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We put forward a broader picture of the effective theory of a spinning particle within the EFT of spinning gravitating objects, through which we derive and establish the new precision frontier at the fifth PN (5PN) order. This frontier includes higher-spin sectors, quadratic and quartic in the spin, which both display novel physical features, due to the extension of the effective theory beyond linear order in the curvature. The quadratic-in-spin sectors give rise to a new tidal effect, and the quartic-in-spin sectors exhibit a new multipolar deformation. We then generalize the concept of tidal operators and of spin-induced multipolar operators, and make conjectures on the numerical values of their Wilson coefficients, and on the effective point-particle action of Kerr black holes. We confirm the generalized actions for generic compact binaries of the NLO quartic-in-spin sectors which were derived via the extension of the EFT of gravitating spinning objects. We first present the corresponding interaction potentials and general Hamiltonians, which consist of 12 distinct sectors, with a new one due to the new multipolar deformation. These Hamiltonians give the full physical information on the binary system, which mostly gets lost in higher-spin sectors, when going to the aligned-spins configuration. Moreover these general Hamiltonians uniquely allow us to find the complete Poincaré algebra at the 5PN order with spins, including the third subleading quadratic-in-spin sectors. We derive consequent observables for GW applications. Finally, to make contact with the scattering problem, we also derive the extrapolated scattering angles for aligned spins. Our completion of the Poincaré algebra provides the strongest validation of our most comprehensive new results, and thus that the 5PN order has now been established as the new precision frontier.