Global minimality of controlled-rail brachistochrones in conformal Kerr and Vaidya spacetimes

A controlled-rail brachistochrone is the fastest trajectory that keeps a prescribed energy relative to a selector field. We study global minimality in conformal Kerr (Thakurta--Kerr) spacetimes with a time-dependent factor and in ingoing Vaidya, for arrival and proper time. On the arrival branch the rails are the null curves of an auxiliary Lorentzian metric, so first arrival is a Fermat problem: where this metric is globally hyperbolic, minimality among all three-dimensional competitors holds exactly up to the null cut point. For conformal Schwarzschild with ʲ/A²≥ 3/2, for any factor, the first event at swept angle π, if reached, is the null cut point; below $3/2$ this holds within an explicit radius, and the threshold is sharp. The rule extends to Vaidya with non-decreasing mass, below $3/2$ under radius and accretion-rate bounds. For conformal Kerr we prove a three-dimensional excursion barrier, a small-spin result, and equatorial results at every spin for constant data and in certified bands otherwise. On the proper-time branch the infimum can fail to be attained through escape toward $r=2M$, or toward the return frontier in Vaidya. We compute the escape price, select constant-data minimizers by a convex-envelope rule, and prove selection near constant data. Under explicit gap conditions, a late drop of the factor destroys attainment, late accretion preserves the minimizer, and early accretion beyond the critical separation prevents it. Computer-assisted steps are explicit inequalities certified in exact or interval arithmetic.