The Midway Residue Theorem: An Area Law Integral Theory on Non-Associative Algebras 中道留数定理: 非结合代数上的面积律积分理论

Abstract

Abstract This paper establishes the closed path integral theory on the non-associative algebra ³—the Midway Residue Theorem. The core finding is that the topological invariant (residue) of traditional complex analysis is not transferable to non-associative algebras. Instead, it is replaced by an area law integral, where the integral value is proportional to the area enclosed by the path rather than the topological winding number. Through systematic research on n=3, 4, 5, 6, it is found that the closed integral exhibits an “asymptotic series with increasing powers” structure: ∮ZdZ = 2πR²·P + 2πR·Q + 2πR·S + O(R). The I₁ component acts as the “vanguard of higher-order corrections,” consistently being the first to exhibit new R correction terms. The constant 11 is proven to be an invariant combination of the multiplication table structure constants of ³, specifically 11 = -(2α+β). The complete unified formula for the A coefficient of n=5 I₂ is confirmed as: A = (3π/64)·z₁²·z₂·[4z₁²+4z₁+11]·α·(α-β). 摘要 本文建立非结合代数 ³ 上的闭合路径积分理论——中道留数定理。核心发现:传统复分析的拓扑不变量(留数)在非结合代数中不可移植,取而代之的是面积律积分——积分值正比于路径包围的面积而非拓扑绕数。通过对 n=3,4,5,6 的系统研究,发现闭合积分呈现”幂次递增的渐近级数”结构:∮ ZdZ = 2πR²·P + 2πR·Q + 2πR·S + O(R)。I₁ 分量是”高阶修正先锋”,总是最先出现新的 R 修正项。常数 11 被证明是 ³ 乘法表结构常数的不变量组合 11 = -(2α+β)。n=5 I₂ 的 A 系数完整统一公式确认:A = (3π/64)·z₁²·z₂·[4z₁²+4z₁+11]·α·(α-β)。