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