  The linear in mass ratio correction to the periapsis advance of equatorial
orbits around a spinning black hole is calculated for the first time and to
very high precision, providing a key benchmark for different approaches
modelling spinning binaries. The high precision of the calculation is leveraged
to discriminate between two recent incompatible derivations of the 4PN
equations of motion. Finally, the limit of the periapsis advance near the
innermost stable orbit (ISCO) allows determination of the ISCO shift,
validating previous calculations using the first law of binary mechanics.
Calculation of the ISCO shift is further extended into the near extremal regime
(with spins up to $1-a=10^{-20}$), revealing new unexpected phenomenology. In
particular, we find that the shift of the ISCO does not have a well defined
extremal limit, but instead continues to oscillate.
