Paper 2-3: Periodic Micro-element Channel: Operator Implementation of Helical Curves and Helical Surfaces Author: Suhang Zhang (Heluo Mathematical School) Abstract Helical structures are among the most ubiquitous periodic forms in n...
Paper 2-2: Dual Rotation Structure: Π Transformation Rules for TorusAuthor: Suhang Zhang (Heluo Mathematical School)AbstractA torus is a 3D solid formed by rotating a circle (generatrix circle) of radius a around a coplanar non-intersecting axi...
Paper 2-1: Dual-Channel Transformation of the Π Operator for Cylindrical Bodies Author: Zhang Suhang (Heluo Mathematical School) Abstract A cylinder is the simplest non-spherical solid of revolution. Its formation intrinsically corr...
Paper E1-4: Euler's Identity as the Minimal Closure Instance of the Π Operator System Author: Suhang Zhang (Heluo Mathematical School) Abstract Renowned as the most elegant formula in mathematics, Euler's identity e^{i\pi}+1=0 unifi...
Paper 1-3: Architecture of Three Major Transformation Channels Based on Multiple Expressions of π
Author: Zhang Suhang (Heluo Mathematical School)
Abstract
The core innovation of the \mathcal{\Pi} operator lies in the fact that differe...
Papers 1-2: Curl-Preserving Axiom and System Self-Consistency Verification
Author: Zhang Suhang Founder of the Heluo Mathematical School
As the second paper in the Π-operator series, this work aims to establish and verify the ...
Paper 1-1: Core Definition and Symbol System of the \mathcal{\Pi} Operator
As one of the most vital constants in the history of mathematics, \pi has long been confi...
URFE Calculates the Hohmann Window
Author: Zhang Suhang
The following directly uses URFE → Newtonian gravity + Keplerian orbits to calculate, without extraneous detail, the launch time and orbital parameters for minimum fuel c...
URFE Calculates the Four Fundamental Forces
Based on the Unified Recursive Field Equation (URFE), this paper selects typical calculable scenarios, applies boundary conditions, performs quantitative simplif...
URFE Calculates Celestial Orbits
Combining the Unified Recursive Field Equation (URFE) with scenario-based calculations for different gravitational field strengths, and following the previously defined symb...
URFE Solves the Three-Body Problem Author: Zhang Suhang 1. First: What Makes the Classical "Three-Body Problem" Difficult Standard Conclusion (holds from 1880 to 2026): · No general elementary analytic solution: The positions cannot be express...
Derivation of the Four Fundamental Interactions from the Unified Recursive Field Equation (URFE)
Abstract: Based on the Unified Recursive Field Equation (URFE), parameter partitioning, group subspace projection, and b...
Complete Derivation of the Unified Recursive Field Equation (URFE) Author: Zhang Suhang
1. Core Prerequisite Theories
Foundation: MOC Multi-Origin Geometry, DOG Discrete Order Geometry, GPCL Geometric Recursive Conservation Chain, Unifie...
--- Integrating All Physical Elements and Factors into the Geometric Recursive Conservation Chain（UGSL）: A Unified Logical Framework
I. General Outline: The Universal Conservation Rule for All Physical Elements
Core ...
Markov Theory as a Strict Special Case of the UPGS Universal Axiom System
Author: Zhang Suhang (Founder of the He-Luo Mathematical School)
This paper independently, purely, and rigorously demonstrates the axiomatic hierarchy a...
--- Mapping–Number Theory Bottom-Layer Unified Explanation Based on Projection, Dynamic Slope, and MOC Curvature Geometry
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In traditional mathematics, mapping ...
--- MOC Coordinate System: Geometric Definition of Multi-Origin Hierarchical Nesting and Classical Coordinate Degeneration
Based on the Multi-Origin Curvature...
--- The Dimensional Mapping Generative Principle of Number Bases: A Unified Geometric Origin of Discrete, Linear, and Periodic Numeral Systems
Founder of the Heluo Mathematical School
This paper ad...
--- The End of the Third Mathematical Crisis: MOC Multi-Origin Curvature Geometry and the Restructuring of Mathematical Foundations
Author: Zhang Suhang (Bosley Zhang)
Three foundation...
A Universal Resolution Theory for Self-Reference Paradoxes Based on MOC Multi-Origin Curvature Geometry
Author: Zhang Suhang (Founder of the Heluo Mathematical School)
Self-reference paradoxes have stood as a core conundrum pla...
--- Resolution of the Barber Paradox within the MOC (Multi-Origin Curvature) Geometric Framework
The Barber paradox (a popular version of Russell's paradox) reveals ...
--- High-Dimensional Truth Gap: The Geometric Root of Gödel Incompleteness and Quantum Uncertainty
Author: Zhang Suhang, Luoyang, Henan
Gödel's incompleteness theorems reveal the existence of undecidable propositions in f...
Chapter 7： Conclusions and Outlook: Classical Logic as a Special Case of MOC Logic — Toward a New Hierarchical Logic System Author: Zhang Suhang, Luoyang, Henan Abstract This is the final chapter of the series Geometric Origin of th...
Chapter 6 ：Reconstruction of the Three Fundamental Laws of Logic: The Law of Identity, the Law of Contradiction and the Law of Excluded Middle as Products of Low-Dimensional Projection Author: Zhang Suhang, Luoyang, Henan Abstract This...
Chapter 5 The Emergence of Binary: Symbolization of the Law of Excluded Middle — Binary is the minimal symbol set of the law of excluded middle, and numeral systems are counting modes after projection Author: Zhang Suhang (Luoyang, Hena...
Chapter 4: Projection Mapping \boldsymbol{\Pi} and State Collapse Mechanism: \boldsymbol{\Pi: \mathcal{T}\to\{0,1\}}, Forced Binarization and Emergence of the Law of Excluded Middle Author: Zhang Suhang, Luoyang, Henan Abstract ...
Chapter 3: Three Non-Classical States of the High-Dimensional Truth Space \mathcal{T} Further elaboration: Definitions of truth-value gap, truth-value glut and indeterminate state Author: Zhang Suhang (Luoyang, Henan)
This pa...
Chapter 2: The Higher-Dimensional "OR" State: The Matrix of the Law of Excluded Middle Author: Zhang Suhang (Luoyang, Henan, China) Abstract This paper is the second installment in the research series Geometric Origin of the Law of Exc...
Chapter 1：Geometric Origin of the Law of Excluded Middle The Emergent Essence of Classical Logical Laws under the MOC Framework
Author: Zhang Suhang (Luoyang, Henan, China)
Chapter 1 Introduction: The Sanctity and Fractures of the L...
Tree Root-Shoot Isomorphism: A Natural Empirical Validation of Information Topology Author: Zhang Suhang --- Abstract This paper does not aim to investigate the growth patterns of plant organs. Rather, it selects the reversible transformation ...