Ordered timelike flow graphs lie on the corresponding geometric side

Fix a stated contact, and write $c=f(p)=g(p)$ and $q=\varphi (c,p)$. At a boundary contact, [F6] applies to exactly the collar, smooth heights, flow domain and variable-time embeddings assumed here. Thus both local images have boundary $S$, pass through $p=q$, and have the same tangent hyperplane and inward image half-space. At an interior contact both images pass through $q$ and have no boundary locally. Their spacelike full tangent spaces contain no nonzero timelike vector.

Choose ambient coordinates $(t,z^1,z^2,z^3)$ at $q$ whose ${\mathrm{\partial }}_t$ at $q$ is future timelike; a linear change of any chart achieves this. Shrink the chart so ${\mathrm{\partial }}_t$ remains future timelike. Projection $\pi (t,z)=z$ has kernel $\mathbb{R}{\mathrm{\partial }}_t$, so its derivative restricts to an isomorphism on each of the spacelike three-dimensional tangent spaces at $q$. In local parametrizations of the two images, extend their smooth coordinate maps across the boundary using [F3]. Their differentials still have rank three at the boundary by [F2]. Apply [F4] to the three-dimensional maps obtained by composing these extensions with $\pi$. This expresses the two images as $t=a(z)$ and $t=b(z)$, with smooth extensions of $a,b$ to open spatial neighborhoods. At the boundary their projected boundaries are the same embedded surface $\pi (S)$, since both fix $S$ and $d\pi$ is injective on their tangent spaces. Their projected inward tangent sides agree by step 1.1. To identify the domains, use the spatial coordinates furnished by the first projected parametrization: its boundary is $z^3=0$, and the second projected boundary is this same surface; its interior has $z^3>0$ by the equal inward tangent sides and continuity of the nonzero transverse derivative. After shrinking, both projected image domains are therefore the same half-neighborhood $D$ with $z^3\ge 0$. At an interior contact their domains instead contain a common open $D$. This construction uses an ordinary timelike coordinate direction, irrespective of the vanishing of $K$ on $S$.

On an ambient neighborhood $W$ of $q$, define $P(t,z)=t-b(z)$ using the smooth extension from step 2.1. Its differential is nonzero, has $dP({\mathrm{\partial }}_t)=1$, and at $q$ has the spacelike kernel $T_q{F}_g(C)$. After shrinking $W$, the kernel remains spacelike: its smooth spanning vectors ${\mathrm{\partial }}_{z^i}+{\mathrm{\partial }}_{i}b{\mathrm{\partial }}_t$ have a positive definite Gram matrix at $q$, and positive definiteness persists near $q$. Consequently $dP$ cannot vanish on any timelike vector anywhere in $W$. On each connected future timelike cone it has constant sign, and that sign is positive because it takes value one on ${\mathrm{\partial }}_t$. Thus $dP(V)>0$ for every future timelike $V$ in $W$, including away from the actual half-graph. In particular $P\ge 0$ describes its local future side; no $K$-adapted chart or positive value of $dP(K)$ on $S$ is asserted.

Choose a compact interval $J=[c-\epsilon ,c+\epsilon ]$ with $\epsilon >0$. At a boundary contact the flow fixes $p$ at all times, since the constant curve at $p$ is an integral curve and [F5] gives uniqueness. Thus $J\times \{p\}$ lies in $\mathcal{D}$ and maps to $q$. At an interior contact choose $\epsilon$ small enough that $J\times \{p\}\subseteq \mathcal{D}$ and $\varphi (J,p)\subseteq W$, using openness and continuity at $(c,p)$. In either case a finite cover of the compact set $J\times \{p\}$ by product neighborhoods contained in $\mathcal{D}\cap {\varphi }^{-1}(W)$ gives a neighborhood $V$ of $p$ with $J\times V\subseteq \mathcal{D}$ and $\varphi (J,V)\subseteq W$. Continuity of $f,g$ permits a smaller collar neighborhood on which both heights lie in $J$. Therefore, for every nearby interior base point $x$, its entire flow segment from $g(x)$ to $f(x)$ stays in $W$. No estimate over an unbounded set of times has been used.

For such an interior $x$, put $v_x(s)=P(\varphi (s,x))$. It is smooth on a neighborhood of $[g(x),f(x)]$, and the flow equation and chain rule give $v_x\mathrm{\prime }(s)=d{P}_{\varphi (s,x)}(K_{\varphi (s,x)})>0$ by the exterior timelikeness hypothesis and step 3.1. Its initial value is zero because $F_g(x)$ lies on $t=b(z)$. If $f(x)>g(x)$, [F7] gives $P(F_f(x))>0$; if they are equal, this value is zero. By continuity, $P(F_f(x))\ge 0$ also holds at nearby boundary base points. Restrict the parametrization neighborhoods further so their image patches in step 2.1 come from the base neighborhood just used. Every point $(a(z),z)$ with $z\in D$ consequently has $P=a(z)-b(z)\ge 0$. This is the claimed future-side ordering on the common whole or half-neighborhood and proves the conclusion.