The Geometry of the $2\pi$ Rolling Circle: A Convergent Phase-Locked Loop Heuristic along the Critical Line

This article introduces a digital signal processing (DSP) heuristic that models the behavior of the Riemann Zeta function $\zeta(s)$ on the critical line $a=1/2$ via a convergent phase-locked loop (PLL) architecture. To bypass the strict divergence of the standard Dirichlet series within the critical strip, we employ the alternating Dirichlet Eta function $\eta(s)$ as a regularized phase tracking engine. By defining the $2\pi$ circumference of a rolling unit circle $C_1$ as a linear reference coordinate $n \in \mathbb{R}^+$, we map non-trivial zeros to localized harmonic consensus points. We resolve the complex 1D update projection paradox by implementing a Gauss-Newton phase-energy minimization routine. Finally, we demonstrate how these tracking coordinates function as the core resonant frequencies required to drive the Möbius bandpass prime isolation filter.