  We develop massive higher-spin theory as a framework for describing dynamics
of rotating compact objects, such as Kerr black holes. In this paper, we
explore gauge interactions up to quartic order and corresponding Compton
amplitudes of higher-spin massive objects coupled to electromagnetism and
Yang-Mills theory. Their classical counterparts are known as root-Kerr
gauge-theory solutions, whose amplitudes are closely related to those of Kerr
black holes. We use three distinct approaches: (i) massive higher-spin gauge
symmetry to introduce cubic interactions for all spins and the quartic
interactions up to spin 3, which is implemented both off shell and via Ward
identities; (ii) a chiral higher-spin approach to construct quartic Lagrangians
with correct degrees of freedom to all spins; (iii) on-shell functional
patterns before and after taking the classical limit to constrain the Compton
amplitudes. As final results, we arrive at simple local formulae for the
candidate root-Kerr Compton amplitudes both in the quantum regime and classical
limit, to all orders in spin. This is a precursor to the gravitational Kerr
case, which is presented in a follow-up paper.
