Abstract This paper systematically investigates the real six-dimensional commutative non-associative algebra 𝒜₆, defined by the basis 1, e₁, e₂, t₁, t₂, t₃ with a quadratic form signature of (+, +, +, −, −, −). Evolving from the high-dimensional extension of the three-dimensional complex number system, this algebra belongs to the class of finite-dimensional real algebras that are commutative but non-associative. Through extensive random sampling, mid-way integration, field-theoretic mapping, cosmological fitting, and fractal lattice fusion simulations, the following core results are obtained: Dimensional Threshold for Non-Associativity: Non-associativity exhibits a dimensional threshold; 1‑2 dimensional subspaces degenerate into associative algebras, while non-associative effects emerge globally in mixed subspaces of dimension ≥3. Non-Degeneracy on Isotropic Elements: The algebra does not degenerate on isotropic elements, forming a critical distinction from the three-dimensional complex algebra 𝒞₃. Trivial Automorphism Group: The automorphism group contains only the identity element, indicating extreme structural rigidity. Algebraic Structure: Non-trivial idempotents exist, but no non-trivial nilpotent elements or proper ideals are present, identifying 𝒜₆ as a simple semi-simple algebra. Deviation Conservation: The deviation conservation axiom of mid-way integration holds to machine precision within this algebra, while iteration exhibits an intrinsic platform error inherent to the algebraic structure. Field Theory Mapping: When mapped to a φ⁴ effective field theory, finite negative entropy phenomena emerge in the low-temperature regime. Cosmological Compatibility: The equation of state parameter at the cosmological level shows a qualitatively compatible interval with DESI‑2024 dark energy observations. Fractal Simulation: Automatic switching between associative and non-associative modes driven by node complexity is achieved in a three-layer nested fractal simulation without a central node. Furthermore, this paper conducts a dual-layer algebraic-numerical stability assessment. Combining the intrinsic properties of the algebra, we discuss the adaptability of 𝒜₆ for next-generation distributed intelligent decision-making, fractal computing-in-memory, and precision control systems. Special emphasis is placed on analyzing its application prospects in high-level intelligent driving situational decision engines, while also outlining current open problems and future research directions. Keywords: Non-associative algebra; Commutative non-associative; Mid-way integration; Deviation conservation; Simple algebra; Fractal computing-in-memory; Intelligent driving; Dark energy 摘要 本文系统研究实六维交换非结合代数, 其基底为, 二次型签名 。该代数由三维复数体系高维扩展演化而来, 属于交换但非结合的有限维实代数。通过大批量随机采样、中道积分、场论映射、宇宙学拟合、分形格点融合仿真, 得到核心结果: (1) 非结合性存在维度阈值, 1‑2 维子空间退化为结合代数, ≥3 维混合子空间全域出现非结合效应; (2) 迷向元上代数不退化, 与三维复数代数 形成关键区分; (3) 自同构群仅含单位元, 具备极高结构刚性; (4) 存在非平凡幂等元, 不存在非平凡幂零元, 无真理想, 为单半单代数; (5) 中道积分的偏差守恒公理在本代数机器精度下成立, 迭代存在由代数本身带来的平台固有误差; (6) 映射为 有效场论, 低温区域出现有限负熵现象; (7) 宇宙学层面状态方程参数与 DESI‑2024 暗能量观测存在定性兼容区间; (8) 在三层嵌套分形无中心节点仿真中实现节点复杂度驱动的结合‑非结合模式自动切换。 本文进一步开展代数‑数值双层稳定性评估, 结合代数固有特性讨论该代数面向下一代分布式智能决策、存算‑一体分形计算、精密控制系统等工程方向的适配性, 重点分析其在高阶智能驾驶态势决策引擎上的应用前景, 同时列出当前尚未解决的开放问题与后续研究路线。 关键词: 非结合代数;交换非结合;中道积分;偏差守恒;单代数;分形存算一体;智能驾驶;暗能量
Zhongqiang Liu (Wed,) studied this question.