Ingoing Kerr: closed form of $\partial_r\Gamma^{r}_{\theta\theta}$

Let $M,a$ be real numbers and $x=(v,r,\theta,\varphi)$ with $\Sigma\neq0$, $\sin\theta\neq0$ ($\Sigma=r^2+a^2\cos^2\theta$). For the ingoing (Kerr's original, advanced) Kerr metric $g^{\rm in}$ of bundle Atlas_Kerr_InDefs, read off from Visser's line element $ds^2=-F\,(dv+a\sin^2\theta\,d\varphi)^2+2\,(dv+a\sin^2\theta\,d\varphi)(dr+a\sin^2\theta\,d\varphi)+\Sigma\,(d\theta^2+\sin^2\theta\,d\varphi^2)$, $F=1-2Mr/\Sigma$, with the half-Hessian convention ($g_{vr}=1$): in coordinates $(v,r,\theta,\varphi)=(x_0,x_1,x_2,x_3)$ and with $A=a\sin^2\theta$, $g_{vv}=-F$, $g_{vr}=g_{rv}=1$, $g_{v\varphi}=g_{\varphi v}=(1-F)A$, $g_{r\varphi}=g_{\varphi r}=A$, $g_{\theta\theta}=\Sigma$, $g_{\varphi\varphi}=(2-F)A^2+\Sigma\sin^2\theta$, all remaining entries $0$; its candidate inverse $\hat g^{\rm in}$ has $\hat g^{vv}=a^2\sin^2\theta/\Sigma$, $\hat g^{vr}=\hat g^{rv}=(r^2+a^2)/\Sigma$, $\hat g^{v\varphi}=\hat g^{\varphi v}=\hat g^{r\varphi}=\hat g^{\varphi r}=-a/\Sigma$, $\hat g^{rr}=\Delta/\Sigma$ with $\Delta=r^2-2Mr+a^2$, $\hat g^{\theta\theta}=1/\Sigma$, $\hat g^{\varphi\varphi}=1/(\Sigma\sin^2\theta)$, all remaining entries $0$ (both matrices are symmetric; every symmetric counterpart is listed), write $U=1/\Sigma$ and let $\Gamma^a_{bc}$ be the coordinate Christoffel symbols of $(g^{\rm in},\hat g^{\rm in})$ (indices $1=r$, $2=\theta$). Then the radial slice derivative of $\Gamma^{1}_{22}$ at $x$ equals

The coordinate Ricci tensor is the one of the definition bundle Atlas_Core_Basic: with $\partial_i f(x)$ the derivative of the slice $u\mapsto f(x[x_i:=u])$ at $u=x_i$,

Here $\hat g$ is the candidate inverse supplied with the metric; whether it is a genuine inverse is a separate obligation.

A generated closed-form entry of the ingoing-Kerr derivative table, promoted to its own node because it is shared by the component lemma $R_{22}$ and the table; it is a rational identity in $r,\sin\theta,\cos\theta,1/\Sigma$.

Formalization Note Everything is coordinate-level on $\mathbb R^4$ (points are functions $\{0,1,2,3\}\to\mathbb R$): no manifold, chart gluing, periodicity of $\varphi$, horizon, causal or global statement is made. Partial derivatives are Mathlib's deriv of coordinate slices; the certificates' $C^\infty$ obligations make every derivative used a genuine derivative. Units $G=c=1$, signature $(-,+,+,+)$.