Certified robustness of quantum spin textures under magnetic-field perturbations

How much can a magnetic field change before a quantum spin texture loses its skyrmion charge? We address this question by combining the geometry of local spin expectation values with spectral perturbation theory. For a triangulated spin-$s$ texture, we determine the smallest change in its local moments that makes the reconstructed topological charge ill-defined, allowing unequal error bounds and fixed boundary moments. This geometric threshold yields explicit bounds on magnetic-field perturbations that preserve the ground-state charge, with $1/N$ scaling when the gap and geometric margin have positive limits. We apply the bounds to finite spin-$1/2$ systems with exchange and Dzyaloshinskii--Moriya interactions. In a rigorously verified seven-spin example, the charge changes at a single certified field while the excitation gap and all local spin polarizations remain bounded away from zero; at such a crossing the geometric margin and the field certificate vanish linearly. The results provide quantitative certificates of texture stability and clarify the limitations of spectral-gap protection.