T₃拓扑弹性场论: 超欧拉旋量代数及其在刚体旋转与多维形变中的应用 T₃ Topological Elastic Field Theory: Hyper‑Euler Spinor Algebra and Its Applications in Rigid‑Body Rotation and Multidimensional Deformation

Abstract Since Hamilton proposed quaternions, with the 4π spinor periodicity and freedom from gimbal lock, quaternions have become the dominant mathematical tool for rigid‑body rotation and three‑dimensional attitude control. Nevertheless, conventional quaternions are purely algebraic constructs. They can only describe attitude transformation, and fail to capture internal scale deformation, topological shear, as well as path memory generated during successive multi‑fold rotations. Based on T₃ topological elastic field theory, this report introduces the novel T₃ hyper‑Euler spinor. Derivation proves that under the pure elliptical phase reduction condition, the T₃ spinor exactly reproduces the 4π periodic property of quaternions. In the general physical regime, the T₃ spinor upgrades classical rigid rotation into an integrated dynamical engine with three channels: scale evolution (α channel), hyperbolic deformation (β channel) and elliptical spin (γ channel). This paper systematically elaborates the mathematical structure of T₃ spinors, the non‑associative topological memory effect, and lists promising applications in aerospace, physics engines, intelligent materials and quantum computation. Keywords: T₃ algebra; hyper‑Euler spinor; quaternion; rigid‑body rotation; topological elasticity; Berry phase; flexible‑body dynamics 摘要 自哈密顿提出四元数以来,凭借4π旋量周期以及无万向节死锁的特性,四元数已经成为刚体旋转与三维姿态控制领域的主流工具。然而传统四元数属于纯代数构造,仅能够描述物体姿态的改变,无法刻画物质内部的尺度形变、拓扑剪切,以及多重旋转过程中的路径记忆效应。 本报告基于T₃拓扑弹性场论,提出全新的T₃超欧拉旋量。推导证明:当系统退化为纯椭圆相位条件时,T₃旋量可以严格复现四元数的4π周期性质;在一般物理状态下,T₃旋量将传统刚体旋转升维为一套统一动力学体系,同时包含尺度演化(α通道)、双曲形变(β通道)与椭圆自旋(γ通道)。本文系统阐述T₃旋量的数学结构、非结合拓扑记忆效应,并列举其在航空航天、物理引擎、智能材料与量子计算领域的应用前景。 关键词:T₃代数;超欧拉旋量;四元数;刚体旋转;拓扑弹性;贝里相位;柔性体动力学