The Weak Coupling Constant from Hopf Projection and Arithmetic Self-Energy: Hopf Projection Below the Dimensional Threshold

We derive the axial-vector coupling constant g_A of the weak interaction from first principles with zero free parameters, achieving g_A = 1.276165 vs. g_A(exp) = 1.2762 ± 0.0005 [PDG 2024], a deviation of 0.003% (0.07σ, i.e. 27 ppm). The only inputs are the prime number sequence, the number of spatial dimensions d=3, and the golden ratio φ = (1+√5)/2 (phase structure). The derivation proceeds in three levels. First, the architectural bifurcation theorem establishes that the weak channel P_W = {11, 13}, with N_W = 2.319 < d = 3, operates in the Hopf projection regime rather than the S³ mode-dilution regime used for electromagnetism and QCD. The tree-level coupling is g_A⁽⁰⁾ = |P_W|/Vol(Hopf) = 2/(π/2) = 4/π. A channel-specific self-energy from the local zeta function ζ11,13(-1) = -1/120 refines the denominator, giving g_A⁽¹⁾ = 2/(π/2 - C_W/N_W). Finally, a vertex correction of order (g_A·C_W)²/(6π) completes the derivation. The final formula is: g_A = 2/(π/2 - C_W/N_W) + (g_A·C_W)²/(2(|P_W|²-1)π) = 1.276165. The result extends the spectral correction programme of the companion papers (derivation of α to 0.06 ppb and α_s to 230 ppm) to the weak sector, using the same local zeta architecture restricted to the weak channel, but with a different geometric regime (Hopf vs. S³). A complete Python validation script reproducing all numerical results is provided in the appendix.