial}\) throughout the right half-plane. Since \(H\) is holomorphic there, \(R_{\partial}\) has no genuine poles in that region. A nontrivial zero \(\rho\) of the Riemann \(\xi\)-function with \(\Re \rho>1/2\) would induce such a pole at \(x=\rho-1/2\) with nonzero residue. Hence no nontrivial zero lies to the right of the critical line, while the functional equation \(\xi(s)=\xi(1-s)\) excludes zeros to its left. The resulting conclusion is that every nontrivial zero lies on the critical line \(\Re s=1/2\).