A Control-Theoretic Approach to the Riemann Hypothesis: State-Space Stability and Spectral Analysis of the Sieve of Eratosthenes

This paper introduces a novel theoretical framework that maps the analytical problem of the non-trivial zeros of the Riemann zeta function into a stability and energy balance problem of feedback control systems. We model the Sieve of Eratosthenes as a discrete, non-linear, time-varying system whose negative feedback loop accumulates pure delays and regulates the density of future prime numbers at an exact rate of \(1/\ln(n)\). Through the formulation of a Lyapunov-Krasovskii energy functional and the application of Barbalat's Theorem, we demonstrate both numerically and analytically that the tracking error energy remains asymptotically bounded under the critical square-root frontier \(\sqrt{t}\). Finally, using the Nyquist Stability Criterion, we argue that the real coordinate \(\sigma = 1/2\) represents the unique axis of dissipative symmetry and pure resonance of the system, thereby confining all closed-loop poles (Riemann zeros) to said critical line.