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

What is the fastest constrained worldline when the spacetime not only evolves but rotates? We study this for the controlled-rail brachistochrone—the time-optimal worldline reaching a fixed target at a free arrival clock—in Thakurta–Kerr, the conformal Kerr geometry $g=A(\eta)^2 g_{\rm Kerr}$. Its foundations—existence and normality, Hamilton–Jacobi verification, the selector hierarchy—are the companion paper's; what rotation adds is frame dragging, a conformal stationary limit and genus-two spectral curves. The domain is delimited at the outset: since $g(\partial_\eta,\partial_\eta)=-A^2(1-2M/r)$, the admissible-velocity set is compact only outside the stationary limit $r=2M$, where the compact-control problem ends, and every curve is labelled an exterior extremal, a limiting contact, or an analytic continuation with no optimality claim. We derive the breathing-indicatrix Hamiltonians and the conformal-time arrival branch $t\equiv\eta$, distinct from proper time $\tau$; equatorial separatrices closed in Weierstrass functions, the exterior retrograde one at $r_d=3.5139M$; a local classification at that limit in which the marginal momenta $\pm J_c$ prove dynamically inequivalent though the shape radical is even in $J$—the retrograde reaches it only asymptotically, the prograde crosses at a finite rate; the rotational and conformal depth inversion, the latter with an existence theorem under a same-launch comparison; and the first-order adiabatic response, on- and off-shell, verified on compact regular subarcs against the true non-autonomous flow to $O(\varepsilon^2)$, its weight-two content a length-two iterated Abelian integral on a genus-two curve. The separatrices, the classification and the on-shell reduction are proved; for frozen $A$ the symmetric fixed-endpoint no-inversion is proved on explicit regions of the coordinate-time turning point, for every spin and, when $E^2\ge3/2$, everywhere beyond the single peak of the proper-time half-angle map, its unrestricted extension remaining conjectural; a rigorous higher-genus polylogarithm framework remains open. v6 — every script the paper names, run and compared with what it prints. The physical separatrix is now checked directly. The elliptic collapse of the off-shell term at a separatrix had been spot-checked only at a prograde root inside the stationary limit and at a negative one. The script cited for it contained only a description of the check. It now runs at thirty digits and includes the exterior retrograde separatrix $r_d=3.514M$: the degeneration, the cancellation of the spurious pole and the collapse of the letters, each with a control that must fail. Stated precision brought to what reproduces.- The $\theta$-ratio form of the third-kind letter had been quoted at $5\times10^{-8}$; the archived double-precision check gives $7\times10^{-5}$. The text now rests that identity on Fay's representation of the normalised third-kind differential, which is a theorem.- The result-to-script table no longer claims that every entry is generated by the provenance command: the slopes and checksums are, and the other residuals are the scripts' printed output, re-run for this version.- The classification script applies the selector criterion $r_d>2M$ that the text states.- Two quoted figures are corrected to the scripts' output. One description completed. On the proper-time branch the drift clock adds third-kind blocks at the seed null surfaces to the block expansion; the text said the formulae carried over unchanged. Every quoted residual traced to a script. Each number the text quotes without naming a script was traced, re-run and now names its script. The evaluation of the $t$-branch separatrix in Weierstrass functions is now archived at forty digits (better than $10^{-46}$ on the five arcs of the figure caption). Values that differed from the printed ones are replaced by the reproduced ones (rail charge $9\times10^{-16}$, colormap $10^{-11}$, design-study drift $3\times10^{-15}$). Two statements with no archived source are withdrawn: the basis-dependent eigenvalues of $\mathrm{Im}\,\tau$ and a Doran-shift residual. The rank-five dilogarithm basis is described by the $10^{-79}$ system residual it actually has. Two statements made precise. The no-conjugate-point proposition cites the certificate its sufficiency direction rests on, and the prograde/retrograde asymmetry of the separation weight is stated pointwise in the sign of the azimuthal velocity. The AI-use statement lists every model used. Corrections after an independent audit of the proofs.- The extension of the no-inversion argument to the quarter regime, and the radius $R_*$ built on it, rested on a convexity step that does not close, and are withdrawn. A new proposition, with its proofs in a new appendix, shows that the proper-time half-angle decreases on explicit regions of the turning radius, for every spin, and has a single peak when $E^2\ge3/2$; for frozen $A$ and the symmetric single-periastron family, the fixed-endpoint no-inversion is a theorem wherever the coordinate-time turning point lies in those regions. At low energy the half-angle map has several maxima, certified by interval arithmetic, so its earlier description as single-peaked is withdrawn. The interval certificates were rerun with their windows extended to the half-potential radius, at seven static configurations, all passing; five of them are now covered analytically.- The no-inversion thresholds carry their powers of $M$, and the key factorization its factor $1/D_E$; the source numerators are given per branch and $\psi$ is defined with the full drift operator; the radial action carries its sheet sign; the sign of $W_{\sup}$ is an algebraic, not a rational, question; the classification states its sector.- After a final end-to-end read: the optical metric of the running problem is shown not to descend by a direct computation of its coefficients, not by importing a Riemannian criterion; the conformal weight of the angular costate is derived from the factorization of the Hamiltonian, and the costate is labelled by branch; the $\tau$-separatrix figure no longer attributes the residual growth near the stationary limit to an orbital instability; the frozen family is described as the instantaneous ingredient of the time-dependent problem, not its solution. Relation to the literature stated. The paper now states how its results meet three lines of work. The separation equation is the Jacobi equation of the rulings of brachistochrone-ruled timelike surfaces (Taş, 2025) in conformal Kerr at frozen $A$; those surfaces are totally geodesic planes only for suitable planar families. For the frozen Schwarzschild optical metric the spectrum of the curvature operator is computed ($K_t$, $K_t$, $K_{\tan}$, with scalar curvature $R_{\rm opt}=2(2K_t+K_{\tan})$) and enters the in-plane and out-of-plane Jacobi equations; it gives local, not global, minimality. The contact with totally geodesic 2-planes of weighted Ricci solitons is stated, with the normalisation of the cited source. Thakurta–Kerr is shown directly not to be Einstein, through a mixed Ricci component proportional to $A'/A$. Unchanged. No closed form or figure changes. The provenance check regenerates the manifest and the adiabatic slopes identically. The computer-assisted certificates were rerun with their new windows and all pass.