T₃拓扑弹性场论: 考拉兹猜想的尺度伸缩与嵌套螺旋动力学分析报告 T₃ Topological Elastic Field Theory: Dynamical Analysis Report on Scale‑Stretching and Nested Spirals for the Collatz Conjecture
Year: 2026
Type: preprint
Abstract: Chapter 1 Introduction — The Dynamical Ghost behind Arithmetic Rules The Collatz conjecture is famous for its extremely simple rules (for odd numbers: ; for even numbers: ) yet extraordinary difficulty of proof. Billions of computational verifications show all positive integers eventually enter the cycle . Classical number theory meets dead‑ends by seeking global invariants from prime factorization and pure algebraic equations. Some researches realize orbit visualization by computer simulation and intuitively show nested spirals, sharp rise and fall trajectories. Nevertheless, visualization is merely post‑processing output without strictly matched algebraic operators, resulting in a split between observed geometric phenomena and mathematical equations: spiral‑inversion can be observed, yet intrinsic algebraic language describing such geometry is absent. From the T₃ perspective, Collatz evolution is far more than an algebraic game: it represents highly complex topological scale dynamics. Meanwhile, as a typical example of discrete limit‑cycle system, Collatz iteration can be used to verify the universal capability of T₃ algebra for describing various closed‑cycle motions. 第一章 引言——算术规则背后的动力学幽灵 考拉兹猜想以其规则极简(奇数乘 3 加 1,偶数除以 2)却极难证明而闻名。几十亿次的计算机验证表明,所有正整数最终都会落入 的循环。 传统数论之所以陷入困境,是因为它试图从“质因数分解”和“纯代数方程”中寻找全局不变量。部分研究通过计算机仿真实现了轨道的可视化,直观展现嵌套螺旋、冲高回落的运动图像,但可视化仅为后处理输出,不存在与之严格匹配的代数算子,造成现象图像与数学方程相互割裂:看得见螺旋反转,却缺少描述该几何行为的内在代数语言。 在 T₃ 理论看来,考拉兹序列的演化根本不是单纯代数游戏,而是一场高度复杂的拓扑尺度动力学运动。同时考拉兹迭代作为离散极限环典型范例,可以用来验证T₃代数对各类闭合循环运动的普适表达能力。
Author: Liu, Zhongqiang
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225461687