The set of boundary points with a corresponding point is open, and the extension preserves the metric, matter fields and connection on the common domain and its corresponding boundary

(The interior past of a future boundary point.) Let $p\in C\cap I^{+}(\mathrm{\Sigma })$. If $x\in I^{-}(p)\cap I^{+}(\mathrm{\Sigma })$, openness of $I^{+}(x)$ and the fact $p\in \mathrm{\partial }U$ give $y\in U\cap I^{+}(x)\cap I^{+}(\mathrm{\Sigma })$. Concatenate a future timelike curve from $\mathrm{\Sigma }$ to $x$ with one from $x$ to $y$. Past convexity [L1] puts that curve, hence $x$, in $U$. Therefore $I^{-}(p)\cap I^{+}(\mathrm{\Sigma })\subseteq U$. Choose a future timelike vector at $p$ and its local geodesic $\gamma$ with $\gamma (0)=p$; for sufficiently small $\epsilon >0$, its segment $\gamma ([-\epsilon ,0))$ lies in $I^{-}(p)\cap I^{+}(\mathrm{\Sigma })$ and hence in $U$. This establishes an actual affinely parametrized timelike geodesic, rather than replacing an arbitrary timelike curve by a geodesic.

(Local normal-coordinate extension.) Let $p^{\mathrm{\prime }}$ correspond to $p$. By [L3] $\phi \circ \gamma \to p^{\mathrm{\prime }}$. Choose convex normal neighborhoods $V$ of $p$ and $V^{\mathrm{\prime }}$ of $p^{\mathrm{\prime }}$; shorten $\gamma$ so its entire segment and its image lie respectively in $V,V^{\mathrm{\prime }}$, and set $a=\gamma (-\epsilon )$. By [L4] the image segment is an affinely parametrized geodesic. In the normal chart based at $\phi (a)$ its inverse-exponential coordinates are its affine parameter times its initial velocity. Continuity at the limit $p^{\mathrm{\prime }}$ therefore gives $d{\phi }_a({\exp }_a^{-1}p)={\exp }_{\phi (a)}^{-1}{p}^{\mathrm{\prime }}$. Both vectors lie in the open normal-chart domains. Consequently, on a sufficiently small neighborhood $W$ of $p$, the formula $F_p(q)={\exp }_{\phi (a)}(d{\phi }_a({\exp }_a^{-1}q))$ is defined and is a diffeomorphism onto its open image. At $q=p$ the vector ${\exp }_a^{-1}p$ is future timelike; shrink $W\subseteq V\cap I^{+}(\mathrm{\Sigma })$ until every ${\exp }_a^{-1}q$ is future timelike as well. Its radial geodesic then joins $a$ to $q$ and stays in $V$. For $q\in W\cap U$, both endpoints lie in $U$, so causal convexity [L1] puts its entire radial segment in $U$. The connection-preserving property [L4] now gives $F_p(q)=\phi (q)$. For past boundary points repeat with reversed time orientation and an interior base point $a\in I^{+}(p)\cap U$.

(Openness and the boundary values.) If $q\in W\cap \mathrm{\partial }U$, take points $q_n\in W\cap U$ tending to $q$ using a countable coordinate-neighborhood basis. Then $\phi (q_n)=F_p(q_n)\to F_p(q)$. The limit cannot lie in $\phi (U)$: continuity of ${\phi }^{-1}$ there would force $q_n\to {\phi }^{-1}(F_p(q))\in U$, contradicting Hausdorff uniqueness of the limit $q\in \mathrm{\partial }U$. Thus $F_p(q)\in \mathrm{\partial }\phi (U)$, and the same sequence witnesses the neighborhood definition of correspondence. Hence $q\in C$ and $F_p(q)$ is its unique corresponding point. Each $p\in C$ has such a $W$, so $C$ is relatively open in $\mathrm{\partial }U$. Define $f$ on $A:=U\cup C$ by $f{\mathrm{\mid }}_U=\phi$ and the unique corresponding values on $C$. Each $F_p$ agrees with $f$ on $W\cap A$.

(Euclidean patching and common jets.) Use the open cover of ${\mathrm{\Omega }}_0:=U\cup \bigcup W$ consisting of $U$ with map $F_0=\phi$ and all admissible pairs $(W,F_p)$ of step 2.1. Index by all such pairs, so no arbitrary-index choice of one map per boundary point is needed. Obtain [L5] and a subordinate locally finite partition $({\rho }_i)$ from [L6]; write $H={\sum }_i{\rho }_i(e\circ F_i):{\mathrm{\Omega }}_0\to {\mathbb{R}}^9$. Each term is extended by zero off its cover member; subordinate support makes the extension smooth. At $q\in A$, every term whose support meets $q$ has $F_i(q)=f(q)$, so $H(q)=e(f(q))$. Moreover fix one local extension $F$ near $q$. On the open set $U$ in each overlapping domain one has $e{F}_i=eF=e\phi$. If $q\in C$, every coordinate derivative of their difference is zero at $q$ by continuity from $U$; if $q\in U$ the equality holds on a neighborhood. Local finiteness and ${\sum }_i{\rho }_i=1$ therefore show that every coordinate derivative of $H-eF$ vanishes at $q$: apply the finite Leibniz expansion to ${\sum }_i{\rho }_i(e{F}_i-eF)$. This proves equality of all jets at every point of $A$.

