Regular stationary event branches supply the local AIK horizon pair

Apply the event-boundary supplier in [F3] with the specified open end $E$, open set $O_0$ and exact identities for $H_{\pm }$ in the Statement. Its smoothness, nullness, spacelike-intersection and transversality hypotheses are precisely assumed here. Thus each full branch is achronal in $O_0$ and, near $S$, is generated by null geodesics orthogonal and transverse to $S$. Achronality remains true when the allowed timelike curves are confined to any smaller open neighborhood.

The supplied ${\mathrm{\Sigma }}_{+}$ is a smooth embedded spacelike three-manifold with boundary $S$ and positive induced metric up to that boundary. The compact normal-tube supplier in [F3] therefore gives the smooth orthonormal frame $(n,\nu )$ along $S$, with $n$ the future unit normal of $\mathrm{\Sigma }$ and $\nu$ the inward unit boundary normal, and the diffeomorphism $\mathrm{\Xi }(s,a,b)={\exp }_s(a{n}_s+b{\nu }_s)$ on $\mathrm{\mid }a\mathrm{\mid }+\mathrm{\mid }b\mathrm{\mid }<ϵ$. Shrink its radius so its image lies in $O_0$. The common end, exact intersection, two-sided hypersurface and inward collar inclusion assumed here are all the hypotheses of the axis lemma in [F4]. After its common shrinking, in the resulting tube $O$ one has $H_{+}\cap O=\{a-b=0\}$, $H_{-}\cap O=\{a+b=0\}$, $I^{-}(E)\cap O=\{a-b<0\}$ and $I^{+}(E)\cap O=\{a+b>0\}$.

The tube and chronological-boundary setting of the quadrant lemma in [F4] have now been established with the same $E$, branches and frame. Put $\alpha =a+b$ and $\beta =a-b$. That lemma gives $I_O^{+}(S)=\{\alpha >0,\beta >0\}$ and $J_O^{+}(S)=\{\alpha \ge 0,\beta \ge 0\}$, including reflexive reachability at $S$; the past sets have both inequalities reversed. Since $\mathrm{\Xi }$ is a diffeomorphism, the relative boundary of the closed future quadrant is its two nonnegative axes, and its difference from the open future quadrant is the same set. The analogous statement holds for the nonpositive axes in the past quadrant. These are the two relative-boundary and $J_O^{\pm }(S)\setminus I_O^{\pm }(S)$ identities required in [F1], with curves confined to $O$ throughout.

In this tube the full axes are parametrized by $t\mapsto \mathrm{\Xi }(s,t,t)$ and $t\mapsto \mathrm{\Xi }(s,t,-t)$, with $\mathrm{\mid }t\mathrm{\mid }<ϵ\mathrm{/}2$. By the exponential-map definition and scaling law in [F6], these curves are ${\exp }_s(t(n_s+{\nu }_s))={\gamma }_{s,n_s+{\nu }_s}(t)$ and ${\exp }_s(t(n_s-{\nu }_s))={\gamma }_{s,n_s-{\nu }_s}(t)$, respectively. Thus $t$ is an affine parameter, with respective initial velocities $n_s+{\nu }_s$ and $n_s-{\nu }_s$. The event-boundary and axis constructions identify their images with the null generators of the branches. Both initial directions are future-directed; the increasing affine parameter is future-directed on each generator. Injectivity of $\mathrm{\Xi }$ makes each generator meet $S$ exactly at $t=0$, an interior point of its connected parameter interval. Thus the nonnegative and nonpositive axes in step 3.1 are exactly the future and past halves of the supplied full sheets, with no extra generator or boundary component.

The tube $O$ is connected because it is diffeomorphic to the product of the connected sphere with the connected normal diamond, and it inherits the ambient orientation. The branches are embedded smooth null hypersurfaces without boundary in $O$, intersecting transversely exactly in $S$, and they are achronal by step 1.1. The assumed zero expansions on both full branches persist under restriction to $O$. Together with steps 3.1–4.1 these verify every condition of the regular bifurcate non-expanding pre-rigidity pair in [F1]. No branch-generating Killing field or global symmetry-extension assertion entered the argument.