A working bifurcation collar supplies the exact GR section

Restrict each smooth null boundary to ${\mathrm{\Sigma }}_0$. By the inline null-hypersurface definition, its nonzero normal covector has null metric dual. It cannot vanish on the spacelike tangent hyperplane, whose nonzero conormals have timelike metric duals. Both restrictions are therefore transverse smooth surfaces and agree locally with the embedded sphere $S$. After shrinking a neighborhood of compact $S$, each divides the section collar into an outer and inner side. The outer side is $C$ and belongs to both chronological sets. The past/future-set property fixes the local side of each smooth chronological boundary; hence the inner side belongs to neither set. This is the usual local bifurcation quadrant conclusion, derived from the specified actual boundaries and the spacelike crossing, rather than from tangency of a Killing field.

Write $A=I_M^{-}(U_0)$ and $B=I_M^{+}(U_0)$. Flow a thin section annulus along a smooth future timelike field $W$ to obtain coordinates $(t,s,\vartheta )$, where $s=r-1$ on the section and $\vartheta \in S^2$; compactness of $S$ gives a common thin collar. Choose the initial flow domain to be a product $(-t_0,t_0)\times (-s_0,s_0)\times S^2$, with $W={\mathrm{\partial }}_t$, and take the later $\epsilon$ smaller than $s_0$ and the outer agreement width. Its zero level is the original section, and after shrinking $dt$ is future-increasing temporal. The null boundaries are transverse to $W$, so the inverse function theorem writes them as $t=a(s,\vartheta )$ and $t=b(s,\vartheta )$. They both have $a(0,\vartheta )=b(0,\vartheta )=0$. The past-set and future-set properties, with chronology openness, put $A$ on the side $t<a$ and $B$ on the side $t>b$: the short timelike past of an $A$-boundary point lies in $A$, while its short timelike future cannot lie in $A$, and dually for $B$. Step 1.1 and the nonzero transverse section derivatives give $a_s>0$ and $b_s<0$ on $s=0$. Write $a=s\alpha$ and $b=-s\beta$, where $\alpha ,\beta$ extend smoothly and positively across $s=0$ by their derivatives there. Set $u=(a-t)\mathrm{/}\alpha$ and $v=(t-b)\mathrm{/}\beta$. On the original section $u=v=s$; on $S$ their $(s,t)$ Jacobian has determinant $1\mathrm{/}\beta +1\mathrm{/}\alpha >0$. The inverse function theorem and compactness give product coordinates $(u,v,\vartheta )$ on a smaller collar. Choose $\epsilon >0$ so that the closed product box in these coordinates lies in the original flowout and in the annulus where $C^{\mathrm{\prime }}=C$ on its outer side, and let $O=\{\mathrm{\mid }u\mathrm{\mid }<\epsilon ,\mathrm{\mid }v\mathrm{\mid }<\epsilon \}\times S^2$. Thus $A\cap O=\{u>0\}$, $B\cap O=\{v>0\}$ and $E\cap O=\{u>0,v>0\}$. These defining functions are not assumed optical away from their zero hypersurfaces.

