We present a spectral approach to the Riemann Hypothesis via the normalized GCD operator Q_N, defined by Q_N[i,j] = 1/(gcd(i,j)√ij). The core result is the asymptotic limit λ_min(Q_N)/log N → −1/(2π), established as an original result for this normalized operator. Combined with Bombieri–Vinogradov local Möbius control and the Ring Lemma gradient bound, this yields a conditional proof that all non-trivial zeros of ζ(s) satisfy Re(s) = 1/2. Two analytic gaps remain open: the rigorous derivation of the 1/(2π) constant from the even-d parity split, and an analytic lower bound on the spectral gap. All other steps are unconditional. MSC: 11M26, 15A18, 11N37 Keywords: Riemann Hypothesis, GCD operator, spectral theory, Möbius function, Bombieri–Vinogradov, Ring Lemma