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…