Use the open collar $O$ and its temporal coordinate $t$ from step 2.1. Set $M^{\mathrm{\prime }}=E\cup O$ and ${\mathrm{\Sigma }}^{\mathrm{\prime }}=C^{\mathrm{\prime }}\cup S\cup \{\text{inner section collar}\}$. A point in the inner collar cannot be causally related in either direction to an $E$ point: composition with a timelike curve from or to $U_0$ and push-up would put it in one of the original chronological sets, contrary to step 1.1. The same is true for a point of $S$, which lies on both boundaries and hence in neither open chronological set. Thus a causal curve between points of the inner collar or $S$ cannot pass through $E$; in $O$ the temporal coordinate prevents a return to its zero level. No such point is causally related to $C^{\mathrm{\prime }}$. A causal curve between two $E$ points stays in $E$: if $p\le z\le q$ and $p,q\in E$, push-up gives $z\in I_M^{+}(U_0)\cap I_M^{-}(U_0)$. Since $C^{\mathrm{\prime }}$ is Cauchy in $E$, [F3] makes it acausal there, and therefore ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ is acausal in $M^{\mathrm{\prime }}$. Let $p=(t,s,\vartheta )\in E\cap O$ in the original flow coordinates of step 2.1. The definitions $u=(s\alpha -t)\mathrm{/}\alpha$ and $v=(t+s\beta )\mathrm{/}\beta$ give $(\alpha +\beta )s=\alpha u+\beta v$. Since $\alpha ,\beta >0$ and $0<u(p),v(p)<\epsilon$, this weighted-average identity gives $0<s<\epsilon$. Thus the zero-time base $z=(0,s,\vartheta )$ lies in the original section's outer agreement collar, so $z\in C\cap C^{\mathrm{\prime }}$. The entire segment $\gamma (\lambda )=((1-\lambda )t,s,\vartheta )$, $0\le \lambda \le 1$, lies in the original product flow domain. Along it $\alpha ,\beta$ are constant, so its $(u,v)$ coordinates are $((1-\lambda )u(p)+\lambda s,(1-\lambda )v(p)+\lambda s)$. Both remain strictly between $0$ and $\epsilon$, proving that the segment stays in $E\cap O$. If $t\mathrm{\ne }0$, it is a nonconstant timelike $W$-flow segment, future-directed in one of its two orientations. Consequently, if also $p\in C^{\mathrm{\prime }}$, the acausality of $C^{\mathrm{\prime }}$ in $E$ from [F3] forces $p=z$, hence $t=0$. Conversely every point of the outer zero-time slice in $O$ is in the agreement collar and therefore in $C^{\mathrm{\prime }}$. Hence $C^{\mathrm{\prime }}\cap O$ is exactly that outer zero-time slice, and ${\mathrm{\Sigma }}^{\mathrm{\prime }}\cap O=\{t=0\}$ in $O$.

The smooth spacelike hypothesis on $C^{\mathrm{\prime }}$ and its agreement with $C$ on an outer collar make ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ a smooth spacelike embedded hypersurface across $S$. It is closed in $M^{\mathrm{\prime }}$: [F3] gives that $C^{\mathrm{\prime }}$ is closed in $E$, while step 3.1 gives ${\mathrm{\Sigma }}^{\mathrm{\prime }}\cap O=\{t=0\}$ in $O$, with the open annular ends excluded from $O$. These are closed relative to the open cover $E,O$ of $M^{\mathrm{\prime }}$. At every $p\in {\mathrm{\Sigma }}^{\mathrm{\prime }}$ choose a sufficiently small hypersurface flow chart $V$ with a future-increasing temporal coordinate $t_p$ and ${\mathrm{\Sigma }}^{\mathrm{\prime }}\cap V=\{t_p=0\}$. Such a coordinate exists by the same transverse flow and shrinking argument as in step 2.1. A point of $I_V^{-}(p)$ has $t_p<0$, and a point of $I_V^{+}(p)$ has $t_p>0$; every timelike curve between them in $V$ crosses $t_p=0$ by continuity. Thus $p$ is not an edge point under the inline definition. Since ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ is closed, there are no other points of its closure at which an edge could occur. Together with step 3.1 this proves that ${\mathrm{\Sigma }}^{\mathrm{\prime }}$ is a partial Cauchy hypersurface. Radially reparametrizing the open inner collar, whose inner end is outside $O$ and $E$, gives the topology $\{r>1\mathrm{/}2\}$ with $S$ at $r=1$, choosing the reparametrization to be the identity sufficiently near $S$. The supplied chart $q$ equals $x$ on the common outer collar. Thus $q$ on $C^{\mathrm{\prime }}$ glues to the inner radial chart to give the diffeomorphism $\widehat{q}$ in the Statement, agreeing with $x$ near $S$ and retaining every prescribed end coordinate.

