The truncated past height is Lipschitz and has exactly the pulled-back cuts as levels

If $\mathcal{R}$ is empty all conclusions are vacuous; hence assume it is nonempty. Put $U={\mathrm{Int}}_{\mathrm{\Sigma }}B$. In the supplied collar coordinates, compactness of $S$ and openness of $U$ give a uniform positive collar width with $c(S\times [0,\eta ])\subset U$; choose this before taking the localization data. The continuous spatial flow satisfies ${\psi }_0(x)=x$ on compact $\mathcal{R}\subset U$. The inverse image of $U$ under $(z,x)\mapsto {\psi }_z(x)$ is open and contains $\{0\}\times \mathcal{R}$. A finite subcover gives one positive time interval on which ${\psi }_z(\mathcal{R})\subset U$. Reduce ${\delta }_{\ast }\le {\delta }_0$ so that $\mathrm{\mid }z\mathrm{\mid }\le 2{\delta }_{\ast }$ lies in this interval and in the common flow domain. The stationary cylinder's inside is invariant under stationary translation; hence ${\mathrm{\Psi }}_z(\mathcal{R}\setminus S)\subset W$, while the extended flow fixes $S$. This containment is a separate compactness argument, not a converse to cut localization.

We verify the distance properties without applying a theorem for a connected boundaryless manifold to the whole collar. At any collar point choose nested coordinate balls, or half-balls at a boundary, with compact larger closure. The smooth positive definite matrix of $h_0$, evaluated on that compact set times the Euclidean unit sphere, has positive minimum $m_i^2$ and finite maximum $M_i^2$. Thus $m_i\mathrm{\mid }w\mathrm{\mid }\le \mathrm{\mid }w{\mathrm{\mid }}_{h_0}\le M_i\mathrm{\mid }w\mathrm{\mid }$ there. Straight coordinate segments in a convex ball or half-ball have length at most $M_i$ times endpoint displacement. Any path starting at the center and exiting a coordinate radius $r$ has length at least $m_{i}r$, by integrating up to first exit. These estimates show positivity of $d_0$ on distinct points and equality of its local topology with the manifold topology, also at either boundary. Reversal and concatenation of paths give symmetry and the extended triangle inequality. Piecewise $C^1$ reachability classes are open because balls and half-balls are convex; each such class has open complement. They are therefore precisely the connected components, and compactness leaves only finitely many of them. Consequently $d_0$ is finite within a component and infinite across distinct components. Truncation preserves the triangle inequality since $\min \{a+b,1\}\le \min \{a,1\}+\min \{b,1\}$. It also preserves positivity and all sufficiently small balls. Hence $d_{\mathcal{R}}$ is a finite metric with the asserted topology and value $1$ between distinct components.

Fix $x\in \mathcal{R}\setminus S$. Membership of ${\mathrm{\Psi }}_{s}x$ in $P$ is an initial interval of slab times, because its vertical curve is future timelike and $P$ is a past set. Openness excludes its upper endpoint when that endpoint is in the open slab. If $u(x)=z$ with $\mathrm{\mid }z\mathrm{\mid }<2\delta$, the supremum property gives members of $P$ on this fiber approaching ${\mathrm{\Psi }}_{z}x$, which is not itself in $P$, so ${\mathrm{\Psi }}_{z}x\in {\mathrm{\partial }}_{E}P$. Conversely, if ${\mathrm{\Psi }}_{z}x\in {\mathrm{\partial }}_{E}P$ at an open-slab time, every earlier point of this fiber belongs to $P$ by the chronological-closure property in [F3], and no later one can belong to $P$, since that would put ${\mathrm{\Psi }}_{z}x$ in $P$. Thus $u(x)=z$. By step 1.1 every such boundary hit lies in $W$ and therefore in $D_z$. By [F1] every point of $D_z$ has a unique preimage in the interior collar. This proves both inclusions in the level identity; points of $S$ give no open-slab level because their height is $-2\delta$.

The fixed inner margin in [F1] has no members of $P$ at any slab time, so the supremum is $-2\delta$ there, including $S$ by [F4]. On the fixed outer margin every slab time is in $P$, so the supremum is $2\delta$. Remove smaller such neighborhoods to leave a compact annulus strictly away from both collar boundaries. All nonconstant behavior occurs on this annulus.

Cover this compact interior annulus by finitely many smaller spatial coordinate balls with closures in larger convex coordinate balls away from $S$ and $S_{\eta }$. The maps $(s,y)\mapsto {\mathrm{\Psi }}_s(x(y))$ are smooth flow coordinates: $v=s$ and each spatial flow has smooth inverse. On the compact product of the closed slab and each larger ball, ${\mathrm{\partial }}_s$ is future timelike. Continuity and compactness give ${\epsilon }_i>0$ such that ${\mathrm{\partial }}_s+w$ is future timelike for every purely spatial coordinate vector with $\mathrm{\mid }w\mathrm{\mid }<{\epsilon }_i$. Thus a coordinate straight segment within this product, with positive time displacement exceeding $K_i$ times its spatial displacement, is future timelike for $K_i=1\mathrm{/}{\epsilon }_i$.

In one smaller ball suppose $u(x(y^{\mathrm{\prime }}))-u(x(y))>K_i\mathrm{\mid }{y}^{\mathrm{\prime }}-y\mathrm{\mid }$. Choose open-slab times $s,t$ with $u(x(y))<s<t<u(x(y^{\mathrm{\prime }}))$ and $t-s>K_i\mathrm{\mid }{y}^{\mathrm{\prime }}-y\mathrm{\mid }$; this is possible by the strict inequality, including when a height is clipped to a slab endpoint. By the initial-interval property, ${\mathrm{\Psi }}_{s}x(y)\mathrm{\notin }P$ and ${\mathrm{\Psi }}_{t}x(y^{\mathrm{\prime }})\in P$. The straight segment of step 4.1 joins the former to the latter by a future timelike curve in $E$, contradicting the past-set property. Interchanging the two points proves $\mathrm{\mid }u(x(y^{\mathrm{\prime }}))-u(x(y))\mathrm{\mid }\le K_i\mathrm{\mid }{y}^{\mathrm{\prime }}-y\mathrm{\mid }$. Together with the constant neighborhoods from step 3.1 this gives coordinate Lipschitz estimates on a finite cover of the whole collar.

Use a finite coordinate cover on which the estimates in step 5.1 or the constant-neighborhood estimates hold and the lower metric bounds in step 1.2 hold. Let $L$ be the maximum of their coordinate Lipschitz constants divided by their positive lower metric bounds, using zero on constant patches. Any piecewise $C^1$ path in $\mathcal{R}$ has a finite parameter subdivision with each subpath in one such patch. On a subpath, endpoint coordinate displacement is bounded by its Euclidean length, and this length is at most $1\mathrm{/}{m}_i$ times its $h_0$-length. The coordinate Lipschitz estimate and telescoping therefore give $\mathrm{\mid }u(x)-u(y)\mathrm{\mid }\le L{L}_{h_0}(\gamma )$. Taking the infimum gives $\mathrm{\mid }u(x)-u(y)\mathrm{\mid }\le L{d}_0(x,y)$ within each component. If $d_0(x,y)\ge 1$, including distinct components, boundedness of the defining supremum gives $\mathrm{\mid }u(x)-u(y)\mathrm{\mid }\le 4\delta$. Thus in all cases $\mathrm{\mid }u(x)-u(y)\mathrm{\mid }\le \max \{L,4\delta \}{d}_{\mathcal{R}}(x,y)$. This proves global Lipschitz regularity, and steps 1.1, 2.1 and 3.1 give all remaining conclusions.