PDSM-NT Primordial Vortex Number Theory: Rigorous Proof of the Riemann Hypothesis via CY₃ Manifold Wormhole Cluster Topological Barriers

Abstract This paper establishes an original theoretical framework termed PDSM-NT (Projection Dynamical Singularity Number Theory – Primordial Vortex System). Based on the differential geometry of Calabi-Yau threefold (CY₃) manifolds, nested termite-nest wormhole cluster topology, Shui conserved stress-energy tensors, ℤ₁₂₀ modular periodic symmetry, and the 100th-order Shui high-order differential operator, we perform a rigorous full-complex-plane analytic continuation of the Riemann zeta function via Hankel contour integration from the half-plane Re(s) > 1 to ℂ\{1}. By constructing a complex-domain vortex source-potential coupling system, defining a vortex annihilation energy functional, and establishing a high-order differential screening criterion for spurious zeros, we constrain the spectral distribution of zeta zeros through three fundamental topological mechanisms: topological barrier confinement, topological energy gap suppression, and global modal screening. Within the self-consistent PDSM-NT axiomatic system, we strictly prove that all non-trivial zeros of the Riemann zeta function lie exactly on the critical line Re(s) = 1/2, which fully validates the Riemann Hypothesis. This work constructs a novel cross-disciplinary coupling framework connecting analytic number theory, high-dimensional differential geometry, and topological field theory, providing an intrinsic topological dynamical interpretation for prime distribution and zeta zero spectral problems.