Brachistochrones in coordinate time and in proper time in the Kerr geometry

Building on Perlick's formulation of the brachistochrone problem in stationary spacetimes~\cite{Perlick1991} and on the optical Randers--Finsler framework for stationary metrics~\cite{GibbonsHerdeiroWarnickWerner2009,Randers1941}, we give a unified treatment of the two brachistochrones of a test particle in the Kerr field: the one minimizing the coordinate time $t$ and the one minimizing the proper time $\tau$. On the equatorial plane $T_t$ is the length of a Randers metric, due to a linear gravitomagnetic term, whereas $T_\tau$ is \emph{purely Riemannian}. We derive the first integrals, exact orbit equations, travel times, and Hamiltonian formulations for both branches. From the equatorial Hamiltonians we obtain a \emph{new} analytical result: for common initial phase-space data the two branches carry different on-shell energies, $H_0^\tau/H_0^t\to E^2/f_0^2$, and this, combined with the gravitomagnetic enhancement of the centrifugal barrier, produces an \emph{inversion} of the radial plunge across a critical angular momentum $J_{\rm inv}(a)$. The resulting closed-form criterion is validated numerically to four significant digits. Off the equator the brachistochrone Hamilton--Jacobi equation is \emph{non-separable}: the conformal factor $n_\tau^2=F/D_E$ obstructs Carter separation at finite energy, while separability is recovered in the Fermat limit. We therefore construct \emph{approximate} quasi-invariants by perturbative Poisson-bracket closure: for $\tau$ a single $O(a^2)$ term reduces the drift, while for $t$ the bare Carter form is already optimal. We also verify the $a\to0$ Schwarzschild limit, develop spin and weak-field expansions, and extend the construction toward the ergoregion using ZAMO variables. These results complement the companion interior treatments~\cite{Rosignoli2026Schw,Rosignoli2026TOV}.