Secular evolution of orbital parameters for general bound orbits in Kerr spacetime

We analytically derive the secular changes of the orbital parameters, i.e., energy, angular momentum, and Carter constant, for general bound orbits in Kerr spacetime, at leading order in the mass ratio, through the 6th post-Newtonian (6PN) order and the 16th order in orbital eccentricity. We validate the formulas against high-precision numerical Teukolsky results and quantify how eccentricity affects both the achievable accuracy and the PN convergence. We then construct and test a simple ``hybrid'' approximation that combines different PN and eccentricity truncations to retain accuracy at reduced computational cost. We also assess the performance of exponential resummation at higher PN orders. These results provide building blocks for fast, (analytic) adiabatic inspiral and waveform models for extreme mass ratio inspirals relevant to space-based detectors such as the Laser Interferometer Space Antenna (LISA).