Supplemental Paper S‑03: Global Generalization of the Π Operator within the Framework of DOG Discrete Order Geometry Author: Suhang Zhang, Heluo School of Mathematics Abstract The conventional Π operator system is constructed upon t...
Paper 11: Computation on UPGS — Algorithms, Complexity, and Numerical Implementation
Author: Zhang Suhang Affiliation: Luoyang, Henan Date: May 25, 2026
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Abstract
UPGS (Étale Probability Schemes), as a unified theoretical frame...
Origins: Sun O_\odot, Earth O_\oplus, Mars , Sun–Mars Lagrange points O_{L1/L2}, …- Lattice points: Earth-orbiting satellites, Mars orbiters, relay satellites, deep-space probes- Curvature coupling coefficient \Omega_i:For each link, accumulate th...
Consciousness State Database and Consciousness Encoding Dictionary 1. Theoretical Necessity of the Consciousness Database Within the framework of the Elementary‑Wave Combinatorial Consciousness Theory, any conscious state is uniquely determined ...
Discrete Order Geometry (DOG) and Knot Topology: From Continued Fraction Recursion to Interdisciplinary Topological Applications
Author: Zhang Suhang Address: Luoyang, Henan
Knot topology has long been regarded as a branc...
--- 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...
Interstellar Corridor (Part I): A Foundational Theory of Low-Energy Interstellar Travel Based on Natural Spacetime Curvature Gradients Author: Zhang Suhang, Luoyang Abstract Current interstellar travel theories face core dilemmas: extremely l...
--- The Dimensional Mapping Generative Principle of Number Bases: A Unified Geometric Origin of Discrete, Linear, and Periodic Numeral Systems
Author: Zhang Suhang
Founder of the Heluo Mathematical School
This paper ad...
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...
Frequency as the Origin of Probability: From Discrete Order Geometry (DOG) to a Quantitative Derivation Where Frequency Difference Determines Probability
Author: Zhang Suhang, Luoyang, Henan
The origin of probability is...
Under the MOC paradigm, the complete analytical equations for the gravitational three-body problem of the Earth, Moon, and Sun. Author: Zhang Suhang (followed by the detailed sections I–IV as in your Chinese text) …Is this acceptable? My res...
Weak Convergence and Ergodic Limit of Stochastic ECS Systems Author: Zhang Suhang Luoyang, Independent Researcher Abstract This paper extends the Extremum-Conservation-Symmetry (ECS) framework to discrete stochastic linear dynamical s...
God makes all things with the greatest economy and efficiency. That is the essence of MlE. MlE provides mathematical support for Darwin's theory of evolution. Redundant structures that serve no purpose and consume resources t...
Paper 5-1: Applications of the Π Operator in Engineering: 3D Modelling, Rotary Mechanisms and Periodic Oscillations Author: Suhang Zhang (Heluo School of Mathematics) Abstract The Π operator framework establishes a unified mathematical...
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 7: Flat Probabilistic Schemes — Migration from Geometric Measure Spaces to Étale Sites
Author: Suhang Zhang, Luoyang, Henan Abstract
Based on the axiomatic system of probability geometry established in previous papers (Paper 1 & P...
Discrete Order Geometry (DOG) and Matrix Algebra: Essential Correspondence and Paradigm Reconstruction （Part One）
Author: Zhang Suhang Affiliation: Luoyang
This paper systematically reveals the intrinsic isomorphic relationship...
MOC Paradigm: Complete Calculation of Earth-Sun-Moon Three-Body Gravitation Author: Zhang Suhang
Constants → Classical Model → MOC Multi-Origin Correction → Full Numerical Precision
I. Fundamental Physical Constants (Astronomi...
Algebraic Isomorphism Between Zhang Group and Continuous Lie Groups Under the Triple Limit Author: Suhang Zhang (Luoyang, Henan, China) Abstract Within the frameworks of Discrete Order Geometry (DOG) and Multi-Origin Recursive Geometry...
Arithmetic Field Steady‑State Solution Determination Rules Author: Zhang Suhang(Independent Researcher, Luoyang) System Affiliation: MOC–MIE–ECS–UCE Unified Mathematical‑Physical ParadigmSeries Number: Eighth Paper --- Abstract This paper is ...
Convergence of the Solution to the Discrete Riccati Equation with Respect to Sampling Period Author: Suhang Zhang Luoyang Abstract
This paper rigorously proves that as the sampling period h\to 0^+, the solution of the disc...
The Significance of Game Theory in Multi-Origin High-Dimensional GeometryIn traditional game theory, a game refers to strategic choices made by rational agents under fixed rules, fixed payoffs, and a single global utility framework. It presupposes...
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...
DOG Discrete Order Geometry: A Minimal Mathematical Paradigm for Unifying the Four Fundamental Interactions
Author: Zhang Suhang, Luoyang
I. Fundamental Unified Substrate
Let the set of all discrete spacetime lattice points in t...
The Combinatorial Axiom Foundation of Discrete Order Geometry (DOG) – Construction of a High-Order Arrangement and Combination System for Geometric Configurations
Author: Zhang Suhang (Luoyang, Henan)
Classical geometry an...
A Minimalist Explanation of Celestial Rotation, Revolution, and Axial Obliquity under the Pure MOC Framework Author: Zhang Suhang (Luoyang, Henan)Affiliation: Independent Researcher Abstract: Relying solely on the core paradigm of Multi-Origin C...
One of the most important areas to focus on when bringing a classic car back to life or maintaining one is its braking performance. Plymouth Valiant Master Cylinder is one of essential part in all classic braking systems. For most, a crucial littl...
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 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
Affiliation: Luoyang, Henan
Continuing the probabilistic-geometric isomorphism fr...