xiaogang shui

Complete Proof of the Riemann Hypothesis Based on PDSM-NT Primitive Vortex Number Theory and CY₃ Manifold Topology

Based on the independently constructed PDSM-NT (Primitive Discrete Symmetric Modular Number Theory) theoretical system, this paper innovatively reconstructs the Riemann zeta function on the basis of interdisciplinary integration of Calabi-Yau three-fold (CY₃) manifold topological...

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

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...

The Geometric Essence of Riemann Zero-Point Distribution: Based on the PDSM‑NT Primordial Vortex Number Theory

The zero-point distribution problem of the Riemann ζ-function serves as the cornerstone of modern analytic number theory, and the Riemann Hypothesis (RH) is a century-old core puzzle spanning mathematics, physics, and cryptography. Classical research frameworks adopt complex anal...

Research on the Deep Correlation Between Prime Gaps and π Based on the Superposition of Eight-Ridge and Π-Ridge Topology

Abstract: Based on the PDSM-NT primitive vortex number theory system, this paper constructs an eight-ridge topological model of primes under the CY₃ Calabi–Yau manifold and introduces the core axial structure, the Π-ridge. Breaking the limitation of traditional number theory that...

PDSM-NT Primordial Vortex Number Theory — Full Equivalence Theory of CY3 Manifold Topology and Riemann Zeta Zeros

This paper establishes a novel primordial vortex number theory system (PDSM-NT). By constructing an eight-ridge topological band and Π-ridge vortex structure on a Calabi–Yau three-fold (CY3) manifold, we define a new topological stress tensor (Shui’s stress tensor). We rigorously...

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

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...

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

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...

Research on the Deep Correlation Between Prime Gaps and π Based on the Superposition of Eight-Ridge and Π-Ridge Topology

Abstract: Based on the PDSM-NT primitive vortex number theory system, this paper constructs an eight-ridge topological model of primes under the CY₃ Calabi–Yau manifold and introduces the core axial structure, the Π-ridge. Breaking the limitation of traditional number theory that...

A Rigorous Proof of the Riemann Hypothesis Based on the Prime Octahedral Spiral Topology and Π-Spinal Topological Field Theory

Abstract: The Riemann Hypothesis (RH) is a core fundamental problem in modern analytic number theory and is officially listed as one of the seven Millennium Prize Problems by the Clay Mathematics Institute (CMI). It essentially governs the intrinsic correlation between the non-tr...

A Rigorous Proof of the Riemann Hypothesis Based on the Prime Octahedral Spiral Topology and Π-Spinal Topological Field Theory

Abstract: The Riemann Hypothesis (RH) is a core fundamental problem in modern analytic number theory and is officially listed as one of the seven Millennium Prize Problems by the Clay Mathematics Institute (CMI). It essentially governs the intrinsic correlation between the non-tr...

Prime Conch Geometry and Riemann Zeta Zero Analysis Based on Modulo-210 Periodic Vortex Topology (PDSM Framework)

Abstract: 1. This paper constructs the Prime Vortex Dynamic System (PDSM) and establishes an eight-ridge prime conch topological structure constrained by dual modulo-30 and modulo-210 periodic boundaries, breaking the coarse-grained limitations of traditional prime geometric mode...

Prime Conch Geometry and Riemann Zeta Zero Analysis Based on Modulo-210 Periodic Vortex Topology (PDSM Framework)

Abstract: 1. This paper constructs the Prime Vortex Dynamic System (PDSM) and establishes an eight-ridge prime conch topological structure constrained by dual modulo-30 and modulo-210 periodic boundaries, breaking the coarse-grained limitations of traditional prime geometric mode...

The Ultimate Coupling Law of 8-Ridge Conch Topology and π³ Third-Order Field — A Complete Closed System of 100 Prime-π Topology Weld Points, 100 Formulas and 100 Primitive Constants (Final PDSM System)

Based on the primitive core system of PDSM (Prime-π Dimension Symmetry Mechanism), this paper systematically constructs a complete ultimate topological welding system of 100 groups of prime-π (π-P) topological coupling nodes, together with a full set of 100 exclusive topological...

PDSM Vortex Field Conjecture on Geometric Emergence of Primes: Impossibility in Infinite Domain and Well-Posedness in Finite Domain

Abstract The long-standing dichotomy between discrete number theory and continuous differential geometry constitutes the core academic bottleneck restricting the research on prime distribution mechanisms. To break through this theoretical dilemma, this paper constructs an origina...

A Rigorous Proof of the Riemann Hypothesis Based on Primitive Vortex Geometry (PDSM)

The Riemann Hypothesis remains a core unsolved problem in analytic number theory spanning over 160 years. Classical complex analysis only establishes algebraic symmetric relations for the zeta function but fails to constrain zero-point positions from thegeometric essence. This pa...

Deep Topological Correlation Between π Ripple Scale, π Decimal Embedded Primes and Prime Gaps (Original PDSM Mechanism)

For more than a century, classical analytic number theory has adhered to the core premise of the random distribution of prime numbers, and the Riemann zeta function system and traditional sieve methods can only realize statistical fitting and approximate estimation of prime distr...

Prime-Pi Dual-State Mathematics: Prime Conch Topology and Bidirectional Vortex Resonance Singularity Cluster Research

Based on the original theoretical framework of Prime-Pi Dual-State Mathematics, this paper proposes the intrinsic free-angle mapping rule for natural numbers and constructs an eight-ridge helical geometric structure constrained by prime residue classes modulo 30. The model define...

Primitive Vortex Mathematics (PDSM): Prime Singularities, Residues and Geometric Theory of the Riemann ζ-Function on Eight-Ridge Spiral Manifolds

Based on Primitive Vortex Mathematics (PDSM) and Vortex Dynamic Differential Geometry (V-DG), this paper constructs an eight-ridge spiral induced manifold as the geometric carrier of prime number distribution. Covariant vortex equations and intrinsic manifold covariant residues a...

Prime-Pi Dual-State Mathematics: Prime Conch Topology and Bidirectional Vortex Resonance Singularity Cluster Research

Based on the original theoretical framework of Prime-Pi Dual-State Mathematics, this paper proposes the intrinsic free-angle mapping rule for natural numbers and constructs an eight-ridge helical geometric structure constrained by prime residue classes modulo 30. The model define...