We study a density-feedback modification of the Faddeev--Niemi (FN) Hopfion functional and show that its self-consistent saddle-point solution carries a rich algebraic structure rooted in the McKay correspondence for the binary icosahedral group $2I$. Three exact algebraic identities follow from $\varphi^{2}=\varphi+1$ alone, without free parameters: the feedback stiffness $K_{fb} = 2\varphi,J_{2a}$, the scale-selection equivalence: $s_{\mathrm{opt}} = 2^{1/6} \Leftrightarrow J_{4}/J_{2a} = 2^{4/3}/\varphi^{5}$, and the topological charge $\mathrm{ord}(q_{2I}) = 10$. The scale-selection value $s_{\mathrm{opt}}=2^{1/6}$ is confirmed numerically to $0.010\%$ and proved in the thin-torus approximation in the companion paper. Conditionally on the $2I/E_8$ group-theoretic input, Standard-Model observables --- $\sin^2\theta W$, the electron Yukawa $y_e$, the Koide ratio $K$, the Koide angle $\theta_K$, and $M_W/M_Z$ --- are predicted at $0.001\%$--$0.56\%$ accuracy with no fitted parameters. A new exact formula for the modified BPS profile, $J_{4}/J_{2a} = C(2C+1)/((C^{2}+1)(3C-1))$, is derived via beta-function integrals. We give a precise statement of what is proven and what remains open, and list falsifiable predictions.The golden ratio $\varphi$ enters through two independent proven routes: as the quantum dimension $\dim_q(\tfrac{1}{2};\,k{=}3)$ of the $\mathrm{SU}(2)_3$ WZW model, and as the character $\chi_{\Gamma_2}(C_5)$ of the fundamental $2I$ representation. The Bogomol'ny parameter $\lambda = \varphi^6$ is the unique value consistent with $Q_{\mathrm{num}} = Q_{\mathrm{group}} = 10$. The self-consistency condition $J_{2\mathrm{iso}}^{\mathrm{fb}}/J_{2a} = \varphi$ is confirmed numerically to $0.00025\%$ and is motivated by this algebraic structure.The 2I/Hopfion structure provides a surprisingly constrained algebraic scaffold that predicts several Standard-Model observables, --- the Weinberg angle, electron Yukawa, Koide ratio, and lepton mass ratios --- at 0.001%–0.56% accuracy with no fitted parameters, and connects them to a single field-theoretic fixed-point condition. It is not yet a derivation: the principal gap is an analytic proof of the virial self-consistency condition $J_{2\text{iso}}^{\text{fb}}/J_{2a} = \varphi$. The numerical agreements are specific and parameter-free enough to constitute a productive research direction.