T₃ Topological Elastic Field Theory: Ontological Identity of Velocity and Frequency T₃拓扑弹性场论: 速度与频率的本体同一性

Abstract For three centuries, translational velocity in classical mechanics and oscillatory frequency in quantum mechanics have been treated as two mutually independent fundamental quantities belonging to two separated scales. Classical mechanics describes the world through continuous trajectories and point-particle dynamics built on velocity; quantum mechanics describes the world through wave functions, superposition and interference built on frequency. Both frameworks are experimentally exact within their domains, yet their underlying axioms cannot be translated into one another: velocity carries the dimension , frequency carries the dimension , and standard physics provides no primitive quantity from which both descend. The de Broglie relation therefore enters quantum theory as an externally grafted postulate, and wave–particle duality retains an irreducible interpretive paradox. This paper, built on the T₃ non-associative topological elastic field theory and its super-Euler mapping, introduces the field element's intrinsic topological characteristic scale to achieve a global dimensional normalization, and establishes a single primitive axiom: the topological phase time evolution rate. On that basis we prove rigorously that macroscopic velocity and microscopic frequency are orthogonal projections of one and the same topological motion onto a spatial stretching channel and a phase curling channel respectively. Through the mid-path conservation constraint, the degree-of-freedom locking mechanism, the three-dimensional vector extension, the connector-field potential equation, the topological derivation of the uncertainty principle, the multi-element superposition rule and the continuous-field limit, we close the entire mathematical and dimensional loop that separates classical from quantum physics, and remove the historical dimensional inconsistency of every naive velocity–frequency unification. The de Broglie relation is thereby upgraded from an empirical assumption to a necessary algebraic identity of the theory; wave–particle duality, the uncertainty principle and the mass hierarchy of elementary particles all acquire a topological origin; and a dimensionally self-consistent, falsifiable topological foundation is laid for the ultimate unification of the gravitational constant and the Planck constant . Keywords: T₃ field theory; super-Euler mapping; topological phase evolution rate; velocity–frequency ontological identity; mid-path conservation; topological characteristic scale; connector locking; wave–particle duality; dimensional normalization; falsifiable predictions 摘要 三百年来,经典力学中的平移速度与量子力学中的振荡频率,一直被当作两个分属不同尺度的、相互独立的基础物理量。经典力学以速度为核心,构建连续轨迹与质点动力学;量子力学以频率为核心,构建波函数、叠加与干涉。两套体系在各自适用域内实验精度极高,但其底层公理彼此不可翻译:速度的量纲是 ,频率的量纲是 ,标准物理学不存在一个由之同时生成二者的原始量。因此德布罗意关系 是以外挂假设的方式进入量子理论的,而波粒二象性始终保留着一个无法消解的诠释悖论。 本文基于T₃非结合拓扑弹性场论及其超欧拉映射,引入场元固有拓扑特征尺度 完成全域量纲归一,确立唯一本源公理——拓扑相位时间演化率,并严格证明:宏观速度与微观频率,是同一个拓扑运动分别在"空间舒展通道"与"相位卷曲通道"上的正交投影。通过中道守恒约束、自由度锁止机制、三维矢量拓展、结合子场势能方程、不确定性原理的拓扑推导、多场元叠加规则与连续场极限,本文将分割经典物理与量子物理的全部数学与量纲闭环彻底合拢,并消除了以往一切"速度—频率朴素统一"所固有的量纲不自洽。 德布罗意关系由此由经验假设升格为体系的必然代数恒等式;波粒二象性、不确定性原理与基本粒子的质量谱获得了同一个拓扑起源;同时为引力常量 与普朗克常量 的最终统一,铺就了量纲自洽、可被证伪的拓扑地基。 关键词:T₃场论;超欧拉映射;拓扑相位演化率;速度—频率本体同一性;中道守恒;拓扑特征尺度;结合子锁止;波粒二象性;量纲归一;可证伪预言