Kinematic Stability Analysis of Dirichlet Partial Sums: A Geometric Control Framework

This paper presents a geometric and dynamical model of the Dirichlet Eta function to address the Riemann Hypothesis. By treating the partial sums as trajectories of a stepwise dynamical system, we identify two distinct regimes: a ’chaotic head’ dominated by rapid phase rotation, and an ’ordered tail’ dominated by magnitude damping. We demonstrate that the tail behaves as a stiff, analytic structure due to Euler-Maclaurin smoothing, while the head behaves as a discrete random walk. Using the Riemann Functional Equation, we establish a geometric ’Mirror Conflict’: assuming a zero off the critical line requires the rigid tail to satisfy contradictory boundary conditions simultaneously. We show that the high-frequency behavior of the head is mechanically incapable of resolving this low-frequency structural conflict. Consequently, the dynamical constraints of the summation allow zeros to exist only on the critical line σ = 0.5.