Maximal bifurcation-boundary section under the original regular hypotheses

The section has one Euclidean end and a compact interior with boundary $S$, and its data agree with the original stationary end. The stationary field is tangent to $S$ and equals ${\mathrm{\partial }}_t$ in the end. Use order five and any fixed Hölder exponent in $(0,1)$: the sixth differentiated end bounds imply the required order-five weighted Hölder bounds. Together with the ambient construction and the identified domain of dependence, every class-(b) and exclusion hypothesis of the existing CW input is supplied. In the source convention, $U=\mathrm{int}{D}_{M_{\ast }}^{\mathrm{t}\mathrm{i}\mathrm{m}}(C_0\cup S)$, with all inextendible timelike curves taken in the glued ambient. The timelike-curve conclusion in F2 gives exactly $U=E$ and $(B\cup W)\cap U=\mathrm{\varnothing }$. This checks the source exclusion for its own domain, without assuming that a causal-curve domain and a timelike-curve domain coincide in general.

Apply the existence input with the hypotheses checked in step 1.1. It gives smooth maximal sections ${\widehat{\mathrm{\Sigma }}}_s$ with fixed boundary $S$, interiors $C_s$ covering $E$ and Cauchy there in the source convention, and the stated end graph bounds. For the Cauchy assertion use only the source conclusion that $C_s$ is achronal and every inextendible timelike curve in $E$ meets it exactly once. Vacuum gives zero Ricci tensor, hence the timelike-convergence condition and the stationary foliation conclusion. This invokes the nonlinear existence theorem as an external prerequisite and does not reconstruct it or infer stronger end gauge from its logarithmic graph estimate.

Fix $s$. The hypersurface $C_s$ is closed in $E$: if $p\in {\overline{C_s}}^E\setminus C_s$, a maximal timelike geodesic through $p$, taken in $E$, meets $C_s$ at a point $q\mathrm{\ne }p$ by the source Cauchy conclusion. Thus $q{\ll }_{E}p$ or $p{\ll }_{E}q$. The corresponding open chronological neighborhood of $p$ meets $C_s$ at another point, contradicting achronality. Moreover $D_E^{\mathrm{t}\mathrm{i}\mathrm{m}}(C_s)=E$ by that same source conclusion. Apply F4 with $W=E$ and $A=C_s$, obtaining $D_E^{\mathrm{c}\mathrm{a}\mathrm{u}\mathrm{s}}(C_s)=E$. By the continuous-curve interface in F5, every inextendible locally absolutely continuous causal curve in $E$ is among the continuous causal curves tested by this imported conclusion: its image is unchanged and its ends still have no point limit in $E$. Consequently every declared causal curve meets $C_s$.

The intersection counts exactly one parameter value. Indeed, near any $p\in C_s$ choose slice coordinates $(f,x)$ with $C_s=\{f=0\}$. Since $C_s$ is spacelike, the coordinates can be chosen with ${\mathrm{\partial }}_f$ future timelike and $\mathrm{\nabla }f$ past timelike at $p$, and both properties persist after shrinking to a product coordinate box. For a future causal curve $\gamma$ through $p$ at parameter $a$, temporal monotonicity in F5 gives $f(\gamma (t))>0$ on a short interval after $a$. A point $z=\gamma (t)$ on that interval is chronologically above some $r\in C_s$, using the coordinate fiber with $x$ fixed from $f=0$ to $f(z)>0$. If a later parameter $b>a$ also gave $q=\gamma (b)\in C_s$, choose $a<t<b$ in that interval. Push-up gives $r{\ll }_{E}z{\le }_{E}q$, hence $r{\ll }_{E}q$, contradicting achronality when $r\mathrm{\ne }q$ and strong causality when $r=q$. Thus two distinct parameter visits are impossible. Together with step 3.1, this establishes the declared causal Cauchy condition.

The local right germ at $S$ is shared with the original spacetime. Now that $C_s$ is Cauchy in the declared sense, the general Cauchy-closure interface applies in the original geometry and excludes further original-ambient accumulation points away from the double boundary $S$. The fixed smooth boundary supplied in step 2.1 already includes all of $S$, so the closure boundary is exactly $S$. All maximality, stationary covariance, foliation and end bounds from step 2.1 are retained.