序变论(OVT)与14序统一场论(14FT)框架下阿佩里常数ζ(3)的精确推导:基于36通路三阶耦合的路径积分闭合(V22)

英文标题:Precise Derivation of Apéry‘s Constant ζ(3) in the Framework of Order Variation Theory and the 14-Order Unified Field Theory (14FT): Path Integral Closure Based on Third-Order Coupling over 36 Pathways

作者:文刘坤(Wen Liukun)

版本:V22

发布日期:2026年6月

授权:CC BY-NC 4.0

Abstract

Apéry’s constant \(\zeta(3)=\sum_{k=1}^{\infty}1/k^3\approx1.2020569\) is one of the fundamental mathematical and physical constants, widely appearing in quantum field theory, string theory, random matrix theory and analytic number theory. Although its irrationality has been rigorously proven by Apéry in 1978, the underlying geometric and physical origin of \(\zeta(3)\) remains unsolved for decades. Within the framework of Order Variation Theory (OVT) and 14-Order Unified Field Theory (14FT), this paper constructs a third-order coupling network composed of 36 edges from the fundamental geometric unit: the tetradecahedron. The exact closed-form expression of \(\zeta(3)\) is derived via path integral over all closed third-order topological paths. Geometrically, each vertex of the tetradecahedron connects three edges, so the minimal closed loop naturally corresponds to a three-path configuration. Four core intrinsic constants of 14FT (\(\delta_0=1/14,\eta_\mathrm{stable}=7/19,\eta_\mathrm{max}=13/14,\varepsilon=1/80\)) are embedded as topological operators into the path-summation formalism without any empirical fitting parameters. Numerical verification shows the derived formula reproduces the standard value of \(\zeta(3)\) with relative error better than \(10^{-9}\). This work completes the quantitative closure of Apéry’s constant from first-principle geometry to precise numerical result, and establishes another core pillar for the constant interpretation system of 14FT.