Three-dimensional spectroscopy of a TUF-M-B Hopf soliton: Goldstone convergence, open-channel tests, and nonlinear mode transfer

We construct the full quadratic operator for perturbations of the self-consistent TUF-M-B Hopf-charge-onebackground, retaining all three spatial coordinates through exact azimuthal Fourier sectors. Theconstrained Hessian H and kinetic matrix M are stored explicitly and solved as H v=omega^2 M v. Sectorsk=0,...,8 supply 540 reported eigenpairs, and k=9,...,16 are repeated in two bases. The only negative rawpair occurs in k=1; it overlaps the rotational Goldstone subspace by 90.4%, its splitting decreases from-5.96e-3 to -2.05e-3 as L grows from 4.5 to 6, and a direct generator frequency softens to 0.1016 at L=8. Nophysical negative mode is found after the symmetry family is separated. Complex absorbing potentials andindependent outgoing/Sommerfeld bases reveal no domain-stable resonance. A cylindrical 3+1 evolutionretaining all azimuthal harmonics shows nonlinear transfer from seeded k=2,3 modes while conserving themonitored energy to 2.72e-8 over T=2. These results support finite-regulator local stability, but notinfinite-time nonlinear stability.