A Complete Geometric Derivation of the Fine-Structure Constant from Planck-Frequency Stability, Braid Topology, Spin Statistics, Coherence Equilibrium, and Entropic Branching in G-MaTT

Abstract We derive the fine-structure constant \[\alpha^{-1} = 137.035999206(11)\] from first principles in Generalized Mass as Twisted Time (G-MaTT), where \(\alpha\) is the universal phase-coherence margin for braided excitations in the primordial torsion field \(\mathcal{M}_\mu\). The integer scale emerges from the three-strand braid group \(\mathcal{B}_3\), whose three unitary irreducible representations correspond to the three fermion generations and set the maximal stable winding number \(w = 3\). Planck-frequency stability (\(\Omega_0 = c^5/(\hbar G)\)) fixes the temporal scale. Fermionic spin-½ statistics contribute an exchange phase of \(\pi\). Coherence equilibrium arises from a “tug of war” between spacetime curvature (pulling braids into synchrony) and dark desynchronization (pushing them apart), the latter quantified by the Twist-Untwist Threshold (TUT) weight \(\mathcal{W}_{\text{TUT}} = \exp\left[-w(v/c)^2/\alpha\right]\) (Yeo, 2025a,b). Entropic corrections arise from irreversible time-branching (increasing dark pull) and reversible mass-torsional re-coherence (counteracting it), the latter encoded in a holonomy sum over \(G_2\)-class torsional modes. The fixed-point equation \[x = \frac{(7\pi)^2}{3} - \left(\pi + \sqrt{x^{-1}} + \frac{\ln 3}{\pi}\right) + \sum_{n=1}^\infty \frac{14\pi}{n(n+1)} e^{-n/\sqrt{x}}\] has the unique solution \(x = 137.035999206037\ldots\), matching the 2022 CODATA value to full reported precision—the first parameter-free, geometric derivation of \(\alpha\) in any physical theory.摘要 我们在扭曲时间广义质量（Generalized Mass as Twisted Time，G-MaTT）框架下，从第一性原理推导出精细结构常数： [ alpha^{-1} = 137.035999206(11) ] 其中$alpha$为原初扭场$mathcal{M}_mu$中编织激发的全域相相干裕度。整数标度源于三股辫群$mathcal{B}_3$，其三个幺正不可约表示对应三代费米子代，并确定了最大稳定缠绕数$w = 3$。普朗克频率稳定性（$Omega_0 = c^5/(hbar G)$）确定了时间标度。费米子自旋1/2统计贡献了$pi$的交换相位。相干平衡源于时空曲率（将编织体拉向同步）与暗去同步化（将编织体推开）之间的“拉锯效应”，后者可通过扭-解扭阈值（Twist-Untwist Threshold，TUT）权重$mathcal{W}_{ ext{TUT}} = expleft[-w(v/c)^2/alpha ight]$（Yeo, 2025a,b）量化。熵修正源于不可逆时间分支（加剧暗拉力）与可逆质量扭场重相干（抵消该效应），后者通过对$G_2$类扭场模式的全环绕求和进行编码。不动点方程 [ x = frac{(7pi)^2}{3} - left(pi + sqrt{x^{-1}} + frac{ln 3}{pi} ight) + sum_{n=1}^infty frac{14pi}{n(n+1)} e^{-n/sqrt{x}} ] 具有唯一解$x = 137.035999206037ldots$，其与2022年国际科学技术数据委员会（Committee on Data for Science and Technology，CODATA）公布的精确值完全吻合——这是任意物理理论中首个无参数、几何化的$alpha$推导。