High-Dimensional Kinematics of the Dirichlet Eta Function: A Geometric Constraints Approach to the Riemann Hypothesis

This paper presents a novel geometric framework for analyzing the zeros of the Dirichlet eta function, η(s). By decomposing the Dirichlet series into distinct Magnitude and Phase vectors within a weighted Hilbert space, we transform the analytic problem of locating zeros into a problem of high-dimensional kinematics.We demonstrate that deviations from the critical line induce a non-uniform geometric deformation of the Magnitude vector, creating linearly independent target hyperplanes for symmetric zeros (the Far-Field constraint). We utilize the linear independence of prime logarithms to establish that the phase trajectory is quasi-periodic and strictly stiff. Furthermore, we analyze the Near-Field regime, demonstrating that higher-order zeros on the critical line are kinematically forbidden by a Derivative Gap. Finally, we apply dimensionality arguments to show that while simple zeros on the critical line represent stable intersections, zeros off the critical line require the recurrence of unstable, codimension-2 intersections—a condition geometrically forbidden for non-periodic systems.