T₃拓扑信号处理: 超欧拉映射与中道约束的代数同构推导报告 T₃ Topological Signal Processing Algebraic Isomorphism Derivation Report of Hyper-Euler Mapping and Middle-Way Constraint

摘要 经典信号处理通过复指数 的旋转投影,提取不随时间变化的频谱不变量。本报告基于 T₃ 三维数代数()及其超欧拉映射,提出一套全新的T₃拓扑信号处理框架。 本报告证明:经典复指数变换仅仅是 T₃ 超欧拉映射在双曲分量为零()时的退化特例。T₃ 体系将时间维度彻底涡旋化,将其分解为标量振幅、双曲形变与椭圆旋转三重正交自由度。通过定义 T₃ 空间的共轭内积与正交投影,推导出物理场的高维频谱展开式。最终严格证明:在这套高维映射下,系统的中道约束(拓扑能量-动量稳态)正是该变换的绝对不变量,对应着宇宙信号的帕塞瓦尔守恒。 Abstract Classical signal processing extracts time-invariant spectral invariants through the rotational projection of the complex exponential . Based on the T₃ three-dimensional number algebra () and its hyper-Euler mapping, this report proposes a brand-new T₃ topological signal processing framework. This report proves that the classical complex exponential transform is only a degenerate special case of the T₃ hyper-Euler mapping when the hyperbolic component is zero (). The T₃ system completely vortices the time dimension and decomposes it into three orthogonal degrees of freedom: scalar amplitude, hyperbolic deformation, and elliptical rotation. By defining the conjugate inner product and orthogonal projection in the T₃ space, the high-dimensional spectral expansion of physical fields is derived. Finally, it is strictly proved that under this high-dimensional mapping, the middle-way constraint (topological energy-momentum steady state) of the system is the absolute invariant of the transform, corresponding to the Parseval conservation of cosmic signals.