A Topological Framework for the Asymptotic Density and Strict Confinement of Riemann Zeros via Geometric Phase Bounding

This paper presents a topological and geometric framework for bounding the non-trivial zeros of the Riemann zeta function, zeta(s), exclusively to the critical line Re(s) = 1/2. By defining a linear geometric baseline utilizing the Euler-Mascheroni constant and the dimensional projection of circle geometry, we isolate the non-linear, logarithmic parametric arc length of the curve zeta(1/2 + it). This isolation yields a continuous phase function, mapped topologically as a bounding solid of revolution. Furthermore, by applying Riemann’s functional equation to this strictly budgeted topological envelope, we introduce a proof by contradiction. We demonstrate that the theoretical existence of off-line zero doublets demands a geometric “Orbit Toll” that mathematically shatters the established amplitude bounds, thereby proving absolute confinement to the critical axis.