Dimensional Scaling Study: Non-Associative Algebras 1D–9D 非结合代数 1D–9D 维度标度系统性研究报告

一、总览与动机 Why sweep 1D to 9D systematically? Because isolated studies cannot reveal scaling laws. 为什么要系统扫描1D到9D? 因为零散研究无法揭示维度标度律。 We construct a unified family of commutative, non-associative algebras with signature (p, q) where p=ceil ( (d+1) /2) positive and q=floor ( (d-1) /2) negative basis elements. 我们构造统一的交换非结合代数族, 签名 (p, q), 其中p=正类基底数, q=负类基底数。 Multiplication rules follow a consistent pattern across all dimensions: 1 is identity, e*e→t (positive×positive→negative), e*t→e (positive×negative→positive), t*t→1 or e (negative×negative→positive). 乘法规则跨维度保持一致: 1是单位元, e*e→t, e*t→e, t*t→1或e。 This unified construction allows us to observe how non-associativity strength, mid-path integral deviation, and isotropic behavior scale with dimension. 这种统一构造让我们能观察非结合性强度、中道积分偏差和迷向行为如何随维度变化。 §2 Unified Algebraic Construction 二、统一代数构造 Basis naming convention: 1, e1. . eE, t1. . tT, s1. . sS 基底命名约定: 1, e1. . eE, t1. . tT, s1. . sS Sign pattern: e-class positive (+1), t-class negative (−1), s-class negative (−1). The number of negative bases grows with dimension. 符号模式: e类为正 (+1), t类为负 (−1), s类为负 (−1) 。负类基底数随维度增长。 Dim Basis Signs Signature (p, q) Notes 1D 1 1 (1, 0) Trivial associative 2D 1, e1 1, 1 (2, 0) Still associative 3D 1, e1, t1 1, 1, -1 (2, 1) First non-associative 4D 1, e1, e2, t1 1, 1, 1, -1 (3, 1) e+t extension 5D 1, e1, e2, t1, t2 1, 1, 1, -1, -1 (3, 2) e+t extension 6D 1, e1, e2, t1, t2, t3 1, 1, 1, -1, -1, -1 (3, 3) e+t extension 7D 1, e1, e2, e3, t1, t2, s1 1, 1, 1, 1, -1, -1, -1 (4, 3) e+t+s extension 8D 1, e1, e2, e3, t1, t2, t3, s1 1, 1, 1, 1, -1, -1, -1, -1 (4, 4) e+t+s extension 9D 1, e1, e2, e3, t1, t2, t3, s1, s2 1, 1, 1, 1, -1, -1, -1, -1, -1 (4, 5) e+t+s extension