Domain Walls Become the Optimal Reversal of a Spin Chain: Free Optimal Control Beyond the Macrospin and an Energy-Barrier Floor on the Switching Cost

Version 2 corrects version 1. Version 1 compared a uniform rotation with a constant-speed domain wall at a short switching time and concluded that uniform rotation is always the field-cost optimum. That conclusion is withdrawn: the constant-speed wall is not the optimal wall, and the free search reported here finds wall-mediated reversals cheaper than uniform rotation at long switching times. Optimal control of magnetization switching has been solved almost entirely for a single rotating moment, and its authors state in print that it remains open under what conditions nonuniform mechanisms such as domain-wall nucleation and propagation become optimal in terms of energy efficiency. We answer this for a ferromagnetic spin chain with nearest-neighbour exchange by minimizing the switching cost over the trajectory of every site, with no assumed reversal mode. First, a lower bound: inverting the Landau-Lifshitz-Gilbert equation and applying the arithmetic-geometric mean inequality gives Phi ≥ 4 alpha dE / (gamma mu) for every pulse at every switching time, with dE the minimum-energy-path barrier. It equals the known infinite-time optimum for a single uniaxial moment. Second, above a crossover length and at long switching time the free optimum is a domain wall entering at an open end, strictly cheaper than uniform rotation. Over 61 chains (J/K = 10, N from 4 to 32, damping 0.5 and 0.1), 28 reverse more cheaply through a wall, by up to 51 percent, and every cost lies above its barrier floor. At fixed switching time the saving is not monotone in length, because a longer wall needs more time to cross. One case was verified by grid refinement, local consistency, and open-loop forward dynamics. Engine (MIT, PyPI spinoct 0.10.0): https://github.com/fsantibanezleal/CAOS_SpinOCT . Product, bake and live instance (MIT): https://github.com/fsantibanezleal/CAOS_RES_Espira ; https://espira.fasl-work.com .