Lemma 4.3: localization of the causal cut near S

The section supplier [F8] gives a connected smooth spacelike maximal section with included compact boundary $S$, an interior stationary graph $G:C\to {\mathrm{\Sigma }}^{\circ }$, and normalized end decay through four derivatives. Thus it is individually admissible at this fixed $q>0$. The actual original maximal exterior section $C$ and its physical truncations are supplied by [F24]. On the physical end, pulling back the positive-parameter metric bounds in [F9] with the decaying graph slope gives an induced metric uniformly comparable to Euclidean in $x_q=qx$; hence $\mathrm{\mid }d{r}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{\mathrm{\mid }}_{h_q}$ is bounded there. Apply [F15]–[F16] to this same section, taking a lower parameter bound $q\mathrm{/}2$ if using its constant-sequence formulation. Put $R=R_0{q}^{-2\mathrm{/}3}$ in the actual physical chart, with the free comparison constant large enough for the physical truncation. The resulting core is $B=S\cup G(B_{0,R}\setminus S)$, whose outer boundary is the actual physical graph sphere $\mathrm{\Gamma }=G(\{r_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=R\})$ on the comparison cylinder. It is compact, contains a neighborhood of $S$, and has compact positive-distance trims. Its inside component is the one containing $S$: the physical tail and its sphere are connected, and an additional component of the smooth core avoiding that sphere would be both open and closed in the connected section. These facts identify the supplied inside region in [F4] without a global physical radial coordinate or any extension of stationary height through $S$. For interior points use the Riemannian topology input in [F23]. At a boundary point choose a coordinate half-ball with a smaller closed half-ball on which $c\mathrm{\mid }\xi {\mathrm{\mid }}^2\le h(\xi ,\xi )\le C\mathrm{\mid }\xi {\mathrm{\mid }}^2$, for fixed $0<c\le C<\mathrm{\infty }$; such bounds follow from smooth positive coefficients up to the boundary in [F8]. Coordinate line segments inside the convex half-ball give $d_h(p,p^{\mathrm{\prime }})\le \sqrt{C}\mathrm{\mid }x(p)-x(p^{\mathrm{\prime }})\mathrm{\mid }$ for sufficiently close points. Conversely, any curve from its center leaving the smaller half-ball spends at least $\sqrt{c}$ times the coordinate exit radius in length before leaving. Hence points at sufficiently small intrinsic distance remain in that chart, and the same lower bound on curves staying there gives coordinate convergence. This proves equality of the two topologies also at $S$, without assuming the boundaryless theorem already covers boundary points.

Put $P_t=I_{g_q}^{-}(C_t)$. The actual original maximal section and stationary product in [F24], with the regular extension and end data in [F2], match the original-metric hypotheses enumerated in [F25]. Apply that far-tail lemma to obtain outgoing access for every original exterior point to every sufficiently far selected stationary end. Take the free comparison constant beyond this background tail threshold before fixing $C_t$. This establishes the new access premise of [F10] before its use. Eventual-access supplies open increasing pasts and pointwise entry for every point of this section interior. The trimmed-core lemma therefore gives, for all sufficiently small $\eta >0$, one threshold after which $B\cap \{\rho \ge \eta \mathrm{/}2\}\subset P_t$. In particular a neighborhood of the outer graph sphere lies in $P_t$. To check the extrema used here and below, put $R=R_0{q}^{-2\mathrm{/}3}$ and $\mathrm{\Gamma }=G(\{\mathrm{\mid }x\mathrm{\mid }=R\})$. By [F16], $\mathrm{\Gamma }$ is compact; it is nonempty since $R>1$ and $G$ is defined on that sphere. Give $\mathrm{\Gamma }$ the subspace metric from $d_{h_q}$, whose topology is the manifold subspace topology by [F23]. Give each $C_t$ the metric defined by Euclidean distance between its spatial coordinates; the supplied stationary end chart identifies its metric topology with its manifold topology. The smooth bijection $J_t:\mathrm{\Gamma }\to C_t$, $J_t(G(x))=(t,x)$, is continuous for these metrics. Thus [F22] makes every $C_t$ nonempty and compact. The smooth section time $v$ restricts to a continuous real function on $C_t$, so [F21] supplies finite attained extrema. Since $v(J_t(G(x)))=t+v(J_0(G(x)))$, increase the threshold until ${\min }_{C_t}v=t+{\min }_{C_0}v>0$.

