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In this work we derive for the first time the complete gravitationalcubic-in-spin effective action at the next-to-leading order in thepost-Newtonian (PN) expansion for the interaction of generic compact binariesvia the effective field theory for gravitating spinning objects, which weextend in this work. This sector, which enters at the fourth and a half PN(4.5PN) order for rapidly-rotating compact objects, completes finite-sizeeffects up to this PN order, and is the first sector completed beyond thecurrent 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 intoaccount additional terms, which arise from a new type of worldline couplings,due to the fact that at this order the Tulczyjew gauge for the rotationaldegrees of freedom, which involves the linear momentum, can no longer beapproximated only in terms of the four-velocity. One of the main motivationsfor us to tackle this sector is also to see what happens when we go to asector, which corresponds to the gravitational Compton scattering with quantumspins larger than one, and maybe possibly also get an insight on the inabilityto uniquely fix its amplitude from factorization when spins larger than two areinvolved. A general observation that we can clearly make already is thateven-parity sectors in the order of the spin are easier to handle than oddones. In the quantum context this corresponds to the greater ease of dealingwith bosons compared to fermions.

Gravitational cubic-in-spin interaction at the next-to-leading post-Newtonian order