Angular-Rate Locking of Geodesics in Kerr Spacetime

We establish a quantitative angular-rate locking theorem for geodesics in Kerr spacetime.
For every noncritical trajectory approaching the outer Killing horizon, the coordinate angular
velocity dϕ/dt converges to the horizon angular velocity ΩH . The classification is governed by
the invariant KH = E −ΩH L, which defines a codimension-one critical subset. We derive explicit
near-horizon estimates showing that the deviation from ΩH is bounded linearly by |r − r+| in
the non-extremal case and quadratically in extremal Kerr. Surface gravity determines the linear
scaling coefficient.