We present a complete proof framework for the Riemann Hypothesis emerging from a divisor-based algebraic model. Starting from the elementary divisor relation xy = n, we derive a quadratic equation whose complex roots lie automatically on the critical line Re = 1/2. Generalizing this structure to operators, we prove that any operator H satisfying H² - H + N = 0 with a self-adjoint N forces its non-real eigenvalues onto Re = 1/2. We define a differential operator T = -i d/dx on L² (R^+) with jump conditions inspired by the von Mangoldt function, and prove via the Krein-Levinson trace formula that its spectral trace matches the Riemann-Weil explicit formula. By the spectral matching theorem, the spectrum of T is precisely the set of ordinates of the non-trivial zeros of zeta (s). Finally, applying Theorem 1 to H = 1/2 + iT proves that all non-trivial zeros lie on Re = 1/2
alyousef et al. (Mon,) studied this question.