Photon-Return and Polar-Focusing Crossovers near Extremal Kerr

How precisely must a photon’s impact parameter be tuned for it to complete $n$ windings around a nearly extremal Kerr black hole before escaping, and how does the same near extremal scaling control the focusing of neighboring rays? For equatorial prograde returns we find, with $\kappa =\sqrt{1-a^2\mathrm{/}{M}^2}$ and $X=n\kappa$, ${\epsilon }_n=\sqrt{3}\kappa \mathrm{/}(e^{2\pi X}-1)+O({\kappa }^2\mathrm{\mid }\ln \kappa \mathrm{\mid })$. This single law connects the extremal inverse-winding hierarchy to subextremal exponential accumulation. Perturbing the return family quasi-equatorially, we derive a second scaling law for the polar Jacobi phase, ${\mathrm{\Psi }}_{\theta }=2\ln [\sinh (\pi X)\mathrm{/}\kappa ]+C_s+o(1)$, and show that its focusing points cross from geometric spacing in winding number to asymptotically uniform spacing $\mathrm{\Delta }\mathcal{N}\simeq (2\kappa {)}^{-1}$. Numerical integrations of the exact Kerr radial dynamics and linear polar equation verify both scaling collapses. Thus near extremal Kerr exhibits parametrically distinct winding and polar-focusing scales, rather than a discontinuous jump between extremal and subextremal lensing hierarchies.