Topological ambiguity in stray-field magnetometry

Abstract Reconstructing the underlying magnetic structure from stray-field measurements is a central inverse problem across spintronics, materials science, medicine, and geology. A growing class of reconstruction methods exploit the topological charge as a strong regularizer to address the ill-posedness of the inverse problem. Here, using analytical and computational arguments, we show that topological charge is not uniquely determined from stray field measurements under realistic magnetometry setups. We apply physics-informed optimization to generate "adversarial pairs": magnetization configurations with distinct topological charge that produce nearly indistinguishable stray fields. Through systematic random sampling, we generate adversarial pairs spanning skyrmions, merons, fractional defects, and uniform domain textures. Multipole expansion analysis identifies the mathematical origin of this degeneracy: topologically distinct structures can share identical low-order magnetic charge distributions that dominate the far-field stray field profile. These results characterize the nature of the non-uniqueness of the inverse problem by explicitly mapping out topologically distinct solutions consistent with the same measurement They provide a benchmark for assessing reconstruction methods and motivate the deployment of strategies such as multiplexed magnetometry capable of resolving topological ambiguity.