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. v7 — a Bakry–Émery identity for the selector, and two sentences corrected. A Bakry–Émery identity. With the weight $\phi=2\ln A$ and the $m$-Bakry–Émery tensor $\mathrm{Ric}^m_\phi=\mathrm{Ric}+\nabla^2\phi-\tfrac1m\,d\phi\otimes d\phi$ at $m=2-n=-2$, the value associated with Einstein metrics in a conformal class, the full Ricci tensor of Thakurta–Kerr gives $\mathrm{Ric}^m_\phi=\mu g$, and the conformal-Killing selector satisfies $\tfrac12\mathcal L_Wg+\mathrm{Ric}^m_\phi=\lambda g$ with $\lambda=A'/A+\mu$. For a non-flat seed ($M>0$), $\lambda$ is constant exactly when $A=A_0e^{c\eta}$, and then $\lambda=c$; along geodesics the weighted drift reproduces $A'/A$. The identity is kept distinct from the conformal rescaling of the metric used in the cited soliton papers, and it characterises the selector without deriving it. Bakry and Émery (1985) and Case (2012, 2013) are cited. New verifier on the full Ricci tensor, with a control that must fail. Corrections.- A Killing selector removes the geodesic drift of the rail charge; a curved rail may still need acceleration orthogonal to it (the text had said the rail costs nothing).- $\mathrm{Ric}(u,u)$ is described as the Ricci focusing term, not as the energy density.- With a trivial weight the weighted curvature operator reduces formally to the ordinary one, whose spectrum the paper computes; this does not place the optical metric in the cited soliton class, and the positivity theorems do not apply. References. The companion paper is cited at its version v8 (DOI 10.5281/zenodo.23071751) and the reproducibility package at release v1.7.3 (DOI 10.5281/zenodo.23071786). Unchanged. The abstract, every theorem and every other result of v6.