PDSM-NT Primordial Vortex Number Theory — The Octa-Ridge Ring Topological Structure Theory of Primes

Abstract

Abstract: This paper establishes a novel unified system of number theory and geometry, namely the PDSM-NT Primordial Vortex Number Theory. Abandoning the traditional one-dimensional number line paradigm for prime research, this theory breaks the inherent limitations of elementary sieve methods, analytic number theory, and random matrix models. All primes greater than 5 are embedded into the octa-ridge ring topological manifold $\mathcal{B}_8$ defined on a three-dimensional compact Calabi-Yau (CY₃) complex manifold. This paper completes the rigorous axiomatic construction, precise differential geometric modeling, systematic establishment of topological field equations, and strict group-theoretic matching of dual periodic structures. A total of 30 hierarchical, quantifiable, and verifiable core conjectures are proposed systematically. The proposed theory innovatively transforms classic intractable problems including the Riemann hypothesis, twin prime conjecture, Goldbach conjecture, prime gap oscillation, prime cluster distribution, and large integer factorization cryptography problems into precise topological geometric and field-theoretic propositions. It fundamentally realizes the complete geometrization of number-theoretic problems, topologization of discrete distributions, manifoldization of analytic properties, and provabilization of classic conjectures. This work reconstructs the underlying logic of prime distribution, reveals the non-random topological essence of prime numbers, and provides a brand-new, complete, and self-consistent theoretical foundation for the traceability of prime distribution laws, breakthroughs in large integer factorization technologies for modern cryptography, and the construction of geometric number theory systems.