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.