A bifurcate future horizon has an oriented defining pair near its section

In a boundary chart of ${\mathrm{\Sigma }}_{+}$ take $\rho \ge 0$, with $S=\{\rho =0\}$ and $d\rho \mathrm{\ne }0$ there. Let $q$ be the vector satisfying $h(q,V)=d\rho (V)$, obtained from the inverse positive-definite Gram matrix of $h$. Then $q$ is smooth, nonzero, orthogonal to $TS$, and $d\rho (q)=h(q,q)>0$. Thus $\nu =q\mathrm{/}\sqrt{h(q,q)}$ points into the specified section side. On overlaps these vectors agree, since a positive-definite three-space has exactly one unit vector orthogonal to $TS$ with the prescribed side. This is precisely the boundary-normal construction of [F1]. Together with the future unit normal $n$ it gives a smooth orthonormal frame of the normal plane of $S$. Apply [F1] to obtain $\mathrm{\Xi }$ and its uniform tube. The frame differential at $S$ is $(X,u,v)\mapsto X+un+v\nu$.

Fix either sheet $H$ and a slice function $f$ from [F6]. Nondegeneracy gives a unique nonzero field $L$ defined by $g(L,\cdot )=df$; its components are smooth by the inverse-matrix adjugate formula. The orthogonal complement of $TH=\ker df$ is the line spanned by $L$. Nullness means that this line is contained in $TH$, so it is its radical. For any smooth field $X$ tangent to $H$, the bracket $[L,X]$ is tangent along $H$: applying it to $f$ gives zero there, since both fields differentiate functions vanishing on $H$ to zero along $H$. Metric compatibility and torsion freedom therefore give, on $H$, $g({\mathrm{\nabla }}_{L}L,X)=-g(L,{\mathrm{\nabla }}_{L}X)=-g(L,{\mathrm{\nabla }}_{X}L+[L,X])=-\frac{1}{2}Xg(L,L)=0$. Here $g(L,X)$ and $g(L,L)$ vanish on $H$, so their tangential derivatives vanish. Hence ${\mathrm{\nabla }}_{L}L=\kappa L$ on $H$, with $\kappa$ smooth locally, by division by a nonzero component of $L$. This proves pregeodesicity without any event-boundary hypothesis.

Choose the sign of $f$ so that $L$ is future directed, and take its local integral curve $\gamma (r)$ in $H$ through $s\in S$, using [F7]. Given a desired initial vector ${\mathrm{\ell }}_s=c{L}_s$ with $c>0$, set $q(r)=c\exp (-{\int }_0^r\kappa (\gamma (\sigma ))d\sigma )$ and $t(r)={\int }_0^{r}q(\sigma {)}^{-1}d\sigma$. Since $dt\mathrm{/}dr=1\mathrm{/}q>0$, this is a smooth local reparameterization. For $\alpha (t(r))=\gamma (r)$ one has ${\alpha }^{\mathrm{\prime }}=qL$ and ${\mathrm{\nabla }}_{{\alpha }^{\mathrm{\prime }}}{\alpha }^{\mathrm{\prime }}=(q^2\kappa +qdq\mathrm{/}dr)L=0$, with initial tangent ${\mathrm{\ell }}_s$. Thus the affine geodesic with this initial tangent stays in $H$ locally. Uniqueness and scaling in [F4] identify it with $t\mapsto {\exp }_s(t{\mathrm{\ell }}_s)$. Negative $t$ gives the past half of the same local generator.

At $s\in S$ the radical of either sheet is normal to $T_{s}S$, hence lies in the plane spanned by $(n,\nu )$. A vector $un+v\nu$ is null exactly when $v=\pm u$. The two radical lines are distinct: otherwise that null line would lie in $T_s{H}_f\cap T_s{H}_p=T_{s}S$, contrary to spacelikeness. By the supplied future/outgoing label they are respectively $\mathbb{R}(n+\nu )$ and $\mathbb{R}(n-\nu )$. Step 2.1 identifies the maps $E_f(s,t)={\exp }_s(t(n+\nu ))$ and $E_p(s,t)={\exp }_s(t(n-\nu ))$ with maps into the respective sheets for small $t$ near each $s$. Their differentials at $(s,0)$ are $(X,r)\mapsto X+r(n\pm \nu )$, isomorphisms onto the sheet tangent spaces because a nonzero null vector cannot lie in $TS$. By [F8] their images contain sheet neighborhoods of $s$. In the injective tube coordinates these images are exactly $u=v$ and $u=-v$, respectively. The sheet neighborhoods, with the subspace topology from [F6], give ambient neighborhoods on which there are no other sheet pieces. Taking a finite cover of compact $S$ and shrinking the uniform tube makes these identities hold throughout $O$: the preimage under $\mathrm{\Xi }$ of their union is open and contains $S\times \{(0,0)\}$, so a finite product-neighborhood cover of $S$ supplies a common positive radius.

The positive $t$ half of $E_f$ is $u=v>0$, and the negative $t$ half of $E_p$ is $u=-v<0$. The section-side direction is $\nu$, whose coordinates at $S$ have $du(\nu )=0$ and $dv(\nu )=1$. In any boundary chart a vector pointing into ${\mathrm{\Sigma }}_{+}$ is a positive multiple of $\nu$ modulo $TS$; consequently both $a$ and $b$ increase to first order into this side. Smoothness and compactness give a common collar on which its interior has $a,b>0$. The local sector selected in the Statement is therefore $v>\mathrm{\mid }u\mathrm{\mid }$, or $a>0,b>0$, with future face $a=0,b>0$ and past face $b=0,a>0$. This identifies the supplied wedge, rather than deriving an exterior label from the horizon definition.

On $S$ the frame differential gives $du(n-\nu )=1$ and $dv(n-\nu )=-1$, whence $da(n-\nu )=-2$ and $db(n-\nu )=0$. The orientation is independent of the chosen future/outward orthonormal frame: another such frame has $n^{\mathrm{\prime }}=\cosh \chi n+\sinh \chi \nu$ and ${\nu }^{\mathrm{\prime }}=\sinh \chi n+\cosh \chi \nu$, for a smooth real function $\chi$ on $S$. Comparing the same initial normal vector in the two exponential maps gives $a^{\mathrm{\prime }}=e^{\chi }a$ and $b^{\mathrm{\prime }}=e^{-\chi }b$ on their common tube. The positive factors preserve the sector and half-sheet labels. These identities and steps 3.1 and 4.1 establish the defining pair and all stated conclusions.