An Optimization-Theoretic Framework for the Riemann Hypothesis: Zeta Spectral Energy Equations and Variational Stability of the Critical Line

The Riemann Hypothesis asserts that all nontrivial zeros of the Riemann zeta function lie on the critical line in the complex plane, i.e., the real part equals 1/2. Despite numerous methods proposed over the past century, this problem remains unsolved. This paper proposes a new optimization theory framework that models the zero-point structure of the zeta function as an energy-minimal system in the complex plane. We construct a new spectral energy functional and propose a new fundamental equation: the Zeta Spectral Energy Equation. Through variational analysis, we prove that this energy function has a stable minimum structure near the critical line and establish a correspondence between the zero-point distribution and a two-dimensional log-interacting particle system. We further construct an optimization dynamical system and analyze its stable structure. This theory transforms the Riemann Hypothesis into a global energy minimization problem, thus providing a new unified perspective of geometry and optimization.