HopfionSeries Paper VI - The Golden-Ratio Tower: Fermion Scale Hierarchy from the Hopfion RG Fixed Point

We derive the fermion mass hierarchy mn = m0 φ−2n of the density-feedback Hopfion framework from first principles, resolving the origin of the φ−2 spacing that was taken as an axiom in earlier papers. The key result is that the golden ratio φ = (1+√5)/2 is the unique RG fixed-point scale ratio of the condensate parent action: under Derrick scaling r → ζr, the fixed-point condition on the k-essence scalar-tensor action forces ζ = φ. Stable bound states of the condensate scalar field therefore exist at and only at the discrete scales mn = m0 φ−2n; the factor φ2 per level is exact, arising from the two transverse dimensions in the sin4θ suppression. A self-contained statement of the four axioms of the framework is given, from which the following are derived: (i) the quantisation of the tower levels; (ii) the identification of the ground state with the condensate scale Λcond; (iii) the WZW T-matrix phase formula Tg = (6−kg)/[8(kg+2)] proved algebraically from the SU(2)kg conformal data, and its connection to the tower; and (iv) the derivation of the electroweak scale vEW from Λcond and φ alone, closing the two-scale gap noted in Paper I. The Bogomolny factor of 2 in the fixed-point proof is supplied by Paper I Theorem 3.4 (sopt6 = 2 at the BPS saddle), completing the proof without appeal to any external conjecture. The tower provides the RG scaffolding underlying the lepton mass ratios, Koide relation, quark mass hierarchy, and WZW T-matrix phases of Papers I–IV; those results are not re-derived here but are shown to follow from the tower structure.