HopfionSeries Paper II - BPS Structure and Fixed-Point Theorem for the Density-Feedback Faddeev--Niemi Hopfion

The companion paper establishes numerically that the Euler--Lagrange (EL) minimum of the density-feedback Faddeev--Niemi energy $E_{\mathrm{fb}} = K_{\mathrm{fb}}\cdot J_4$ satisfies $K_{\mathrm{fb}}/J_4 = \lambda/2^{1/3}$ to within $O(h^2)$ discretisation error ($+0.020\%$ on a $256\times256$ lattice), where $\lambda=\varphi^6$ and $\varphi$ is the golden ratio. Here we address this analytically in three steps. We introduce the generalised functional $\mathcal{G}[f;\lambda_{\mathrm{eff}}] = K_{\mathrm{fb}}[f] + \lambda_{\mathrm{eff}}\,J_4[f]$ and define the ratio map$H(\lambda_{\mathrm{eff}}) := K_{\mathrm{fb}}[f_{\lambda_{\mathrm{eff}}}]/J_4[f_{\lambda_{\mathrm{eff}}}]$ where $f_{\lambda_{\mathrm{eff}}}$ is…