HopfionSeries Paper IV - The Hopf Spoke, WZW Fermion Mass Renormalization, and Three- and Four-Term Formulas for the Fine Structure Constant

Seven interconnected results extend the density-feedback Hopfion framework and close the residual gaps in the fine structure constant and electron mass formulas left by Paper III. The single experimental input is TCMB = 2.7255 K; all Standard-Model outputs follow from condensate geometry and φ = (1+√5)/2 alone. Compton-wavelength identification (exact) Under the Witten bosonization dictionary of Paper III, the Stieltjes resolvent parameter β* equals exactly the squared inverse mass of the WZW condensate fermion: β* = 1/mWZW2. This is a proved consequence of the resolvent interpretation, not an assumption. One-loop WZW mass renormalization (exact) The condensate background renormalizes the WZW fermion mass by the exact factor mWZW2/m02 = V*/V(0) = 8φ/15, where V* = φ and V(0) = 15/8 are the IR and UV virial values proved in Paper III. A three-term α−1 formula (unconditional) The QED running from mWZW to me contributes δQED = 1/(9φ6), giving α−1 = 360/φ2 − 3/(2π) + 1/(9φ6) = 137.036 49, with a residual of only 0.0004% from the measured value 137.035 999 177. The formula is unconditional: mWZW/me is derived from condensate geometry alone via the modular T-matrix correction T1/2/Q = 3/400 of the j = 1/2 spoke carrier. A four-term α−1 formula (unconditional) The Chern–Simons background of the condensate reduces the effective QED vacuum polarisation by a factor 1/(4π) per unit WZW level, adding a fourth term −1/(36πφ6) = −T3/(4π) to the series. This gives α−1 = 360/φ2 − 3/(2π) + 1/(9φ6) − 1/(36πφ6) = 137.035 998 486, with a residual of 5×10−7% from the measured value 137.035 999 177 — a factor-of-700 improvement over three terms. The series converges geometrically with ratio T4/T3 = 1/(4π) ≈ 0.080. Electron mass: pole-mass scheme correction (new) The spoke formula computes me in the MS-bar renormalisation scheme native to WZW bosonization, while the measured 0.511 MeV is the pole mass. The one-loop QED conversion mpole = mMS-bar (1 + α/π) closes the previous 0.22% residual, giving the corrected formula me = Λcond φ20 exp(4π − 3/400 − α/π) with a residual of 0.013%. The same α that appears in the four-term formula corrects its own mass: as α is refined experimentally, me tracks via δme/me = −δα/(πα), an amplification of 43.6×. This is a zero-parameter consistency condition among TCMB, α, and me with no analogue in the Standard Model. Hopf-spoke formula for vEW (corrected) The electroweak scale is expressed entirely in terms of TCMB alone — no Standard Model parameters: vEW = Λcond φ20 exp(49π/6 − 3/400) ≈ 247.9 GeV, accurate to 0.69%. The exponent 4π − 3/400 corrects the Abelian Chern–Simons result 4π by the modular T-matrix phase T1/2/Q = 3/400 of the j = 1/2 spoke carrier traversing the Q = 10 topological sectors. The α/π pole-mass correction does not apply to vEW: the scheme conversion in me and the Yukawa coupling ye cancel in the ratio vEW = me/ye. The golden-angle cascade and running of α Two of the four-term formula's terms have distinct origins that should not be conflated. The leading term, the golden angle 360/φ2 = 137.508, is reached from the geometric UV value 150.3 not by running but as Δ1, the exact computable difference between two independently-exact quantities established in Paper III (its deeper origin is Paper III's Open Problem 4). The subleading term −k/(2π) is a genuine one-loop WZW anti-screening effect: the SU(2)3 condensate is non-abelian and, in analogy with asymptotic freedom, virtual flux-knot pairs enhance the coupling at lower energies, moving α−1 from 137.508 down toward 137.036. Above me, standard QED and electroweak running takes over, reaching α−1 ≈ 128.9 at the Z-boson scale in agreement with measurement. The three-correction structure of the α−1 formula — dominant non-abelian WZW anti-screening, subleading one-loop QED screening with weight 1/(9φ6), and the Chern–Simons correction −1/(36πφ6) reducing the QED loop by the condensate's magnetic flux — recurs in Paper V for the CP-violating phase δCP, where the same condensate phase quantum Q/(k h(E8)) = 1/9 corrects a dominant WZW topological spin sum to 65.0°, within 0.12σ of the PDG value.