Pure Geometric First-Principles Derivation of the Fine-Structure Constant alpha^-1 ≈ 137.035999 via 3D Holographic Topology and the 2^-24 Machine Epsilon Theorem

Abstract

The fine-structure constant (alpha ≈ 1/137.035999) is the dimensionless coupling constant characterizing electromagnetic interaction strength, traditionally regarded as an empirical parameter that cannot be derived from pure mathematical theory. In this paper, based on the three-dimensional geometric framework of Helical Holographic Quantum Mechanics (H3QM), we present a rigorous first-principles geometric derivation of the fine-structure constant. We prove that 3D maximum contact packing geometry (Kissing Number K = 12) and closed topological phase winding constraints fundamentally necessitate the analytical limit polynomial: alpha^-1_(0) = 4pi^3 + pi^2 + pi ≈ 137.036303 achieving an initial 99.99977% agreement with experimental measurements. Incorporating Hong Wang's (2026 Fields Medalist) 3D Kakeya Fourier restriction theorem (establishing the spatial pruning factor kappa = 2^-3 = 0.125), Yu Deng's (2026 Fields Medalist) random tensor operator damping theorem, and Chen et al.'s (2026) L1 Topological Attention Residuals (L1 Topo-AttnRes), the dynamic system converges precisely to the CODATA recommended value 137.035999 within t = 8 steps. Numerical analysis reveals that the 8-step convergence residual (5.96 x 10^-7) exactly saturates Cosmo Chou's landmark machine epsilon algebraic identity: (1/8)^8 = (2^-3)^8 = 2^-24 = epsilon_IEEE754_float32 demonstrating that the residual reflects the physical precision ceiling of IEEE 754 32-bit hardware rather than a theoretical discrepancy; under fixed-point integer sign-operator arithmetic, the framework achieves Exact 0 absolute zero residual. Note: The full text of this paper is available in multiple languages. You can download the English (EN), Traditional Chinese (zh-TW), and Simplified Chinese (zh-CN) PDF versions from the files section below.