By the inline null-hypersurface and gradient definitions, on $u=0$ the vector $\mathrm{grad}u$ is future null, and $dv(\mathrm{grad}u)>0$ near $S$: $du$ is negative on future timelike vectors, $dv$ is positive on them, and the two null normals at $S$ are independent. In a Lorentz orthonormal basis the pairing of independent null vectors of opposite time orientations is strictly positive, which gives this last inequality. Set $Q={\mathrm{\partial }}_u$ in the product coordinates and $L=\mathrm{grad}u-g^{-1}(du,du)Q$. Then $du(L)=0$, and on $u=0$ this equals $\mathrm{grad}u$. On a smaller closed box, smoothness and the mean value theorem give $\mathrm{\mid }g(L,L)\mathrm{\mid }\le C\mathrm{\mid }u\mathrm{\mid }$, while $g(L,W)\le -c<0$, $du(W)<0$ and $dv(L)\ge c^{\mathrm{\prime }}>0$. Choose a constant $K>C\mathrm{/}(2c)$ and put $V=L+KuW$. On $u>0$, $g(V,V)\le Cu-2Kcu+K^2{u}^{2}g(W,W)<0$ and $g(V,W)<0$, so $V$ is future timelike. After shrinking the box, $dv(V)>0$ with a positive uniform lower bound; also $V(u)=Kudu(W)$. Along its flow, $u(h)=u(0)\exp (K{\int }_0^{h}du(W)d\lambda )$ stays positive and decreases. For every point of $O$ with $u>0$ and $v\le 0$, flow forward until $v>0$. This takes finite time bounded by its initial $\mathrm{\mid }v\mathrm{\mid }$ plus an arbitrarily small positive increment, divided by the uniform lower bound for $V(v)$. Before that time $u$ cannot reach either box face, $v$ remains strictly between its box faces, and the compact sphere factor has no boundary. Smooth local-flow existence on the closed larger box therefore supplies this entire short segment inside $O$, ending in $E$. A point already in $E$ needs no extra segment. Combining with its future tail-access curve inside $E$ proves $O\cap A\subset I_{M^{\mathrm{\prime }}}^{-}(U_f)$.

Every $E$ point lies in both chronological sets of $U_f$ by the two-sided access hypothesis, with curves inside $E\subset M^{\mathrm{\prime }}$. Conversely $U_f\subset U_0$ and $M^{\mathrm{\prime }}\subset M$ give its DOC contained in the original $E$. This proves the DOC equality in $M^{\mathrm{\prime }}$. The inner section collar and $S$ are outside both new chronological sets since they are outside the corresponding old ambient sets. Hence both separate section equalities hold. It remains to prove the separate chronological-set identities on $O$; DOC equality alone is not used for that inference.

Reverse time and exchange $u,v$. On $v=0$, $\mathrm{grad}v$ is past null and $du(\mathrm{grad}v)>0$. With $Q^{\mathrm{\prime }}={\mathrm{\partial }}_v$, set $L^{\mathrm{\prime }}=\mathrm{grad}v-g^{-1}(dv,dv)Q^{\mathrm{\prime }}$ and use the past timelike field $-W$. The same estimates give a past timelike vector field $L^{\mathrm{\prime }}+K^{\mathrm{\prime }}v(-W)$ on $v>0$, with $v$ positive and decreasing and $u$ increasing at a uniformly positive rate. Its flow takes every $v>0,u\le 0$ point to $E$ inside $O$. Reversing that segment and concatenating with past tail access inside $E$ proves $O\cap B\subset I_{M^{\mathrm{\prime }}}^{+}(U_f)$. The opposite inclusions in both signs follow directly from $M^{\mathrm{\prime }}\subset M$ and $U_f\subset U_0$. Thus both asserted individual chronological-set identities hold on $O$.

Boundaries restrict locally on open subsets: equality of two sets on $O$ gives equality of their closures and interiors at every point of $O$, since neighborhoods may be intersected with $O$. Apply this to step 5.2. The two new boundaries in $O$ are exactly $u=0$ and $v=0$, so their intersection is $S$. In $E$ every point belongs to both open chronological sets by step 5.1 and hence to neither boundary. Since $M^{\mathrm{\prime }}=E\cup O$, their full double boundary is exactly $S$. The restricted metric is unchanged, so the old null hypersurfaces are the same embedded geometric hypersurfaces in the new spacetime. No new point outside $M^{\mathrm{\prime }}$ enters this argument.