Complex‑Analysis and Geometric‑Structure of 3‑Dimensional Complex Numbers: From Algebraic Axioms to Conformal Field Theory 三维复数的复分析与几何结构 从代数公理到共形场论

This paper systematically establishes the frameworks of complex analysis, differential geometry and conformal field theory over the 3‑dimensional complex number . The 3‑dimensional complex number is a 3‑dimensional commutative associative algebra over the real number field with basis satisfying . It naturally induces the ‑dimensional Minkowski metric , and its non‑invertible region forms the zero‑divisor cone . Through purely algebraic derivation and systematic numerical verification, this paper proves the following core results: The 3‑dimensional Cauchy integral theorem holds exactly: inside any simply‑connected domain of 3‑dimensional complex numbers, the closed contour integral of a holomorphic function equals zero, and the deviation converges at the rate . The residue theorem recovers exactly on the subspace : under the standard complex‑plane embedding, 3‑dimensional complex numbers are fully compatible with classical complex analysis. The zero‑divisor cone produces a surface‑residue effect: the integral jumps when a contour crosses the zero‑divisor cone, and the jump amplitude is determined by the local curvature of the cone. A complete analytic proof of this effect is given, including the explicit computation of the Gaussian curvature of the cone. Conformal‑transformation rules are uniquely determined: holomorphic functions on 3‑dimensional complex numbers satisfy under Möbius transformations, with no extra weight parameters required. The zero‑divisor cone is invariant under Möbius transformations: the cone is mapped into itself under holomorphic functions of 3‑dimensional complex numbers, corresponding to the light‑cone structure of the 3‑dimensional conformal group . Topological triviality: both the first Chern class and the second Chern class of the 3‑dimensional complex manifold vanish, and it is topologically equivalent to . Core Proposition: The 3‑dimensional complex number is the natural generalization of standard complex analysis into 3‑dimensional space. Its non‑triviality lies at the algebraic and differential‑geometric levels rather than the topological level. 摘要 本文系统建立了三维复数 上的复分析、微分几何与共形场论框架。三维复数是实数域上的三维交换结合代数,基底 满足 ,天然诱导 维闵可夫斯基度规 ,其不可逆区域构成零因子锥面 。 本文通过纯代数推导与系统性数值验证证明以下核心结果: 三维柯西积分定理精确成立:在三维复数的任意单连通区域内,全纯函数的闭合围道积分为零,偏差以 速率收敛。 留数定理在 子空间精确还原:三维复数在标准复平面嵌入下与经典复分析完全兼容。 零因子锥面产生曲面留数效应:围道穿越零因子锥面时积分发生跳跃,跳跃幅值由锥面局部曲率决定。本文给出该效应的完整解析证明,包含锥面 Gauss 曲率的显式计算。 共形变换规则唯一确定:三维复数上的全纯函数在 Möbius 变换下满足 ,无需额外权重参数。 零因子锥面在 Möbius 变换下不变:锥面 在三维复数全纯函数下映射到自身,对应三维共形群 的光锥结构。 拓扑平凡性:三维复流形 的第一陈类与第二陈类均为零,其拓扑等价于 。 核心命题:三维复数是标准复分析在三维空间的自然推广,其非平凡性存在于代数与微分几何层面,而非拓扑层面。