The IK continuation also fixes one global reciprocal invariant

Let $U=M^{\mathrm{e}\mathrm{n}\mathrm{d}}$ and let $\mathrm{\Phi }:\mathcal{D}\to M$ be the ambient maximal flow of $T$. By AF completeness, $\mathbb{R}\times E\subseteq \mathcal{D}$. The set $E=I_M^{-}(U)\cap I_M^{+}(U)$ is open, since each chronological set of $U$ is a union of the open chronological sets of its points. On $U$, uniqueness of integral curves and $T={\mathrm{\partial }}_t$ give ${\mathrm{\Phi }}_s(t,x)=(t+s,x)$ for every real $s$, so $U$ is invariant. By the Killing equation ${\mathcal{L}}_{T}g=0$ in [F1] and the tensor-flow criterion in [F4], each sufficiently short local time map satisfies ${\mathrm{\Phi }}_s^{\ast }g=g$. Composing these local isometries along a finite orbit segment makes each ambient time map an isometry on its open domain. For $q\in E$ and a future timelike $v\in T_{q}M$, $d{\mathrm{\Phi }}_s(v)$ remains timelike and depends continuously on $s$. Its pairing with $N_{{\mathrm{\Phi }}_s(q)}$ is nonzero and initially negative, hence negative for all $s$. Thus these differentials preserve future timelike vectors at every point of $E$, without assuming that the image already lies in $E$.

Fix $p\in E$ and choose future timelike curves $a\to p\to b$ with $a,b\in U$. Since $U$ is open, trim their end portions to obtain $a^{\mathrm{\prime }}\to p\to b^{\mathrm{\prime }}$ with $a^{\mathrm{\prime }},b^{\mathrm{\prime }}\in U$, strictly between $a$ and $p$ and between $p$ and $b$, respectively. Every point of the trimmed curves, including their endpoints, is chronologically after $a$ and before $b$ by restriction and finite concatenation, and therefore belongs to $E$. Consequently ${\mathrm{\Phi }}_s$ is defined on both entire trimmed curves for every real $s$. Smooth composition and step 1.1 show that their images are future timelike curves from ${\mathrm{\Phi }}_s(a^{\mathrm{\prime }})$ through ${\mathrm{\Phi }}_s(p)$ to ${\mathrm{\Phi }}_s(b^{\mathrm{\prime }})$. Both endpoints lie in $U$ by its invariance, so ${\mathrm{\Phi }}_s(p)\in E$. This proves ${\mathrm{\Phi }}_s(E)\subseteq E$ for every $s$; applying $-s$ and the inverse-flow law gives equality. Thus the restriction is a smooth complete flow $\mathbb{R}\times E\to E$ with diffeomorphic time maps.

Apply the analytic completion to obtain $1-\sigma \mathrm{\ne }0$ and $\mathcal{S}=0$ on ${\mathrm{\Sigma }}_0\cap E$, and retain its initial neighborhood of the bifurcation sphere. By step 2.1, the complete stationary flow preserves $E$. For every $p\in E$, the AF orbit assumption gives $q\in {\mathrm{\Sigma }}_0\cap E$ and $t\in \mathbb{R}$ with $p={\mathrm{\Phi }}_t(q)$. This uses the exact IK orbit hypothesis after exterior preservation has been proved.

Antisymmetry gives $T(\sigma )=d\sigma (T)=2\mathcal{F}(T,T)=0$. Integrating along each stationary orbit gives $\sigma ({\mathrm{\Phi }}_t(q))=\sigma (q)$, with the same supplied normalized potential and no additive shift. Hence $1-\sigma (p)\mathrm{\ne }0$ for every $p\in E$ by step 3.1. In particular the Mars–Simon formula now defines a smooth tensor on all of $E$, before any tensor transport is applied.

The local time maps preserve the metric by step 1.1. Differentiating the flow law ${\mathrm{\Phi }}_t({\mathrm{\Phi }}_s(p))={\mathrm{\Phi }}_s({\mathrm{\Phi }}_t(p))$ at $s=0$ gives $d{\mathrm{\Phi }}_t{\mathrm{\mid }}_p(T_p)=T_{{\mathrm{\Phi }}_t(p)}$. Their differentials are invertible by [F6] and vary continuously from the identity, so their orientation sign remains positive. The real and imaginary parts of $\mathcal{F}$ have vanishing Lie derivatives by [F4]; applying its real tensor-flow criterion to each part gives ${\mathrm{\Phi }}_t^{\ast }\mathcal{F}=\mathcal{F}$. The same criterion gives ${\mathrm{\Phi }}_t^{\ast }F=F$. Thus these maps preserve the metric, $T$, orientation and the forms without any completeness assumption at this local stage. For such a flow map $\psi$, transport the connection by ${\tilde{\mathrm{\nabla }}}_{X}Y=(d\psi {)}^{-1}({\mathrm{\nabla }}_{{\psi }_{\ast }X}{\psi }_{\ast }Y)$. The connection rules and the chain rule show it is metric-compatible; its torsion vanishes because diffeomorphisms preserve the vector-field bracket. Uniqueness in [F4] gives $\tilde{\mathrm{\nabla }}=\mathrm{\nabla }$. Substitution into the curvature formula therefore gives ${\psi }^{\ast }R=R$ in the same convention. The preserved oriented metric also preserves its volume tensor and Hodge operation. Thus every factor in the Mars–Simon formula is preserved, including $\sigma$ by step 4.1, so ${\mathrm{\Phi }}_t^{\ast }\mathcal{S}=\mathcal{S}$. Composing local maps along the complete orbit gives this identity for every real $t$. The transport lemma in [F5] now applies on all of $E$ and yields $\mathcal{S}=0$ there.

On this connected zero region, source (4.33) and (7.1) give $d({\mathcal{F}}^2(1-\sigma {)}^{-4})=0$. The initial sphere neighborhood and its assumed sphere value identify the constant as $-1\mathrm{/}(4M^2)$, giving $-4M^2{\mathcal{F}}^2=(1-\sigma {)}^4$ throughout $E$. Thus the full hypotheses for the local differential computation in source Lemma 7.1 hold in every overlapping zero-region chart.

That computation says $dB=0$ for the same smooth scalar $B=z^2+4M^2(y^2+z^2)g^{-1}(dz,dz)$. These local identities glue without coordinate or branch choices because $y,z$ come from the one normalized nonvanishing denominator, with the fixed inverse-metric contraction of [F5]. Connectedness makes the value global. Its value agrees with the initial neighborhood at $S$, where source Lemma 8.1 gives $0\le B<1\mathrm{/}4$ and $y_S=(1+\sqrt{1-4B})\mathrm{/}2$; source Lemma 7.1 gives $z^2\le B$, including on $S$ by continuity. Thus its horizon and end values are the same constant; no global chart-to-Kerr map has been assumed.