An exposed boundary point: the last boundary point of a boundary null geodesic has no other boundary point in its past

(An initial boundary generator.) Choose $p_0\in C\cap J^{+}(\mathrm{\Sigma })$. Because $\mathrm{\Sigma }\subseteq U$, the splitting in [L1] gives $p_0\in I^{+}(\mathrm{\Sigma })$. If the displayed exposed-point property already holds, take $p=p_0$. Otherwise choose a distinct $q_0\in J^{-}(p_0)\cap \mathrm{\partial }U\cap J^{+}(\mathrm{\Sigma })$ and a causal curve joining them. Achronality [L2] and the null-versus-timelike alternative [L3] make this a null geodesic segment. Parametrize its maximal extension $\gamma :(\alpha ,\beta )\to M$ in the past direction with $\gamma (0)=p_0$ and $\gamma (1)=q_0$. By [L4] it meets $\mathrm{\Sigma }$ once, at a parameter $T>1$: a crossing at $T\le 1$ would imply $q_0\in J^{-}(\mathrm{\Sigma })\cap J^{+}(\mathrm{\Sigma })=\mathrm{\Sigma }$, impossible. All points $\gamma (t)$ for $0\le t<T$ lie in $I^{+}(\mathrm{\Sigma })$, because they are causally after $\gamma (T)\in \mathrm{\Sigma }$, are not on $\mathrm{\Sigma }$, and [L1] excludes the past side.

(No interior complement point below a future boundary.) Let $x\in \mathrm{\partial }U\cap I^{+}(\mathrm{\Sigma })$ and suppose $r\in \mathrm{int}(U^c)\cap J^{+}(\mathrm{\Sigma })$ with $r\le x$. Since $\mathrm{\Sigma }\subseteq U$, one has $r\in I^{+}(\mathrm{\Sigma })$. Choose a sufficiently small past timelike perturbation $r_{-}\ll r$ inside the open set $\mathrm{int}(U^c)\cap I^{+}(\mathrm{\Sigma })$. Push-up gives $r_{-}\ll x$. The open neighborhood $I^{+}(r_{-})\cap I^{+}(\mathrm{\Sigma })$ of $x$ meets $U$, at $s$ say. Concatenate a timelike curve from $\mathrm{\Sigma }$ to $r_{-}$ with one from $r_{-}$ to $s$. Past convexity [L2] puts this entire curve in $U$, contradicting $r_{-}\mathrm{\notin }U$. Thus no such $r$ exists. The perturbation is to the past, so the push-up direction is valid.

(Boundary-to-boundary causal segments stay on the boundary.) Let $y\le x$ be future boundary points and let $\eta$ be a future causal segment from $y$ to $x$. Every point $z$ of the segment lies in $I^{+}(\mathrm{\Sigma })$: some $\sigma \in \mathrm{\Sigma }$ has $\sigma \ll y\le z$. If $z\in U$, concatenate a timelike curve from $\mathrm{\Sigma }$ to $y$ with the segment from $y$ to $z$; past convexity [L2] would give $y\in U$. If $z\in \mathrm{int}(U^c)$, step 1.2 with $z\le x$ gives a contradiction. Thus every point lies in $\mathrm{\partial }U$. Achronality and [L3] further make every nonconstant such segment a null geodesic.

(The actual last boundary parameter before the Cauchy crossing.) Set $B:=\{t\in [0,T]:\gamma (t)\in \mathrm{\partial }U\}$. It is closed in the compact interval $[0,T]$, contains $[0,1]$ by step 2.1, and misses a neighborhood of $T$ because $\gamma (T)\in \mathrm{\Sigma }\subseteq U$. Whenever $t\in B$, step 2.1 applied to $\gamma (t)\le \gamma (0)$ puts $[0,t]$ in $B$. Therefore $B=[0,b]$ for its maximum $b$, with $1\le b<T<\beta$. In particular the whole segment $\gamma ([0,b])$ is future boundary, and no parameter in $(b,T]$ is a boundary parameter. This establishes the interval property instead of assuming it for a closed preimage.

