xiaogang shui

Elevation of Prime Gaps to Vector Topological Distances on Calabi-Yau Three-Folds: Discrete-Continuous Geometric Embedding within the PDSM-NT Framework

Abstract As a core research object in analytic number theory, the prime gap $$g_n=p_{n+1}-p_n$$ has long been regarded as a one-dimensional non-negative integer scalar. Within the axiomatic framework of the PDSM-NT primitive vortex number theory and wormhole space theory, this pa...

Elevation of Prime Gaps to Vector Topological Distances on Calabi-Yau Three-folds: A Discrete-Continuous Geometric Embedding Study Based on the PDSM-NT Framework

Abstract The prime gap $$g_n=p_{n+1}-p_n$$ is a fundamental physical and mathematical quantity that characterizes the asymptotic distribution behavior of prime numbers in analytic number theory. In traditional research systems, prime gaps are simply defined as one-dimensional dis...

A Spectral Framework for Prime-Generating Operators and an Exploratory π-Polynomial Relation for the Fine Structure Constant

Euler’s formula ζ(2) = π2/6 reveals a deep connection between primes (via the Euler product)and the transcendental number π. This paper elevates the fact that “primes generate π”to a rigorous operator-spectral language: we define a prime-generating operator Agen,shuiwhose discret...

PDSM Geometrization of the Fine-Structure Constant: Six- Transcendental Basis Fitting and CY₃ Topological Correspondence

AbstractThe fine-structure constant α ≈1 /137 is a dimensionless fundamental constantcharacterizing the strength of electromagnetic coupling in quantum electrodynamics. Itsvalue cannot be derived from first principles and was called “one of the greatest damnmysteries of physics”...

PDSM Geometrization of the Fine-Structure Constant: Six- Transcendental Basis Fitting and CY₃ Topological Correspondence

AbstractThe fine-structure constant α ≈1 /137 is a dimensionless fundamental constantcharacterizing the strength of electromagnetic coupling in quantum electrodynamics. Itsvalue cannot be derived from first principles and was called “one of the greatest damnmysteries of physics”...

he Prime Tree: A Geometric Framework for Riemann Zeta Zeros withNumerical Verification

AbstractWe propose the Shui Prime Tree —a geometric framework embedding the distribution of primes into three-dimensional Euclideanspace. Each prime is mapped to a node on a helical curve, , where the radiusencodes the Riemann critical line , the vertical coordinate is base- , an...

Steady-State Characteristics of Normalized 11th-Order Prime Differences and Its Gradient Correlation with the Decimal Sequence of π

Abstract Classical number theory and transcendental number theory generally hold that the high-order differences of prime sequences are unbounded and disordered without finite steady-state periodic characteristics. As a typical transcendental number, the decimal expansion of π is...

Research on the Underlying Analytical Mechanism of 11th-Order Optimality and 64-bit Steady-State Scale and Prime Number Correlation

Abstract To reveal the deep coupling law between the high-order difference optimality criterion and the 2-adic scale steady-state system, this paper conducts interdisciplinary research integrating number theory difference operators, p-adic geometry, cyclotomic field theory, and C...

Steady-State Characteristics of Normalized 11th-Order Prime Differences and Its Gradient Correlation with the Decimal Sequence of π

Abstract Classical number theory and transcendental number theory generally hold that the high-order differences of prime sequences are unbounded and disordered without finite steady-state periodic characteristics. As a typical transcendental number, the decimal expansion of π is...

Research on the Underlying Analytical Mechanism of 11th-Order Optimality and 64-bit Steady-State Scale and Prime Number Correlation

Abstract To reveal the deep coupling law between the high-order difference optimality criterion and the 2-adic scale steady-state system, this paper conducts interdisciplinary research integrating number theory difference operators, p-adic geometry, cyclotomic field theory, and C...

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...

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...

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 — 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...