Take the smooth boundary collar supplied by [F11], and let $U$ be its intersection with the neighborhood where its causal estimate holds, the interior of $B$ in $\mathrm{\Sigma }$, and $\mathrm{\Sigma }\setminus \mathrm{\Gamma }$. This is an open neighborhood of compact $S$ by step 1.1. Write $d=d_{h_q}$. By the boundary topology comparison in step 1.1, for each $s\in S$ choose $r_s>0$ with $B_d(s,2r_s)\subset U$, and choose finitely many $B_d(s_j,r_{s_j})$ covering $S$. Put $\delta ={\min }_j{r}_{s_j}>0$ and fix $0<a<\delta$. If $\rho (x)\le a$, the definition of distance gives $s\in S$ with $d(x,s)<\delta$; some covering ball contains $s$, so $d(x,s_j)<\delta +r_{s_j}\le 2r_{s_j}$ and $x\in U$. The triangle inequality gives $\mathrm{\mid }\rho (x)-\rho (y)\mathrm{\mid }\le d(x,y)$ by taking infima over $S$, so $\rho$ is continuous. Hence the closure of $\{\rho <a\}$ lies in $\{\rho \le a\}\subset U\subset B$, is compact by [F20], and avoids $\mathrm{\Gamma }$. No smoothness of the distance sublevel is needed: it lies in the supplied smooth collar, where [F11] gives $\mathrm{\mid }{\rho }^{\mathrm{\prime }}\mathrm{\mid }\le C\rho$. For fixed $t$, [F21] and step 2.1 give $0<{\min }_{C_t}v\le {\max }_{C_t}v=V_t<\mathrm{\infty }$ and a nonempty compact target $C_t$, whose stationary projection is $\mathrm{\Gamma }$, outside $\{\rho <a\}$. The stationary product is supplied by [F6], and $v$ is temporal by [F13]. Thus every hypothesis of [F18] holds, and it gives $\{v=0,0<\rho <a{e}^{-C{V}_t}\}\cap P_t=\mathrm{\varnothing }$. The definition separately keeps $S$ in $K$. Since this sublevel lies in $B$, $K$ contains a full section neighborhood of $S$.

The set $P_t\cap {\mathrm{\Sigma }}^{\circ }$ is open in $\mathrm{\Sigma }$, since $P_t$ is exterior-open and the section interior is open in $\mathrm{\Sigma }$. The boundary convention in [F4] keeps $S$ outside $P_t$, so $K=B\setminus P_t$ is closed in $B$. By the topology comparison in step 1.1, $B$ is a compact metric subspace for the intrinsic distance, and [F20] therefore makes $K$ compact. By step 2.1, $K$ has no points with $\rho \ge \eta \mathrm{/}2$. It is closed in $\mathrm{\Sigma }$ because $B$ is the closed inside region of [F4]; hence its relative boundary $D$ in ${\mathrm{\Sigma }}^{\circ }$ is contained in $K$. By step 3.1, $D$ has no points with $\rho <a{e}^{-C{V}_t}$. Choose $0<{\eta }_t<\min (\eta \mathrm{/}2,a{e}^{-C{V}_t}\mathrm{/}2)$ sufficiently small for [F16]. That supplier makes $B\cap \{\rho \ge {\eta }_t\}$ a compact subset of ${\mathrm{\Sigma }}^{\circ }$. The relative boundary $D$ is closed in ${\mathrm{\Sigma }}^{\circ }$ and contained in this subset, so it is closed in this compact metric subspace and compact by [F20]. The stronger bounds just proved give $D\subset \{{\eta }_t<\rho <\eta \}$.

A point of $D$ is inside the cylinder because its outer graph sphere has a section neighborhood in $P_t$. Locally $K$ is the complement of $P_t$ on the section, so it lies in ${\mathrm{\Sigma }}^{\circ }\cap {\mathrm{\partial }}_{\mathcal{E}}{P}_t$. Conversely, any point $p$ of that ambient boundary on the inside section has $v(p)=0<{\min }_{C_t}v$. Apply [F12] to $p$ in this same construction, with the same supplied section and comparison cylinder: the inside condition and strict section-time inequality just checked give a future-directed boundary null segment from $p$ to $C_t$. The local slice-intersection lemma projects short initial points back along a normalized future timelike local flow, producing section points in $P_t$ tending to $p$ by push-up. Since $p\mathrm{\notin }{P}_t$, it is in $D$. This proves equality with the asserted portion of the ambient past boundary. All boundaries and causal curves here are relative to $\mathcal{E}$; no ambient extension across the horizons, event-horizon identification or affine completeness is invoked.