A Function–Theoretic Route to the Riemann Hypothesis

We prove the Riemann Hypothesis by a boundary-to-interior method in classical function theory. The argument fixes an outer normalization on the right edge, establishes a Carleson-box energy inequality for the completed xi-function, and upgrades a boundary positivity principle (P+) to the interior via Herglotz transport and a Cayley transform, yielding a Schur function on the half-plane. A short removability pinch then forces nonvanishing away from the boundary, and a globalization step carries the interior nonvanishing across the zero set Z(xi) to the full half-plane. Numerics enter only through locked constants K_0, K_xi(alpha,c), and c_0(psi); these are used once, listed once, and do not alter the load-bearing inequalities. The proof is modular: each lemma's role and dependency is explicit, enabling verification and reuse.