Dimensional Insularity in Multi-Vector Prime Dynamics: A Clifford Geometric Algebra ($\mathbb{G}_2$) Resolution to the Hilbertian Coordinate Bleeding of CMI-RH

This paper presents a definitive geometric resolution to the coordinate instability surrounding the non-trivial zeros of the Riemann Zeta function within the Clay Mathematics Institute's Millennium Prize formulation (CMI-RH). We demonstrate that the standard representation over the complex scalar field ($\mathbb{C}$) suffers from a systemic structural vulnerability—the Russian Doll Paradox—wherein abstract algebraic operations allow frequency variations to bleed back into, and shift, the real spatial baseline ($a=\frac{1}{2}$). To enforce absolute coordinate rigidity, we reformulate Hans von Mangoldt’s explicit prime-counting formula utilizing Clifford Geometric Algebra over a two-dimensional space ($\mathbb{G}_2$). By mapping the spatial baseline to a linear vector ($e_1$) and the spectral frequency to an orthogonal bivector ($e_1e_2$), we prove that the structural grammar of Clifford multiplication ($e_1 \cdot e_2 = 0$) physically paralyzes variable cross-contamination. Under this framework, the critical line is revealed to be an un-deformable geometric node of a standing wave pinned at exactly $a = \frac{1}{2}e_1$, yielding a structurally isolated, non-Fourier counting of exact prime numbers.