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

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. Two results follow. First, a lower bound: inverting the equation of motion and applying the arithmetic-geometric mean inequality gives Phi >= 4 alpha Delta E / (gamma mu) for every pulse at every switching time, with Delta E 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, two damping values) 28 reverse more cheaply through a wall, by up to 51 percent, and every cost lies above its barrier floor. The saving at fixed switching time is not monotone in length: a longer wall needs more time to cross. One case was verified by grid refinement, local consistency, and open-loop forward dynamics. A comparison restricted to two fixed modes, uniform rotation against a constant-speed wall, reaches the opposite conclusion, because that wall is not the optimal one. Code and data: engine (MIT) https://github.com/fsantibanezleal/CAOS_SpinOCT ; product, committed artifacts and live instance (MIT) https://github.com/fsantibanezleal/CAOS_RES_Espira ; https://espira.fasl-work.com .