Brachistochrones in conformal Kerr spacetimes: control domain, exterior separatrices, and adiabatic response

We develop the relativistic controlled-rail brachistochrone---the time-optimal constrained worldline reaching a fixed spatial target at a free arrival clock---in the rotating, axisymmetric Thakurta--Kerr spacetime, the conformal Kerr geometry $g=A(\eta)^2 g_{\rm Kerr}$. The controlled-rail principle, its existence/normality and non-autonomous Hamilton--Jacobi verification, and the Killing, conformal-Killing and Kodama selector hierarchy are established in the companion paper (with the spherically symmetric FLRW andVaidya cases); here frame dragging, a conformal stationary limit and genus-two spectralcurves appear. Since $g(\partial_\eta,\partial_\eta)=-A^2(1-2M/r)$, the admissible-velocity set is compact only for $r>2M$: the compact-control problem \emph{ends} at the stationary limit, and every curve is labelled an exterior controlled-rail extremal, a limiting contact, or an analytic continuation carrying no optimality claim. We derive the breathing-indicatrix Hamiltonians and the conformal-time ($\eta$) brachistochrone---the arrival branch $t\equiv\eta$, distinct from proper time $\tau$; closed-form equatorial separatrices in Weierstrass functions; a complete local classification at the stationary limit, in which the two marginal momenta $\pm J_c$ prove \emph{dynamically inequivalent} although the shape radical is even in $J$---the retrograde one has a finite limiting tangent and reaches $r=2M$ only asymptotically, the prograde one crosses at a finite rate; the exterior retrograde separatrix at $r_d=3.5139M$; therotational and conformal plunge inversion, with an existence theorem for the latter under a same-launch comparison; and the first-order adiabatic response, on-shell and off-shell, canonically re-derived from the extended Hamiltonian on compact regular subarcs and verified against the true non-autonomous flow to $O(\varepsilon^2)$, its weight-two content a length-two iterated Abelian integral on a genus-two hyperelliptic curve. The separatrices, the local classification and the on-shell reduction are proved; the fixed-endpoint no-inversion is established in the stated analytic domain($r_0\le R_\ast(E,a)$) and machine-certified at representative $r_0$, while its unrestricted extension remains conjectural; a rigorous higher-genus polylogarithm framework remains open.