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We confirm the generalized actions of the complete NLO cubic-in-spininteractions for generic compact binaries which were first tackled via anextension of the EFT of spinning gravitating objects. We first reduce thesegeneralized actions to standard actions with spins, where the interactionpotentials are found to consist of $6$ independent sectors, including a newunique sector that is proportional to the square of the quadrupolar deformationparameter, $C_{\text{ES}^2}$. We derive the general Hamiltonians in anarbitrary reference frame, and for generic kinematic configurations. With thesemost general Hamiltonians we construct the full Poincar\'e algebra of all thesectors at the fourth and a half post-Newtonian (4.5PN) order, including thethird subleading spin-orbit sector, recently accomplished uniquely via ourframework, thus proving the Poincar\'e invariance of all relevant sectors. Wethen derive the binding energies with gauge-invariant relations useful forgravitational-wave applications. Finally, we also derive the extrapolatedscattering angles in the aligned-spins configuration for the scatteringproblem. Yet, as made clear already as of quadratic-in-spin sectors, thealigned-spins simplification inherent to the scattering-angle observable,entails a great loss of physical information, that is only growing withhigher-spin sectors. Our completion of the full Poincar\'e algebra at thepresent $4.5$PN order provides strong confidence that this new precisionfrontier in PN theory has now been established.

From the EFT of Spinning Gravitating Objects to Poincar\'e and Gauge Invariance at the 4.5PN Precision Frontier