Fig. 3: Skyrmion numbers and matter-field configurations of a Néel-type AGB skyrmion under increasing perturbation.

a Gaussian noise with standard deviation (\(\sigma\)) ranging from 0 to \(\pi\) was introduced to create controlled perturbations (see details in Methods 2). At each noise level, the mean skyrmion number \({N}_{{\rm{sk}}}\) (data points) and its standard deviation \({\sigma }_{{N}_{{\rm{sk}}}}\)(shaded region) were determined through repeated measurements. b Representative AGB skyrmion patterns at ten equally spaced noise levels (i–x, corresponding to markers in panel a). In the low noise regime (i-iii), the internal axis field is distorted while the boundary remains intact, and the skyrmion number stays topologically protected. When the noise reaches a moderate level (iv), the skyrmion number begins to fluctuate, indicating that topological protection is no longer guaranteed to hold. Under higher noise (v–x), the boundary conditions are disrupted, resulting in the collapse of topological protection