Proper stationary projection for continued slices

The existing section is already a global stationary graph $j(x)=(u_{\ast }(x),x)$ in the complete stationary product of [F1]. Its tangent vectors are $(d{u}_{\ast }(v),v)$, so $T=(1,0)$ cannot be tangent: its zero spatial component would force $v=0$ and hence a zero time component. By the normal decomposition in [F6], $N_{\ast }=-g_{q_{\ast }}(T,n_{\ast })$ therefore never vanishes in the interior. It is smooth, and that interior is connected as the graph over the connected original exterior. On the supplied end $Y=T$ is future timelike, so its contraction with the future unit normal gives $N_{\ast }>0$ there. A continuous nowhere-zero real function on a connected space cannot take both signs: its positive and negative sets would be disjoint nonempty open sets partitioning that space. Thus $N_{\ast }>0$ everywhere inside. This establishes the reference sign using the supplied end field and graph formula before applying any compact-core or transformed-lapse theorem. The first clause of the existing-graph helper in [F1] also gives the complete product based on ${\mathrm{\Sigma }}_{\ast }$ and its global projection, independently of collar estimates or any new graph conclusion for ${\mathrm{\Sigma }}_q$.

For compact exhaustion, apply the collar clause of the existing-graph helper in [F1], with reference positivity from step 1.1 and original positivity from the original-lapse result. The completed boost model in [F2] represents the original and existing sections as bounded-rapidity graphs. The local boost/Taylor argument in that helper makes $K$ future timelike on the small outer collar of either spacelike section; positive boost rate is unchanged in future/outward normal bases. Thus the positive stationary lapses give the unique crossing on each, and their full small boundary collars correspond under the stationary quotient with comparable radial boundary coordinates. Use the adapted identification $\chi$ of [F9] on the original section; it extends smoothly across $S$ to the usual closed spherical exterior. Thus the usual-compactification hypothesis of the collar helper holds in these coordinates. Write $a=\mathrm{\mid }\chi (\cdot )\mathrm{\mid }$ on $B$ and $J:C\to {\mathrm{\Sigma }}_{\ast }^{\circ }$ for the existing graph map from step 1.1. The adapted annuli lift under $J$ to compact annular truncations between the full reference collar and the adapted graph end $J(\{a>A\})$. This conclusion concerns adapted coordinates only.

Match the independent end charts without equating their maps. For a sufficiently large physical radius $R$, $B_R$ is compact by [F9], so the continuous $a$ has a finite maximum there. Any adapted tail beyond that maximum lies in $V_R$. Conversely, $K_A=\{a\le A\}\subset B$ is compact for every $A>1$ by [F9]. Cofinality of the physical truncations gives an $R$ with $K_A\subset B_R$, hence $V_R\subset \{a>A\}$. Transporting both inclusions by $J$ proves cofinality of the two graph ends; rescaling $x_{q_{\ast }}=q_{\ast }x$ changes only the positive radius threshold. Removing a full reference collar and a sufficiently far physical graph end now leaves a closed subset of one adapted compact annular truncation from step 2.1, so it is compact by [F9]. Adding the compact closed smaller reference collar, including $S$, gives the compact physical-end complement required by the lapse theorem. For the later one-sheet application choose a smaller original boost collar $Q_0$ with compact closure, as in the collar helper. The continuous $a$ is bounded on that closure by [F9]. The full correspondence in step 2.1 takes $Q_0$ onto a reference collar $Q_{\ast }$, so $a_{\ast }=a\circ J^{-1}$ is bounded on $Q_{\ast }$. This is precisely a bounded target-collar radius in the adapted coordinates, regardless of its physical coordinate description. These facts use no transformed-lapse conclusion.

The reference is supplied smooth up to $S$, connected, spacelike and maximal. Step 3.1 verifies its compact-core exterior structure, while the assumed reference derivative estimates in [F3] imply its normalized end gradient tends uniformly to zero. Thus the transformed-lapse theorem in [F4] now applies to the reference, supplying its lapse equation, nonnegative potential, zero boundary value and positive end limit. The boundary-control supplier in [F4] gives a strictly positive inward derivative and, on a fixed smaller collar, $N_{\ast }\ge c{\rho }_{\ast }$ for some $c>0$. These constants belong to the fixed supplied reference.

