How does hard magnetic soft matter deform?

Hard magnetic soft materials, i.e. elastomers seeded with hard magnetic particles, convert magnetic energy into complex motion and have become central to the design of untethered soft robots, biomedical devices, and, in general, for enabling fast, wireless actuation. Their behavior depends critically on how the surrounding soft matrix deforms and transmits torque to each magnetized inclusion, yet there is no consensus on the correct mapping between macroscopic strain and microscopic motion. Competing models predict conflicting particle rotations and magnetic responses under identical loads. Here, we construct counterexamples that show the key shortcomings in these models, and then establish a first-principles framework that leads to an optimal deformation mapping. The resulting closed-form law unifies conflicting models and shows that they are, in fact, simply different limits of our general approach. Our work permits the rational design of magnetically actuated soft devices and a unified view of deformation in magnetic soft matter, and broadly, in similar architectured materials.