--- Part III: Subordination Relations, Nesting Rules, and Cross-Layer Mapping Systems for Hierarchical Sets Author: Zhang Suhang, Luoyang, Henan Abstract Building upon the establishment of the hierarchically nested set paradigm and the three-l...
--- Part II: Beyond the Flat Static Set View: Establishing a Hierarchically Nested Set System Author: Zhang Suhang, Luoyang, Henan Abstract Classical set theory, with its flat, single-layered, statically fixed conception of sets as its foundat...
Part One：Structural Limitations of Classical Set Theory and the Necessity of Expanding the Hierarchically Nested Set Paradigm Author: Zhang Suhang, Luoyang, Henan Abstract Since the emergence of naive set theory and the establishment of the ...
Renormalization Group Equations for Quantum Gravity Based on the Discrete Parent GroupAuthor: Suhang Zhang, Luoyang, HenanAbstractThe ultraviolet divergences in quantum gravity originate from the assumption of infinitely differentiable continuo...
Proof that Continuous Lie Groups Are Special Cases of Discrete Groups from a Recursive‑Geometric PerspectiveAuthor: Suhang Zhang(Luoyang, Henan, China)AbstractClassical group theory divides discrete groups and continuous Lie groups into two mutual...
Geometric Recursive Conservation Chain System: Reframing and Extending Classical Group Symmetry Theory
Author: Zhang Suhang, Luoyang, Henan
Abstract
Classical physics established a binding paradigm of continuous symmetry group – Noethe...
So Mighty is the Pull: Reflections on Mechanics
So mighty is the pull, so vast its spirit. For millennia, humanity's quest to understand the forces of the universe has carried this indomitable resolve.
Starting from the sages who came before...
Geometric Progressive Conservation Law and the Complementary Architecture of Noether Theorem — Dual Paradigm Unification of Geometric Origin and Symmetric Origin
Author: Zhang Suhang Affiliation: Luoyang, Henan
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In class...
--- Geometric Progressive Conservation Chain in Curved Spacetime: Spacetime Curvature and Conservation Transmission Mechanism
Classical general relativity establishes a...
Lorentz Covariant Extension of the Geometric Progressive Conservation Law: Unification of Classical and Relativistic Conservation Author: Zhang SuhangAffiliation: Luoyang, Henan --- Abstract The Geometric Progressive Conservation Law (GPCL)...
--- Theoretical Support of Poisson Mechanics for the Geometric Progressive Conservation Law (GPCL)
Author: Zhang Suhang
Affiliation: Luoyang, Henan
Based on the core geometric axiom that "linear motion is equivalent to circul...
--- Complete Proof of the Gilbert-Pollak Conjecture Based on the MOC-DOG-MIE-ECS Native Framework Author: Zhang Suhang, Luoyang, Henan --- Abstract The Gilbert-Pollak conjecture is a core classical problem in planar extremal geometry. Traditi...
--- Transcendental Continued Fraction Embedding and Multi-Origin Completion: The Grand Unified Framework of FCFG Fractal-Continued-Fraction Geometry Author: Zhang Suhang (Bosley Zhang) System Affiliation: FCFG Fractal-Continued-Fraction Geometr...
Dynamical Substrate of FCFG: Continued-Fraction Expansions of Complex Exponentials and Spin-Fractal Recursive IsomorphismAuthor: Zhang SuhangCore Axiomatic Support: FCFG Recursive Isomorphism Axiom (r_n = S_n); Multi-Origin Curvature (MOC) Framewo...
On the Central Position of Geometry in Mathematics
For a long time, many people merely regard mathematics as a tool for calculation and problem-solving, which greatly narrows its true meaning. Mathematics is essentially an organic and vibran...
--- Part Two A Rigorous Global Proof of the Kakeya Conjecture via DOG-MOC Author: Zhang Suhang (Bosley Zhang) Location: Luoyang, Henan, China Date: May 2026 Abstract This paper, based on the self-developed Discrete Order Geometry (DOG) cov...
--- Part One Title: The Geometric Foundation and Model Transformation of the Kakeya Conjecture via DOG-MOC
Author: Zhang Suhang (Bosley Zhang) Location: Luoyang, Henan, China Date: May 2026
Under the framework of Multi-Origin...
Frequency, Probability and Geometric Measure: A Trinity Author: Zhang SuhangAffiliation: Luoyang, HenanAbstractThis paper establishes the connection between Discrete Order Geometry and flat schemes, thereby bridging the logical and mathematica...
Paper 12: Étale Differential Equations on UPGS — Continuum Limit of Random Walks and Heat Kernel, with Derivation of the Schrödinger Equation Author: Zhang SuhangAffiliation: Luoyang, HenanDate: May 25, 2026 --- Abstract In Papers 7 and 8, ...
Paper 11: Computation on UPGS — Algorithms, Complexity, and Numerical Implementation
Author: Zhang Suhang Affiliation: Luoyang, Henan Date: May 25, 2026
UPGS (Étale Probability Schemes), as a unified theoretical frame...
Paper 10: The Weil Conjectures under UPGS — Étale Random Walks, Curvature Duality, and ECS Stability
Within the UPGS (Étale Probability Schemes) fram...
Paper 9: The Riemann–Roch Theorem under UPGS — A Geometric Proof Based on the Étale Heat Kernel, MIE Variational Principle, and ECS Conservation Laws
...
UPG and the Connection to ∞-Toposes: Construction of Weighted ∞-Toposes (UPGT) Author: Suhang ZhangAffiliation: Luoyang, Henan --- Abstract This paper establishes a connection between the Probability–Geometry Unified Framework (UPG) and Lurie'...
Applications of the UPG Framework: Central Limit Theorem, Statistical Inference, Machine Learning and Quantum Physics Author: Zhang Suhang Affiliation: Luoyang, Henan Abstract This paper serves as an application article for the Unifie...
Paper 6: Dynamic Midpoint Extremum Theorem: The Dynamic Tripartite Unification of Probability–Geometry–Algebra (UPG) Author: Zhang Suhang Affiliation: Luoyang, Henan Abstract This paper establishes the Dynamic Midpoint Extremum Theore...
For the longest time, I honestly thought all volleyball nets were basically the same. As long as the ball could go over and the poles stayed upright, I figured that was enough. A few months ago, my friends and I started playing more regularly on ...
Paper 5: Stochastic Processes and Geometric Flows: From Random Walks to Brownian Motion and Quantum Probability Author: Zhang Suhang Affiliation: Luoyang, Henan Abstract This paper extends the tripartite unifying framework of probabil...
Paper 4: Geometric Reconstruction of Probability Axiom System: Kolmogorov Axioms Equivalent to Geometric Measure Axioms
This paper accomplishes a landmark unification between pr...
Paper 3：Geometric Embedding of Multivariate Random Variables: A Unified Triple Operation of Joint Distribution, Marginalization, and Conditioning Author: Zhang Suhang Affiliation: Luoyang, Henan, China Abstract Building upon the algebraic-prob...
Paper 2: Geometric Realizations of One-Dimensional Probability Distributions: Bell Curve, Staircase, Lattice, and Fractal
Continuing the probabilistic-geometric isomorphism fr...