T₃ Topological Vortex Unified Field Theory T₃ 拓扑涡旋统一场论

Abstract This paper constructs a complete vortex-unification framework on the original T₃ algebra (e²=1, i²=−1, e·i=0, non-associative). It is the full merged edition of v4.0 (algebra-kinematics), v5.0 (topological geometry), v6.0/v6.1 (golden-ratio mass spectrum and S³ Clifford rings), v6.2 (fine-structure constant and nucleon-neutrino coupling), and v7.0 (quantum phase-space high-order corrections). While retaining all existing results, v7.1 rewrites the multi-vortex-state (trefoil-knot) chapter: it establishes the geometric mean-path constraint of the three-component native basis, locks the trefoil-knot baryon steady state at the pairwise-orthogonal √3C of the three components with a potential upper limit of 3C, and explicitly abandons the out-of-bounds 2√3C conception of the old version. Core result matrix: Algebra layer: defines the associator and non-associativity, derives the Cayley-Hamilton theorem and the symplectic-area norm identity. Kinematics layer: establishes the two-account topological transition, with the critical velocity V_c=c√2 as the two-dimensional topological threshold; the associator field locks the excess as rest mass. Dynamics layer: self-consistently derives the mass-energy equation E=mc² from the Newtonian kinetic-energy template, and establishes the unified inertia framework p=m_eff·v. Field-theory interface: strictly proves the helical form of electromagnetic field motion from Maxwell's equations via the Riemann-Silberstein vector, deriving the 1:1 equipartition of the two accounts. Topological geometry layer: leptons correspond to Möbius single vortices (two components), with half-angle twist explaining the 4π periodicity of spin 1/2; baryons correspond to trefoil-knot composite vortices (three components). Three-component geometric mean-path constraint (v7.1 core): the three native photon vortex filaments each have modulus c; the three-component resultant velocity satisfies |w|²=3c²+2c²(cosθ₁₂+cosθ₁₃+cosθ₂₃); when pairwise orthogonal it reaches the steady state √3C. Symmetry breaking (v7.1 core): the two-dimensional orthogonal √2C is the death boundary of the Möbius closure, while the three-dimensional orthogonal √3C is the life steady state of the trefoil knot. Golden-ratio layer: the mass spectrum is governed by M=M₀φ^Φ, with integer exponents for leptons and fractional exponents for baryons. S³ Clifford-ring layer: the geometric origin of the golden ratio φ is the Clifford ring on the S³ hypersphere. Fine-structure-constant layer: α=3²/(5³π²), matching CODATA 2022 to the order of 10⁻¹³. Quantum phase-space high-order correction layer: α⁻¹=4π³+π²+π, a multi-loop phase-space expansion. Four frontier applications: precise calculation of the QED anomalous magnetic moment, geometrization of hydrogen energy levels, the origin of the proton-electron mass ratio, and the topological projection of the Weinberg angle. Keywords: T₃ algebra; two-account topological transition; Möbius strip; trefoil knot; geometric mean-path constraint; three-component orthogonal steady state √3C; golden ratio; S³ Clifford ring; fine-structure constant; quantum phase-space correction; Weinberg angle 摘要 本文在原始T₃代数(e²=1,i²=−1,e·i=0,非结合)上建立完整的涡旋统一框架,是v4.0(代数-运动学)、v5.0(拓扑几何)、v6.0/v6.1(黄金比例质量谱与S³ Clifford环)、v6.2(精细结构常数与核子-中微子耦合)与v7.0(量子相空间高阶修正)的完全合并版。v7.1在保留全部既有成果的基础上,重写多涡旋态(三叶结)篇章:确立三分量原生基底几何中道约束,将三叶结重子稳态锁定于三分量两两正交的√3C,潜能上限为3C,并明确废弃旧版2√3C的越界构想。 核心成果矩阵: 1. 代数层:定义结合子与非结合性,导出Cayley-Hamilton定理、辛面积范数恒等式。 2. 运动学层:建立双账户拓扑跃迁,临界速度V_c=c√2作为二维体系拓扑阈值;结合子场将超出部分锁定为静质量。 3. 动力学层:从牛顿动能模板自洽推导质能方程E=mc²,建立统一惯量框架p=m_eff·v。 4. 场论接口:从麦克斯韦方程经Riemann-Silberstein矢量严格证明电磁场螺旋运动形式,导出双账户1:1均分。 5. 拓扑几何层:轻子对应莫比乌斯单涡旋(二分量),半角扭转解释自旋1/2的4π周期律;重子对应三叶结复合涡旋(三分量)。 6. 三分量几何中道约束(v7.1核心):三根原生光子涡旋丝模长均为c,三分量合速度满足|w|²=3c²+2c²(cosθ₁₂+cosθ₁₃+cosθ₂₃);两两正交时达到稳态√3C。 7. 对称性破缺(v7.1核心):二维正交√2C为莫比乌斯闭环的死亡边界,三维正交√3C为三叶结的生命稳态。 8. 黄金比例层:质量谱受M=M₀φ^Φ支配,轻子整数指数,重子分数指数。 9. S³ Clifford环层:黄金比例φ的几何起源为S³超球面上的Clifford环。 10. 精细结构常数层:α=3²/(5³π²),与CODATA 2022吻合到10⁻¹³量级。 11. 量子相空间高阶修正层:α⁻¹=4π³+π²+π,多圈相空间展开式。 12. 四大前沿应用:QED反常磁矩精确计算、氢原子能级几何化、质子-电子质量比起源、温伯格角拓扑投影。 关键词:T₃代数;双账户拓扑跃迁;莫比乌斯带;三叶结;几何中道约束;三分量正交稳态√3C;黄金比例;S³ Clifford环;精细结构常数;量子相空间修正;温伯格角