  In this work we derive for the first time the complete gravitational
cubic-in-spin effective action at the next-to-leading order in the
post-Newtonian (PN) expansion for the interaction of generic compact binaries
via the effective field theory for gravitating spinning objects, which we
extend in this work. This sector, which enters at the fourth and a half PN
(4.5PN) order for rapidly-rotating compact objects, completes finite-size
effects up to this PN order, and is the first sector completed beyond the
current state of the art for generic compact binary dynamics at the 4PN order.
At this order in spins with gravitational nonlinearities we have to take into
account additional terms, which arise from a new type of worldline couplings,
due to the fact that at this order the Tulczyjew gauge for the rotational
degrees of freedom, which involves the linear momentum, can no longer be
approximated only in terms of the four-velocity. One of the main motivations
for us to tackle this sector is also to see what happens when we go to a
sector, which corresponds to the gravitational Compton scattering with quantum
spins larger than one, and maybe possibly also get an insight on the inability
to uniquely fix its amplitude from factorization when spins larger than two are
involved. A general observation that we can clearly make already is that
even-parity sectors in the order of the spin are easier to handle than odd
ones. In the quantum context this corresponds to the greater ease of dealing
with bosons compared to fermions.