By [F3], $F_w$ is an admissible flow graph and equals stationary translation on the end. On a compact interior core, sufficiently small graphs remain in the open exterior. At the boundary, the scaled-collar helper in [F5] places $\mathrm{\mid }s\mathrm{\mid }\le \eta {\rho }_{\ast }$ between the two smooth null event-boundaries, so $\mathrm{\mid }w\mathrm{\mid }\le ϵ{\rho }_{\ast }$ keeps the punctured graph in the exterior there as well. The bound on $w$ follows by integrating its small radial derivative from $w{\mathrm{\mid }}_S=0$. Hence the global projection from step 1.1 is defined on all of ${\mathrm{\Sigma }}_q^{\circ }$. The supplied graph has $C^{4,\alpha }$ regularity up to the fixed smooth boundary, solves the maximal equation in the smooth metric $g_q$, and has $w{\mathrm{\mid }}_S=0$. Thus [F7] makes it smooth up to $S$ and throughout the interior. It is connected and has one exterior end because it is an embedded flow graph over the connected reference with the physical compact truncations from step 3.1. On the end, $Y=T$ means its original-time height is $u_{\ast }+w$. The assumed reference derivative estimates and the weighted bound $Dw=O(r^{\beta -1})$ imply that its normalized height has spatial gradient tending uniformly to zero; the positive time and spatial normalization factors are constant on each slice. Its end is therefore an explicit stationary graph with the end-slope condition required by [F4]. The positive-lapse theorem now applies to this smooth supplied maximal candidate without assuming its interior is already a stationary graph. It makes $T$ transverse. In the product coordinates from step 1.1, the differential of the stationary projection has kernel exactly $\mathbb{R}T$. Its restriction to the three-dimensional tangent space of the candidate is injective by transversality and therefore is an isomorphism onto the three-dimensional target tangent space. The smooth inverse function theorem in [F8] consequently makes this restriction a local diffeomorphism.

In the reference $Y$ chart, retain the tangential term: $T=a{\mathrm{\partial }}_s+b{\mathrm{\partial }}_{{\rho }_{\ast }}+W$. The scaled coefficient and uniform-return lemmas give $a\ge (c\mathrm{/}2){\rho }_{\ast }$ and $\mathrm{\mid }b\mathrm{\mid }\le B{\rho }_{\ast }$ throughout the chosen wedge. They follow the moving $y$ on compact $S$, control ${\rho }_{\ast }$ by Grönwall, and force a meeting with $s=0$ within $4ϵ\mathrm{/}c$, with uniformly comparable boundary distances. The global existing-graph projection identifies this actual local meeting with $P(x)$. Thus a base sequence approaching $S$ has projected points approaching $S$, even though its stationary flow moves tangentially.

On the physical graph end, $P(x)=x$ exactly because $Y=T$. Step 3.1 puts a sufficiently far adapted graph end inside this physical end, so the identity holds on the adapted end required by the helpers as well. The compact truncations from steps 2.1 and 3.1 and the boundary comparison in step 6.1 give properness by the properness helper. The target is the actual connected stationary graph, with adapted diffeomorphism $J\circ {\chi }^{-1}:\{\mathrm{\mid }y\mathrm{\mid }>1\}\to {\mathrm{\Sigma }}_{\ast }^{\circ }$. Apply the finite-sheet theorem in these adapted coordinates. Shrink the source collar until its return image lies in the bounded-adapted-radius target collar $Q_{\ast }$ from step 3.1. The remaining middle annulus is compact, and $a_{\ast }\circ P$ has a finite maximum there by [F9]. A target point beyond both this maximum and the collar bound, and in the adapted identity end, therefore has no preimage in the collar or middle annulus and has exactly itself as preimage in the end. These are the explicit hypotheses of the one-sheet helper in [F5]. The sheet number is consequently one, and the smooth local inverses give a global diffeomorphism in the sense of [F8]. Composing with the known existing graph and original stationary product gives the asserted global $T$-graph of the continued slice.