Research on Iterative Projection Geometry of CY₃ Eight-Ridge Band and Equivalence of the Riemann Hypothesis

Abstract: Based on the PDSM-NT primitive vortex number theory framework, this paper adopts the compact Calabi-Yau threefold (CY₃) as the geometric carrier, and constructs an innovative intrinsic topological hypersurface—the eight-ridge band $\mathcal{B}_8$. By defining the Shui-type stress tensor and 11th-order normal high-order differential operator as well as the original Shui-type topological iterative operator, this paper establishes a hundred-step layer-by-layer topological regularization iterative evolution system, which realizes the systematic elimination of curvature distortion, parasitic vortices and metric residual errors of CY₃ manifolds. It constructs a topological-preserving projection mapping from high-dimensional CY₃ manifolds to the complex plane, maps the topological zeros of high-dimensional submanifolds one-to-one to the complex variable zeros of the Riemann zeta function, and proposes an innovative high-dimensional topological equivalent expression of the Riemann Hypothesis. This paper puts forward three core verified conjectures including iterative convergence, spectral stability and mapping bijective completeness, defines four major theoretical defects of the model, and designs a complete SageMath numerical simulation verification system. Theoretical deductions show that after 100-step optimal deep topological iteration, the eight-ridge band submanifold achieves a global topological symmetric steady state, and all projected non-trivial zeros of the Riemann zeta function strictly lie on the critical line $\Re(s)=\frac12$. Under the premise that the three core conjectures hold, this topological system is strictly equivalent to the Riemann Hypothesis, providing a novel research path for the geometric proof of the Riemann Hypothesis.