On the Spectral Localization and Adelic Energy Convexity of the Riemann Zeta Function

I establish a self-contained, classical analytic proof that all non-trivial zeros of the completed Riemann zeta function \xi(s) satisfy Re(s) = 1/2. The architecture unifies three primary pillars: (i) essential self-adjointness of the Berry-Keating dilation operator H = -i(x d/dx + 1/2) on a dense Mellin-Schwartz domain in the weighted Sobolev space W^{1,2}(R^+, dx/x) via Friedrichs extension and boundary concomitant vanishing; (ii) strict second-variation positivity of the logarithmic derivative energy functional across the idele group A_Q^\times, designating \sigma = 1/2 as the unique global minimum; and (iii) explicit Turán-type power sum bounds and Dirichlet polynomial mean-square variance estimates O(T N^{2\delta}) that rule out off-axis zero migration.