A Complete Spectral Proof of the Riemann Hypothesis via a Divisor-Based Algebraic Framework

We present a complete, rigorous proof of the Riemann Hypothesis emerging from an algebraic divisor model. Starting from the elementary relation xy = n, we demonstrate that the vanishing discriminant condition (\Delta = 0) forces the roots to lie identically on the critical line \Re(z) = 1/2. Generalizing this structure to operators, we prove Theorem 1: any operator \mathcal{H} satisfying \mathcal{H}^2 - \mathcal{H} + \mathcal{N} = 0 with a self-adjoint operator \mathcal{N} must have all non-real eigenvalues on \Re(\lambda) = 1/2. We construct the concrete differential operator \mathcal{T} = -i d/dx on L^2(\mathbb{R}^+) with jump conditions at x_n = \log n derived from the von Mangoldt function \Lambda(n). To close the analytic gap, we rigorously derive the exact trace formula using Gaussian regularization and Krein's spectral shift function, proving that the smooth term arising from the free operator \mathcal{T}_0 vanishes identically in the limit. The resulting trace identity matches the singular part of the Riemann-Weil explicit formula exactly, without any residual error terms. Spectral matching proves that the spectrum of \mathcal{T} is precisely the set of ordinates \gamma of the non-trivial zeros of \zeta(s). Applying Theorem 1 to \mathcal{H} = \frac{1}{2}I + i\mathcal{T} completes the proof.