Multi-Scale Self-Referential Field Equations: Renormalization Group Flow of High-Dimensional Modes onto Low-Dimensional Effective Constants —— Scale Dependence and Topological Origin of the Fine-Structure Constant in Yuanxian Theory

Yuanxian Theory establishes the ontological topology of the cosmos as a sixty-four-dimensional compact torus (T64), whose global dynamics are governed by the self-referential mind field (PsiSR). Four-dimensional spacetime and its associated Standard Model physics emerge as an effective low-energy projection of PsiSR. However, non-linear couplings across mode scales (indexed by winding numbers n belonging to Z64) induce a scale-dependent "running" of low-dimensional effective constants—such as the fine-structure constant alpha—with respect to the observation scale. Crucially, this running does not originate from vacuum fluctuations as in conventional quantum field theory, but stems from the statistical contribution of high-dimensional modes to the low-dimensional projection. This paper establishes the mathematical architecture of the multi-scale self-referential field equations, linking the modal decomposition of PsiSR on T64 to a systemic renormalization group flow. First, we derive the hierarchical system of mode-coupling equations, demonstrating how high-winding modes renormalize low-energy effective actions via "modal dilution" and "topological polarization" mechanisms. Second, we compute the explicit running equation for alpha (mu) and prove that its fixed point, alpha = 1/137. 036, corresponds to the globally optimal geometric ratio of the T64 manifold, where any local deviation triggers an inherent topological restoring force driving alpha back to the attractor point. Finally, we provide verifiable empirical predictions: the infinitesimal drift of alpha in high-energy collisions and its cosmological spatial fluctuations must satisfy a specific correlation spectrum uniquely dictated by the dimensionality of T64. This work builds a rigorous dynamical bridge linking microscopic high-dimensional topology to macroscopic observational constants, while effectively resolving the long-standing Landau pole problem in Quantum Electrodynamics (QED). 元宪理论将宇宙的本体拓扑建为六十四维紧致环面 (T64), 全域动力学由自指心场 (PsiSR) 描述。四维时空及其中的标准模型物理被视为 PsiSR 在低能投影下的有效理论。然而, 不同尺度的模式 (由属于 Z64 的环绕数 n 标记) 之间存在非线性耦合, 导致低维有效常数 (如精细结构常数 alpha) 随着探测能标的变化而发生“流动”——这一流动并非由传统量子场论的真空涨落圈图产生, 而是源于高维模对低维投影的统计贡献。 本文旨在建立一套多尺度自指场方程的数学架构, 将 PsiSR 在 T64 上的模态分解与尺度重正化群流联系起来。首先, 我们导出模式耦合的层级方程组, 证明高环绕数模式通过“模式稀释”效应和“拓扑极化”机制影响低能有效作用量。其次, 计算精细结构常数 alpha (mu) 的流动方程, 并证明其固定点 alpha = 1/137. 036 对应于 T64 几何比值的全局最优值——任何局部偏离都会产生一种固有的拓扑恢复力, 迫使 alpha 回归该吸引子点。最后, 给出可检验的经验预言: 高能对撞中 alpha 的微小漂移与宇宙学观测中精细结构常数的空间涨落之间, 应满足由 T64 维数决定的特定关联谱。本文为元宪理论提供了一个联系微观高维拓扑与宏观观测常数的严格动力学桥梁, 同时也为彻底解决量子电动力学 (QED) 中长期存在的朗道极点问题提供了新路径。