Complexified 3D Numbers: An Algebraic Framework for the Riemann Hypothesis 复化三维数:研究黎曼猜想的代数框架

Abstract

Abstract 摘要 English:We introduce and systematically investigate the algebra of complexified 3D numbers , a six-dimensional real algebra (three-dimensional complex algebra) obtained by extending the base field of the non-associative 3D phase-geometric system from to . Through five interlocking experimental series (K1–K5), we establish: (1) the algebraic self-consistency of the complexified framework, including distributivity, commutativity, and conjugation compatibility; (2) a complete phase diagram of non-associativity over the complex -plane, revealing a left-half-plane stable region and a universal transition-band structure; (3) the topological structure of the isotropic cone, proving the non-existence of non-trivial isotropic subspaces and discovering an unexpected isotropic suppression effect on non-associativity (); (4) the complex-geometric stability of the critical line , showing that it is the unique dynamic balance line between time-like and space-like regions, with Hessian oscillations suggesting a deep connection to the GUE spacing of Riemann zeros; and (5) the closure of the non-associative residue gap in the complexified framework, proving that scalar contour integration is immune to non-associativity, thereby validating Cauchy's theorem and the residue theorem in . These results collectively suggest that complexification acts as a “non-associativity isolation layer,” preserving analytic tools while revealing new geometric constraints that converge on the critical line. 中文:本文引入并系统研究复化三维数代数,该代数为六维实代数(三维复代数),由非结合三维相位几何系统将基域从实数域扩张至复数域得到。通过五组相互关联的实验序列(K1–K5),本文得到如下结论:(1) 复化框架具备代数自洽性,满足分配律、交换律与共轭相容性;(2) 绘制复平面上非结合度完整相图,发现左半平面稳定区域与通用过渡带结构;(3) 刻画迷向锥拓扑结构,证明不存在非平凡迷向子空间,并观测到非结合效应存在约50倍的迷向抑制作用;(4) 分析临界线的复几何稳定性,证明其是类时区域与类空区域之间唯一动力学平衡线,海森矩阵振荡特征揭示其与黎曼零点GUE间距分布存在深层关联;(5) 复化框架下非结合留数缺口被完全闭合,标量围道积分不受非结合性干扰,柯西定理与留数定理在内均成立。综合结果表明,复化操作相当于一层“非结合隔离层”,在完整保留解析分析工具的同时,导出一系列收敛至临界线的全新几何约束。