HopfionSeries Paper III - Two Constants from One Knot: The Fine Structure Constant and the Linking Scale as Exact Predictions of the Icosahedral WZW Condensate

Two problems left open by Papers I and II are identified and closed: the analytic proof of the Linking Scale Conjecture (LSC) for the density-feedback Hopfion, and the derivation of the fine structure constant α from the same framework. Both follow from the identification of the density-feedback model as the SU(2)3 WZW model of three free Dirac fermions (Witten non-abelian bosonization). Golden-angle cascade The fine structure constant arises via a two-step cascade of different character: φ6/sin4(π/5) ≈ 150.3 and 360/φ2 ≈ 137.508 (the golden angle) are both independently exact; their gap is a computed difference Δ1, not a WZW running effect (its physical origin is an open problem). The second step, α−1 ≈ 137.036, is a genuine one-loop WZW correction −k/(2π) to the golden angle. The formula 360/φ2 − k/(2π) = 137.030 agrees with the measured value to 0.004% with no fitted parameters; the residual 0.006 is resolved in Paper IV. LSC as conditional theorem Two exact algebraic lemmas are proved: μ* = √(k+2)/φ and 1 + μ*φ = 2φ (both exact at k = 3). Together they reduce the LSC to the single virial condition V* = φ: if Jfb/J2a = φ at the EL minimum, then J4/J2a = 24/3/φ5 exactly. Three-fermion bosonization (LSC closed) The density-feedback model with μ* = √5/φ is exactly the SU(2)3 WZW model of k = 3 free Dirac fermions. The Hopfion curvature equals the WZW metric (BS identity); the Q = 2 Hopfion is the j = 0 singlet; the Verlinde formula gives V* = S0,1/2/S0,0 = φ exactly in the thin-torus. A Brouwer/Rellich–Kondrachov fixed-point argument extends V* = φ to all finite torus radii, closing the LSC completely. LSC for all finite R0 The thin-torus cancellation J4/J2a = 24/3/φ5 holds exactly for all torus radii R0: the winding correction 1/(2R02) that shifts the tube term fills the deficit precisely, so the WZW target value is recovered for every R0 without an asymptotic limit. The remaining 0.006 residual in the α formula is a genuine formula-level gap — independent of lattice precision — resolved in Paper IV by the three-term formula α−1 = 360/φ2 − k/(2π) + 1/(9φ6) = 137.036 49 (0.0004% from the measured value).