Self-consistent TUF-M-B Hopf solitons: energetics, first-order structure, and local hyperbolicity

We formulate a dynamical soft-homeostasis extension of the Faddeev-Skyrme model and compute anequivariant Hopf-charge-one stationary state by relaxing the map field and the homeostatic scalarsimultaneously. The explicit exponential coupling is bounded below, the scalar has a positive kinetic term,and the principal kinetic form is positive on the computed background. At alpha6=0.10, mu6=1, andZmin=0.25, the largest-domain calculation gives E=268.110319, QH=0.989866, maximum density41.389145, root-mean-square radius 1.788732, and a local minimum characteristic speed squared 0.105814.Energy and residuals are reported under grid, domain, and initial-data refinement. The dominantuncertainty is the finite boundary: E changes by 0.636250 between L=5 and L=6 and by 0.465212 betweenthe later L=7 and L=8 backgrounds. The model is therefore a well-defined ghost-free soft theory on thetested background, but the chosen smooth family is not a proved limit of a hard KKT density constraint.