On the Pure Geometric Origin of the Fine-Structure Constant via Spontaneous Topological Doubling — Version 4 (Extended & Axiomatized) —

Version 4 — Extended the residual ≈ 0.005 is interpreted as the QED radiative correction. New material in v4 (relative to v2): Formal axiomatization — Definition 1 of the Arc Vortex (γL, γM, χ); Axiom 1 of R=D weightlessness; Explicit five-fold doubling-symmetry enumeration S1,...,S5 underlying 𝒩SD = 25; Pythagorean closure 12 + (√15)2 = 42 fixing the inertial-plane radius; Precise figure of the arc-vortex geometric construction (Fig. 2), accompanying the v2 (p,q) mode-family figure (Fig. 1) which now explicitly identifies the (1,2) mode as the e-Arc-Helix γE — the topological body of the electron; Explicit treatment of the common "128π3" misconception; Three falsifiability predictions on running α, symmetry count, and dimensional translation; Full Discussion section linking the residual 0.005 to QED radiative corrections. Preserved from v2 (Supplementary Material): Geometric origins of c and h from the 24:(24−1) spacetime asymmetry: cgeo = 15/16, hgeo = 1/16, cgeo + hgeo ≡ 1; Unified geometric coupling identity αgeo = (3/23) h2geo β4; The Big-G entropy formula 𝒯G = Ggeo · ln(24|Tr(ρ̂1,2Ĉ)|), expressing gravity as a logarithmic-entropic projection of temporal residue overlap. A closure paragraph in §S.3 demonstrates that the three pillars of low-energy physics — α, (c, h), and G — are not independent inputs but coupled projections of one self-doubling topological vacuum. 第四版(扩展且公理化) 本记录为精细结构常数 α 之纯几何推导论文第四版,整合 v2 已发布版(DOI: 10.5281/zenodo.18991171)与 v3 工作稿。基于自发倍律(SDL)作用下真空之弧旋子(环面纽结 T(1,2))拓扑,导出: α−1geom = 27π/3 + 3 ≈ 137.041286 与 CODATA 实测值相对误差小于 4×10−5。残差 ≈ 0.005 被解释为 QED 辐射修正。 v4 相对 v2 新增:(i) 形式化公理(弧旋子定义 1 + R=D 失重公理 1);(ii) 五重倍律对称 S1–S5 显式枚举;(iii) 勾股闭合 12+(√15)2=42;(iv) 弧旋子精确几何构造插图(图 2)+ v2 之 (p,q) 模态家族图(图 1,明确 (1,2) 模态即电弧旋(e-Arc-Helix)γE,亦即电子之拓扑本体);(v) 对常见 "128π3" 误读的勘正;(vi) 三条可证伪性预言;(vii) 完整讨论节连结残差 0.005 与 QED 辐射修正。 保留 v2 补充材料:(viii) 由 24:(24−1) 时空非对称导出 c、h 之几何起源;(ix) α 之统一几何耦合恒等式;(x) 大 G 熵公式(Big-G formula)。