The Tripartite Ontological Divergence of Prime Extraction: Historical Horizons from Riemann to Hilbert and the Geometric Mechanics of the Critical Strip

This paper establishes a rigorous historical and structural critique of the epistemological shifts surrounding the non-trivial zeros of the Dirichlet-Riemann Zeta system. We demonstrate that the contemporary mathematical stalemate regarding the Riemann Hypothesis (RH) stems from a conflation of three fundamentally incompatible objectives across a historical timeline: Bernhard Riemann's wave-harmonic correction engine (1859), Hans von Mangoldt's analytical arithmetic purification (1895), and David Hilbert's axiomatic operator formalism (1900). Furthermore, we clarify the ontological status of the Zeta function, defining the critical strip ($0 < a < 1$) not as an arbitrary scalar domain, but as an observer-dependent rotational zone governed by Euler’s identity, where the continuous spatial baseline allows for perfect circular phase trajectories.