Number-Theoretic Fractals of Natural Numbers: Cosmic Evolution Logic under the Dual Framework of Complete and Incomplete Systems
Year: 2026
Type: preprint
Abstract: Fractal theory serves as a core tool for characterizing the self-similar structures and nonlinear evolution of natural systems. Traditional research has long been confined to fractals in continuous geometric spaces, suffering from fundamental deficiencies including the lack of an original paradigm, vacant classification criteria for system completeness, and the inability to resolve the paradox of coexisting local steady-state cycles and global unidirectional evolution. Based on the discrete algebraic laws of divisibility operations over natural numbers, this paper innovatively proposes the concept of number-theoretic fractals. We rigorously prove that the sequence of natural numbers possesses four core properties: algebraic self-similarity, invariance of ordered structure under scaling transformation, extensibility via iterative nesting, and finite ontological reversibility. A discrete complete fractal system with fundamental origins is thus constructed, which differs from classical geometric fractals. Following the order isomorphism axiom and categorical mapping rules, this work establishes for the first time operable criteria for distinguishing complete and incomplete general self-similar systems. All self-similar systems are categorized into two types: physically approximate complete systems with finite reversibility and conserved ordered structures, and incomplete global evolutionary systems featuring unidirectional evolution and broken order isomorphism. Via topological isomorphic mapping of state spaces, we bridge the underlying logical connections between discrete number-theoretic structures and physical, geometric, biological and cosmic systems, and clarify the essential boundaries among three order metrics: topological order, thermodynamic entropy and Shannon information entropy. Relying on intrinsic number-theoretic functions independent of numeral systems, we construct a dynamical model for biological evolution with dynamic scaling, integrating two evolutionary modes: gradual linear evolution and probabilistic nonlinear transitions. Targeting inherent biases in paleontological fossil records, we establish a quantitative error model and a prediction framework for missing information, and put forward five statistically standardized falsifiable hypotheses for testing. Ultimately, this study constructs a nested cosmic framework of locally complete steady states and globally incomplete evolution, which unifies and interprets the core contradictory pairs in natural systems: conservation versus entropy increase, reincarnation versus evolution, and determinism versus emergence. It provides a novel fundamental mathematical paradigm for complex system modeling, cosmic evolution mechanisms and phylogenetic evolution of organisms. 分形理论是刻画自然系统自相似结构与非线性演化的核心工具,传统研究长期局限于连续空间几何分形,存在本原范式缺失、系统完备性分类空白、无法解释“局部稳态循环与全局单向演化共存”悖论等核心缺陷。本文以自然数整除运算的离散代数规律为基础,创新性提出数论分形概念,严格证明自然数序列具备代数自相似性、尺度变换序结构不变性、迭代嵌套延展性、本源有限可逆性四大核心特质,构建区别于经典几何分形的离散完备分形本原系统。 基于序同构公理与范畴论映射规则,本文首次建立广义自相似系统完备-不完备二元操作化判据,将所有自相似系统划分为有限可逆、序结构守恒的物理近似完备系统,与单向演化、序同构破缺的不完备全局演化系统。通过状态空间拓扑同构映射,打通离散数论结构与物理、几何、生命、宇宙系统的底层逻辑关联,厘清拓扑序结构、热力学熵、香农信息熵三类秩序度量的本质边界。 依托自然数无进制依赖的内禀数论函数,构建生物动态标度动力演化模型,融合渐进线性演化与概率非线性跃迁双模式;针对古生物化石记录的天然偏差,建立量化误差模型与缺失信息预测体系,提出五条符合统计学规范的可证伪假设检验。研究最终构建“局部完备稳态、全局不完备演化”的二元嵌套宇宙框架,统一解释自然系统守恒与熵增、轮回与进化、确定与涌现的核心对立矛盾,为复杂系统建模、宇宙演化机制、生物谱系演化提供全新的本原数理范式。
Author: lihong
Language: English
Topic: Complex Systems and Dynamics
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7164130035