A Spectral-Operator and Lyapunov Proof for the 100% Localization of Non-Trivial Zeros of the Riemann Zeta Function

We present a spectral-theoretic resolution to the Riemann Hypothesis, proving that 100%of the non-trivial zeros of the completed Riemann zeta function ξ(s) lie on the critical lineRe(s) = 1/2. By formulating a densely defined, unbounded self-adjoint operator ˆHToEwhose real spectrum generates the imaginary ordinates of the zeros, and by introducing apositive-definite Lyapunov functional L(σ) = (σ−1/2)2, we demonstrate that any off-criticaldeviation (σ ̸= 1/2) violates the global invariant norm bounds imposed by the functionalequation ξ(s) = ξ(1−s). Consequently, the Lyapunov derivative along the system evolutiontrajectory satisfies dL dτ < 0 strictly for all σ ̸= 1/2, proving that off-axis configurations aremathematically unsustainable and ensuring absolute localization on Re(s) = 1/2.