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

We confirm the generalized actions of the complete NLO cubic-in-spin interactions for generic compact binaries which were first tackled via an extension of the EFT of spinning gravitating objects. We first reduce these generalized actions to standard actions with spins, where the interaction potentials are found to consist of $6$ independent sectors, including a new unique sector that is proportional to the square of the quadrupolar deformation parameter, $C_{\text{ES}^2}$. We derive the general Hamiltonians in an arbitrary reference frame, and for generic kinematic configurations. With these most general Hamiltonians we construct the full Poincar\'e algebra of all the sectors at the fourth and a half post-Newtonian (4.5PN) order, including the third subleading spin-orbit sector, recently accomplished uniquely via our framework, thus proving the Poincar\'e invariance of all relevant sectors. We then derive the binding energies with gauge-invariant relations useful for gravitational-wave applications. Finally, we also derive the extrapolated scattering angles in the aligned-spins configuration for the scattering problem. Yet, as made clear already as of quadratic-in-spin sectors, the aligned-spins simplification inherent to the scattering-angle observable, entails a great loss of physical information, that is only growing with higher-spin sectors. Our completion of the full Poincar\'e algebra at the present $4.5$PN order provides strong confidence that this new precision frontier in PN theory has now been established.