(Retraction and local invertibility.) The open set ${\mathrm{\Omega }}_1:=H^{-1}(T)$ contains $A$. Define $\mathrm{\Phi }=e^{-1}\circ r\circ H$ there. It is smooth and agrees with $f$ on $A$. For a local extension $F$ from step 4.1, the identity $e^{-1}\circ r\circ e\circ F=F$ holds near $q$; chain and Leibniz rules and the common jets from step 4.1 give equality of all jets of $\mathrm{\Phi }$ and $F$ at $q\in A$. In particular $d{\mathrm{\Phi }}_q=d{F}_q$ is invertible. Restrict to the open set $\mathrm{\Omega }\subseteq {\mathrm{\Omega }}_1$ where $d\mathrm{\Phi }$ is invertible; it contains $A$, and $\mathrm{\Phi }$ is a local diffeomorphism by [L7]. For each $p$, replace $W_p$ by $W_p\cap \mathrm{\Omega }$. The original exponential formula remains a local diffeomorphism there and agrees with $\mathrm{\Phi }$ on $W_p\cap A$. This proves (b), without asserting that the different exponential formulas agree on exterior overlaps. Countable Choice supplies [L5]-[L6], while Countable and Dependent Choice also accompany the boundary-structure supplier [L3].

(Metric, matter and connection.) On $U$, $\mathrm{\Phi }=\phi$ preserves the metric and matter by [L4], and the connection by its Koszul formula. The smooth tensor fields ${\mathrm{\Phi }}^{\ast }{g}^{\mathrm{\prime }}-g$, ${\mathrm{\Phi }}^{\ast }{\psi }^{\mathrm{\prime }i}-{\psi }^i$, and ${\mathrm{\Phi }}^{\ast }{\mathrm{\nabla }}^{\mathrm{\prime }}-\mathrm{\nabla }$ vanish on $U$. The pullback connection is defined because $\mathrm{\Phi }$ is a local diffeomorphism on $\mathrm{\Omega }$. All these tensors, and all their coordinate derivatives, vanish at every point of $C$ by continuity from $U$. Thus (c) holds, and the matter identities also hold to first covariant order at each point of $A$. No metric identity is asserted on the exterior of $\bar{U}$.

(Homeomorphism on the closure side.) The map $f$ is injective on $U$; on $C$ it is injective by applying uniqueness of corresponding points to the reversed common subdevelopment, and its images of $U$ and $C$ are disjoint because they lie respectively in $\phi (U)$ and its boundary. Write $C^{\mathrm{\prime }}$ for all corresponding boundary points on the primed side. Symmetry of the definition gives $f(A)=\phi (U)\cup C^{\mathrm{\prime }}$. Apply the local construction of steps 1.1-2.1 to the reversed pair. Near every point of $C^{\mathrm{\prime }}$ it supplies a continuous local extension agreeing with $f^{-1}$ on its part of $\phi (U)\cup C^{\mathrm{\prime }}$; on the open interior, the ordinary inverse ${\phi }^{-1}$ supplies it. Hence $f^{-1}$ is continuous in the subspace topology, while $f$ is continuous as the restriction of $\mathrm{\Phi }$. Thus $f:A\to f(A)$ is a homeomorphism. Restricting this homeomorphism to any embedded submanifold $S\subseteq A$ and using the invertibility of $d\mathrm{\Phi }$ proves that $\mathrm{\Phi }{\mathrm{\mid }}_S$ is an embedding.

(Geodesic transfer.) In coordinates, the acceleration of $\mathrm{\Phi }\circ \gamma$ is the image under $d\mathrm{\Phi }$ of the acceleration of $\gamma$, with the difference term precisely the pullback-connection difference of step 6.1. For a geodesic whose entire image lies in $A$, both vanish at each parameter value, so its image is an affinely parametrized geodesic. The metric identity of step 6.1 preserves the zero squared velocity of a null geodesic. This proves (d).

(Induced data.) The metric identity of step 6.1 restricts to the induced metric of $S$. Any unit normal $n$ is carried to the unit normal $d\mathrm{\Phi }(n)$ of $\mathrm{\Phi }(S)$; time orientation follows on $U$ from [L4] and at $C$ by continuity. Connection compatibility gives $g^{\mathrm{\prime }}({\mathrm{\nabla }}_{d\mathrm{\Phi }X}^{\mathrm{\prime }}d\mathrm{\Phi }n,d\mathrm{\Phi }Y)=g({\mathrm{\nabla }}_{X}n,Y)$ for tangent vectors $X,Y$, which is equality of the second fundamental forms. The matter identities and all their coordinate derivatives from step 6.1, together with the full jets of connection compatibility, identify every finite-order covariant derivative along $S$. Thus any prescribed complete finite-jet Cauchy data of the fixed model are carried as well. This proves (e), in particular the full restart-hypersurface data identification.