The Fer expansion of a driven spin under su(2) operations cannot terminate beyond first order

The Fer expansion writes the propagator of a driven quantum system as an ordered product of exponentials, each generated by the time integral of an effective Hamiltonian that governs the evolution left over by the previous factors. It can be stopped after finitely many factors whenever this leftover evolution is known in closed form, for instance when the effective Hamiltonian commutes with itself at all times, or precesses about a fixed axis with constant angular velocity. For su(2), in any representation, from a qubit to a spin of any size, we resum the Fer recursion into a closed map on curves in R cubed. We prove that, for a real-analytic drive, in both cases the expansion stops beyond the first factor only trivially, when the first exponential is already exact.