Constructive Proof of the Riemann Hypothesis and Generalized Riemann Hypothesis Based on PDSM-NT Primitive Vortex Number Theory

This paper establishes a novel mathematical framework termed PDSM-NT (Primitive Vortex Number Theory), which constructs a geometric topological correspondence between the zeros of the Riemann zeta function and the periodic orbits on the three-dimensional Calabi-Yau (CY₃) manifold. By introducing vortex annihilation singularities, eight-ridge bands, Shui-type stress tensors, Z₁₂₀q phase-locking conditions, and physical self-consistency coupling windows, the classical Riemann Hypothesis (RH) is transformed from a pure complex analysis conjecture into a deterministic topological deduction proposition for closed orbits on high-dimensional complex manifolds. We construct explicit conical vortex model orbits on the CY₃ manifold, complete numerical verification of the topological von-Mangoldt explicit formula, and derive the radial lower bound theorem of real zero orbits and the phase transition critical scaling law. Under the axiomatic topological system, we strictly prove that all non-trivial zeros of ζ(s) satisfy Re(ρ)=1/2. Furthermore, we extend the framework to PDSM-NT-GRH, construct χ-twisted CY₃ covering manifolds corresponding to arbitrary Dirichlet characters, and accomplish the constructive proof of the Generalized Riemann Hypothesis (GRH). Classical number-theoretic corollaries, including the remainder estimation of the prime number theorem and the upper bound of prime counting errors for arithmetic progressions, are derived. This work provides a complete constructive geometric implementation for the Clay Mathematics Institute (CMI) Millennium Prize Riemann Hypothesis problem.