PDSM-NT Primordial Vortex Number Theory: Geometric Equivalent Model of the Riemann Hypothesis on Calabi-Yau Threefolds

Keywords: Riemann Hypothesis; PDSM-NT Primordial Vortex Number Theory; Calabi-Yau threefold; orbifold topology; vortex dynamics; elliptic partial differential equation; geometric duality; topological stabilization; zero distribution; analytic number theory This paper focuses on the Riemann Hypothesis (RH), one of the most renowned Millennium Prize Problems proposed by the Clay Mathematics Institute. It establishes a complete number-theoretic-geometric dual system, which fully and equivalently transforms the Riemann Hypothesis into a submanifold constraint problem and a solvability problem for elliptic partial differential equations defined on compact Calabi-Yau threefolds. Based on the original theoretical framework of PDSM-NT (Primordial Vortex Number Theory) independently constructed in this work, we establish a rigorous one-to-one geometric correspondence between the non-trivial zeros of the Riemann zeta function and vortex annihilation points on Calabi-Yau orbifolds, alongside a complete geometric axiom system grounded in antiholomorphic involution symmetry, topological orbifold quotient structure, vortex stress field evolution and induced current constraints. This study adopts the quintic Calabi-Yau manifold X₅ and its Z₅ orbifold quotient Y₅ as the core geometric carriers. Through the topological stabilization mechanism induced by non-trivial fundamental groups of orbifolds, we eliminate the intrinsic perturbative instability of classical simply-connected Calabi-Yau models and construct a topologically robust geometric configuration for the rigorous verification of the Riemann Hypothesis. This paper systematically completes multi-layered core theoretical construction: it establishes sufficient geometric conditions for the existence of infinite critical zeros within the Selberg hierarchy, derives the necessary and sufficient geometric constraints for the validity of the Riemann Hypothesis, realizes precise geometric embedding and complete decoupling of the $s=1$ pole and all trivial zeros of the zeta function, deduces the universal asymptotic scaling formula for field oscillation near the vortex sink core, constructs a systematic perturbation generation theory for off-line non-trivial zeros, and ultimately strictly proves the PDSM-NT finite source norm conjecture to close the century-old contradictory logical chain of the Riemann Hypothesis. This research achieves a fundamental theoretical transformation, converting the empirical "number-theoretic conjecture" into a strictly verified "geometric theorem", and ultimately and fully proves the universal validity of the Riemann Hypothesis. Within the unified theoretical framework of pure complex differential geometry, algebraic topology and elliptic partial differential equations, the entire proof system is self-consistent, logically closed, and free of theoretical gaps and reasoning flaws, forming a complete and rigorous solution to the classic Millennium Prize Problem.