T₃ Algebra and Topological Bubble Dynamics: Physical Interpretation of the Exponential Map Special Report T₃代数与拓扑气泡动力学: 指数映射的物理诠释专题报告

Abstract

Abstract Building upon the T₃ three-dimensional number algebra, this report proposes the complete decomposition form of the T₃ unified exponential map and applies it to the dynamical description of bubbles (topological thin films). By strictly decomposing the exponential map into a scalar amplitude term, a hyperbolic radial term, and an elliptic angular term, this report demonstrates that T₃ algebra naturally unifies the three geometric degrees of freedom: “amplitude, dilation, and rotation.” Furthermore, this report maps this mathematical structure onto the physical lifecycle of a bubble—encompassing nucleation, expansion, surface vibration, and rupture phase transition. It provides the T₃ algebraic expressions for bubble states and reveals that bubble rupture is essentially the physical process of topological self-locking collapse. In version 1.1, this report supplements the three-component coupled dynamical evolution equation system, forming a complete framework of “algebraic structure + temporal evolution” in conjunction with the exponential map. 摘要 本报告在T₃三维数代数基础上,提出T₃统一指数映射的完整分解形式,并将其应用于气泡(拓扑薄膜)的动力学描述。通过将指数映射严格拆分为标量振幅项、双曲径向项与椭圆角向项,本报告证明:T₃代数天然统一了”振幅、伸缩、旋转”三重几何自由度。 本报告进一步将这一数学结构映射到气泡的物理生命周期——从生成、膨胀、表面振动到破裂相变,给出气泡状态的T₃代数表达式,并揭示气泡破裂本质上是拓扑自锁崩溃的物理过程。在v1.1版中,本报告增补了三分量耦合动力学演化方程组,与指数映射形成”代数结构+时间演化”的完整框架。