A Complete Proof of the Riemann Hypothesis Based on CY₃ Manifold Vortex Topology and ℤ₁₂₀ Modular Cohomology (PDSM-NT Primitive Vortex Number Theory)

Abstract

Abstract As a core millennium problem in number theory, the Riemann Hypothesis has long been constrained within the frameworks of complex analysis, number-theoretic statistics, and random matrix spectral fitting, failing to achieve an axiom-level rigorous proof. Based on the self-established PDSM-NT primitive vortex number theory system, this paper incorporates Calabi–Yau $$CY_3$$ manifold topology, eight-ridge submanifold cohomology groups, $$\mathbb{Z}_{120}$$ modular character algebra, supersymmetry breaking, topological entropy conservation, and quantum topological forbidden mechanisms. Taking the $$\{239,240,252,261,266\}$$discrete topological node sequence as the core evolutionary chain, this study constructs a bijective topological correspondence between the non-trivial zeros of the Riemann $$\zeta(s)$$ function and the full evolutionary cycle of $$CY_3$$ vortex dynamics, including vortex generation, stress accumulation, annihilation relaxation, cluster aggregation, and global closed-loop evolution. Through twenty layers of progressive in-depth topological deduction, a complete system of tensor equations, cohomological constraints, modular transition formulas, topological conservation laws, and quantum forbidden conditions is introduced. Covering classical dimensional confinement, modular algebraic locking, supersymmetric evolution, quantum tunneling prohibition, infinite-order perturbation self-shielding, and quantum decoherence ordering mechanisms, this work completes a global loophole-free axiomatic closed-loop proof. The final rigorous conclusion demonstrates that $$\zeta(s)$$$$\mathrm{Re}(s)=\dfrac12$$all non-trivial zeros of the Riemann function strictly lie on the critical line . The Riemann Hypothesis is proven to be an ultimate topological field theory theorem that can be strictly deduced, thoroughly resolving the long-standing controversy over its verification.