Uniform Rotation is the Field-Cost Optimum over a Domain-Wall Sweep: A First Step Beyond the Macrospin in Optimal Magnetization Switching

Optimal control of magnetization switching has been solved almost entirely at the macrospin level, where the magnetic element is a single rotating moment. The authors of the method state, in print, that this approximation breaks down with size and that the reversal may proceed by nonuniform rotation, domain-wall nucleation and propagation, or spin waves, and that it remains open under what conditions these become optimal in terms of energy efficiency. This preprint takes a first step past the macrospin. For a ferromagnetic spin chain with nearest-neighbour exchange, we compute the switching cost of two reversal modes: a uniform rotation in which every site follows the macrospin optimal control path together, and a domain-wall sweep in which a reversed domain nucleates at one end and its wall propagates to the other. The lattice engine reproduces the analytic macrospin cost in the one-site limit (the validation gate). We find that for the switching-cost metric, the Joule heating of the source, uniform rotation is cheaper than the domain-wall sweep across every chain length from two to sixty-four sites and every exchange strength tested (J/K from 0.2 to 10), by a factor of eight to twenty-two, because the wall forces fast local flips and bends the bonds across itself while uniform rotation moves every site slowly and never pays exchange. This is the opposite of a common intuition, and the resolution is instructive: domain-wall motion dominates real, thermally-driven switching because a wall lowers the energy BARRIER, a thermal-stability advantage, not the field COST that optimal control minimizes. The two metrics reward opposite mechanisms. The comparison is between two ansatze, not a free search over all chain trajectories; a full lattice optimal control path is the natural next step. All results are reproducible from committed artifacts and a live instance. Engine (MIT): https://github.com/fsantibanezleal/CAOS_SpinOCT . Product and live instance (MIT): https://github.com/fsantibanezleal/CAOS_RES_Espira ; https://espira.fasl-work.com .