(Follow only the initial connected component of corresponding points.) Let $J:=\{t\in [0,b]:\gamma (t)\in C\}$. It is relatively open and contains $0$ by [L5]. Define $s:=\sup \{t\in [0,b]:[0,t]\subseteq J\}$; relative openness at $0$ gives $s>0$, and every $0\le t<s$ belongs to $J$. On the interval $[0,s)$, the curve $f\circ \gamma =\mathrm{\Phi }\circ \gamma$ is a past null affine geodesic by [L5]. Let $\eta$ be its unique maximal geodesic extension in $M^{\mathrm{\prime }}$, with the same parameter and initial data at $0$. Local uniqueness and connectedness give $\eta (t)=f(\gamma (t))$ for all $0\le t<s$; equivalently the interval solution itself is a restriction of the maximal solution. Each such value is a corresponding future boundary point, and hence belongs to $K^{\mathrm{\prime }}:=J^{\mathrm{\prime }-}(f(p_0))\cap J^{\mathrm{\prime }+}({\mathrm{\Sigma }}^{\mathrm{\prime }})$, which is compact by [L6]. No connectedness of the entire open set $J$ has been assumed.

(Continuation of the image at the first possible exit.) The right endpoint ${\beta }^{\mathrm{\prime }}$ of the maximal domain of $\eta$ satisfies ${\beta }^{\mathrm{\prime }}\ge s$. If ${\beta }^{\mathrm{\prime }}=s$, its tail $\eta ([0,s))$ lies in $K^{\mathrm{\prime }}$. On the component containing that curve choose the auxiliary complete proper metric $h^{\mathrm{\prime }}$ from [L6]; $K^{\mathrm{\prime }}$ intersected with this component is compact, since connected components of a manifold are closed. Strong causality gives a uniform length bound for every subsegment $\eta ([0,t])$, $t<s$. Taking the supremum of these integrals gives finite $h^{\mathrm{\prime }}$-length of the entire tail. Since $s\le b<\mathrm{\infty }$, the compact-length affine extension theorem [L6] extends $\eta$ beyond $s$, a contradiction. Thus ${\beta }^{\mathrm{\prime }}>s$, and $\eta (s)$ is an actual smooth geodesic value with $\eta (t)\to \eta (s)$ as $t\uparrow s$.

(The first exit is still corresponding.) As $t\uparrow s$, the corresponding pairs $(\gamma (t),\eta (t))$ converge to $(\gamma (s),\eta (s))$. Their source values lie in $\mathrm{\partial }U$ and their target values in $\mathrm{\partial }\phi (U)$; both boundaries are closed. Step 3.1 already gives $\gamma (s)\in \mathrm{\partial }U$, and closedness gives $\eta (s)\in \mathrm{\partial }\phi (U)$. For any neighborhoods of these two limits, sufficiently late values of the pair lie in both neighborhoods; [L7] then supplies an interior witness for the same neighborhoods. Hence the limit pair is corresponding, and $s\in J$. If $s<b$, relative openness gives an interval to the right of $s$ in $J$, which together with $[0,s]\subseteq J$ contradicts the definition of $s$. Therefore $s=b$, and $p:=\gamma (b)$ belongs to $C\cap I^{+}(\mathrm{\Sigma })$. In particular all of $[0,b]$ lies in $J$. The map used at boundary points throughout is $f$ or $\mathrm{\Phi }$, never the interior map $\phi$ evaluated off its domain.

(Expose the final boundary point.) Suppose a distinct $q\in J^{-}(p)\cap \mathrm{\partial }U\cap J^{+}(\mathrm{\Sigma })$ existed. By step 2.1 a past causal segment from $p$ to $q$ is a null geodesic lying in the future boundary. Concatenate the past segment $\gamma {\mathrm{\mid }}_{[0,b]}$ from $p_0$ to $p$ with this past segment from $p$ to $q$. If the two tangent rays at $p$ differ, the resulting corner and [L3] yield $q\ll p_0$, contradicting achronality [L2]. If they agree, rescale the second affine parameter by a positive constant to match the nonzero velocities at $p$. Uniqueness of the geodesic equation [L4] identifies a short initial portion of the second segment with $\gamma ((b,b+\epsilon ])$, for some $0<\epsilon <T-b$. That portion lies in $\mathrm{\partial }U$ by step 2.1, contradicting step 3.1. Both alternatives are impossible. Thus $J^{-}(p)\cap \mathrm{\partial }U\cap J^{+}(\mathrm{\Sigma })=\{p\}$, with $p\in C\cap J^{+}(\mathrm{\Sigma })$ as required.