Group Structures in Discrete Order Geometry（DOG）Author: Zhang Suhang(Luoyang, Henan)AbstractDiscrete Order Geometry (DOG) takes finite discrete lattice points, nested order and local correlations as its ontology. This paper constructs group struct...
Proof of Existence and Uniqueness of Smooth Solutions to Navier-Stokes Equations within DOG/MOC/ECS/MIE Framework (Supplemented Revised Version) Author: Zhang Suhang Luoyang, Henan Abstract The global existence of smooth solutions...
In today’s competitive industrial landscape, manufacturers are under constant pressure to increase productivity, reduce operational costs, maintain product consistency, and meet strict environmental and safety standards. One of the most overlooked...
Combinatorial Foundations of Discrete Order Geometry (DOG): Lattices, Order, and Combinatorial Counting of Neighborhoods Author: Zhang Suhang (Luoyang, Henan) --- Abstract Discrete Order Geometry (DOG) takes finite discrete lattices, order ne...
Basic Coupling Equation of Discrete Order Geometry (DOG): Generative Construction from Zhang's Matrix to FCE
Author: Zhang Suhang (Luoyang, Henan)
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Abstract
Discrete Order Geometry (DOG) takes discrete lattice points, hierarchical ...
Proof of the P vs. NP Problem within the DOG/MOC/ECS/MIE Framework
Author: Zhang Suhang Address: Luoyang, Henan
The P vs. NP problem is a central open question in theoretical computer science, asking whether every prob...
Proof of the BSD Conjecture within the DOG/MOC/ECS/MIE Framework
The BSD (Birch and Swinnerton-Dyer) conjecture is a Millennium Problem in number theory concerning the equal...
The Continuous Limit of Discrete Order Geometry (DOG): A Study of the Correspondence Between Discrete Evolution and Differential Equations
Discrete Order Geometry (DOG) constructs a p...
The Sea of Paradigms: From Grothendieck to the Dissolution of Problems in a New Geometric Framework
Author: Zhang Suhang Address: Luoyang, Henan, China
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Grothendieck once likened his mathematical method to a "rising tide": n...
Recasting the Hodge Conjecture in an Extended Geometric Framework Author: Zhang SuhangAddress: Luoyang, Henan --- Abstract The Hodge conjecture is usually stated as follows: on a smooth complex projective algebraic variety, every (p,p)-type ra...
Foundations of Discrete Order Geometry (DOG): System Definition, Morphisms and Order Isomorphism — First Cornerstone Draft of the DOG Framework Author: Zhang Suhang (Luoyang, Henan) Remarks This paper lays the first foundati...
ECS-Hodge Correspondence Categorical Equivalence Between Symmetric Conservation Modes and Hodge Classes Author: Suhang Zhang Address: Luoyang, Henan Abstract Hodge classes serve as core objects in algebraic geometry, defined as ra...
MOC Embedding Theorem: Embedding Representation of Complex Projective Algebraic Varieties in Multi-Origin Curvature Geometry Author: Zhang SuhangAddress: Luoyang, Henan --- Abstract Traditional complex projective algebraic varieties are built ...
--- A Study of Fractal Order and Chaos Emergence Mechanisms Based on Finite-Level Continued Fraction Sequences Author: Zhang SuhangAddress: Luoyang, Henan --- Abstract Fractal geometry and continued fractions belong respectively to the two ma...
Lattice Structure and Matrix Representation in Discrete Order Geometry Author: Zhang Suhang, Luoyang, Henan
Discrete Order Geometry (DOG) takes finite discrete lattice points and order nesting as its ontology, withou...
--- Conclusion: Low-Energy Approximation in the DOG System Simultaneously Conforms to the MIE Extremum Criterion and the ECS Symmetric Equilibrium Steady State — A Triple Self-Consistent Closed Loop Author: Zhang Suhang, Luoyang, Henan --- I. ...
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From the largest to the smallest — from celestial bodies down to particles — all things are discrete. Order, or law, connects all things. ———SuHang Zhang Within DOG, this connection is precisely encoded as a three‑...
The Combinatorial Axiomatic Basis of Discrete Order Geometry (DOG)
— A Preliminary Exploration of a Combinatorial Generative Framework for Geometric Configurations
Classical geomet...
Discrete Order Geometry (DOG) and Topology: An Open Research Framework for Non-Connected Ordered Spaces
Author: Zhang Suhang Affiliation: Luoyang, Henan
Discrete Order Geometry (DOG) does not take spatial connectivity as ...
Riemannian Manifold Structures in Discrete Order Geometry (DOG) ——Continuity of Lattices and Discreteness Between Lattices
Author: Zhang Suhang
(Luoyang, Henan)
Discrete Order Geometry (DOG) takes discrete lattice poin...
Operator Algebra Structures in Discrete Order Geometry (DOG)
Discrete Order Geometry (DOG) takes finite discrete lattices, order nesting, and local correlation as its ontology. This pap...
--- DOG Discrete Order Geometry and Its Coupling with Discrete Mathematics Author: Zhang SuhangLuoyang Abstract Modern mathematics has long been divided into two fundamental branches: continuous mathematics and discrete mathematics. There exis...
Discrete Order Geometry (DOG): A Discrete Hierarchical Geometric Framework Based on Fractal Nesting and Continued-Fraction Scales Author: Zhang Suhang(Luoyang, Henan) Abstract Euclidean geometry and Riemannian geometry take continuous connec...
System-Wide Data Verification and Framework Application Summary
Author: Zhang Suhang (Independent Researcher, Luoyang)
Series No.: 16
This paper is the concluding installment of a systematic argument concerning the Rie...
Complete Exclusion of All Counterexample Paths at the Curvature Level Author: Zhang Suhang(Independent Researcher, Luoyang) System Affiliation: MOC–MIE–ECS–UCE Unified Mathematical-Physical ParadigmSeries Number: Fifteenth Paper --- Abstract ...
The Complete Closed Proof of the Riemann Hypothesis (UCE Coordinated Version) Author: Zhang Suhang(Independent Researcher, Luoyang) System Affiliation: MOC–MIE–ECS–UCE Unified Mathematical-Physical ParadigmSeries Number: Fourteenth Paper --- A...
Complete Self‑Consistency of Spatial, Evolutionary, and Constraint Triple Curvatures Author: Zhang Suhang(Independent Researcher, Luoyang) System Affiliation: MOC–MIE–ECS–UCE Unified Mathematical‑Physical ParadigmSeries Number: Thirteenth Paper ...
Direct Derivation of the Functional Equation from Curvature Dual Symmetry Author: Zhang Suhang(Independent Researcher, Luoyang) System Affiliation: MOC–MIE–ECS–UCE Unified Mathematical-Physical ParadigmSeries Number: Twelfth Paper --- Abstract...
Proof that the Critical Line is the Principal Axis of Global Curvature Equilibration
System Affiliation: MOC–MIE–ECS–UCE Unified Mathematical-Physical Paradigm Series Number: Eleventh ...