Lemma 3.2: compact stationary-height bounds

The case $K=\mathrm{\varnothing }$ is immediate; assume $K\mathrm{\ne }\mathrm{\varnothing }$. Fix $\mu \in (0,1)$ and the uniform normalized end radius $R$ from [F4]. The original product supplier in F1 applies to the actual exterior section $C={\mathrm{\Sigma }}_0\cap \mathcal{E}$, whose original-metric maximality is an explicit Statement premise. It gives the stationary product over that section; the supplied original graph convention identifies its base with ${\mathrm{\Sigma }}_1^{\circ }$. Independent topological and physical end charts are not equated. Choose $R_f$ beyond the threshold of the two-sided access premise, inside the specified physical end, and with $R_f>R\mathrm{/}\delta$, and put $U_f=\mathbb{R}\times \{r>R_f\}$ using the physical stationary chart. By the chart match in [F4], the stationary-product projection of a point of $U_f$ is the physical end point $(0,y)$ with the same spatial coordinate $y$ and $\mathrm{\mid }y\mathrm{\mid }>R_f$. This is also the coordinate used to evaluate the normalized graph height. The outgoing half of that premise gives a future original-metric timelike path from every $(0,x)$ to $U_f$, entirely in $\mathcal{E}$. This uses the explicit access premise rather than treating access to all far tails as part of regularity.

The function ${\tau }_q(t,x)=t-u_q(x)$ has spacelike level sets, since these are stationary translates of the given graph. Its differential annihilates the level-set tangents and evaluates to $1$ on $T$. Extending the future unit normal by stationary translation, positive lapse gives $d{\tau }_q=-g_q(n_q,\cdot )\mathrm{/}{N}_q$, where $N_q=-g_q(T,n_q)>0$. Consequently $d{\tau }_q(v)>0$ for every future timelike $v$: in a Lorentzian orthonormal frame the inner product of two future timelike vectors is negative. Hence ${\tau }_q$ strictly increases along each smooth piece of a transferred curve, and therefore along the full finite concatenation.

The incoming half of the explicit two-sided access premise gives, for every $(0,x)$, a future original-metric timelike path from $U_f$ to $(0,x)$, entirely in $\mathcal{E}$. Thus both directions of path access used in the compact-family argument are supplied in the same metric and unchanged exterior. No time-reversal deduction of a missing access premise, and no equality of separate coordinate maps, is used.

All curves just obtained are contained in $\mathcal{E}$ by the explicit access premise. Also, any ambient future timelike curve between two exterior points stays in the exterior: an end point chronologically preceding its first endpoint and an end point following its last endpoint can be concatenated to its subarcs, putting every intermediate point in both chronological sets defining $\mathcal{E}$. For either of the curves at a fixed $x$, cover its compact parameter interval by inverse images of strictly convex normal neighborhoods contained in the open $\mathcal{E}$. A Lebesgue number gives a finite subdivision whose subarcs each lie in one neighborhood. Restriction and positive affine reparametrization put each subarc in the curve class of [F7]; its cone characterization makes the joining initial vector future timelike. Replace that subarc by the unique joining geodesic in the same neighborhood. Finite concatenation gives a piecewise smooth future timelike curve in $\mathcal{E}$ with the original endpoints.

For the incoming curve at $x$, fix its penultimate vertex and vary its final endpoint from $(0,x)$ to $(0,z)$ in the last convex normal neighborhood. For the outgoing curve, vary the first endpoint to $(0,z)$ and fix the next vertex in the first such neighborhood. The smooth two-point inverse and geodesic evaluation in [F7] give smooth families of these terminal and initial geodesic pieces. Their initial vectors remain future timelike for $z$ close to $x$, since the future timelike cone is open. Keep all other pieces fixed. Choose an open coordinate neighborhood $V_x$ of $x$ whose closure is a closed coordinate ball contained in this common validity region; it may also be chosen inside the chronological endpoint neighborhoods of [F5]. These are incoming and outgoing piecewise smooth curves for every $z\in {\bar{V}}_x$, with the end endpoints fixed.

By compactness choose finitely many $V_x$ covering $K$, and retain the corresponding pairs of curve families. Every varying piece is smooth on a neighborhood of the compact parameter set $[0,1]\times {\bar{V}}_x$; fixed pieces have compact parameter interval. Their images form a compact subset of $\mathcal{E}$: pull an open cover back to each parameter set, take a finite subcover, and then take the finite union over pieces and families. The stationary time coordinate on these curves is uniformly bounded. Moreover the scalar evaluations $W_0({\gamma }^{\mathrm{\prime }})$ and $T^{\mathrm{♭}}({\gamma }^{\mathrm{\prime }})$ are continuous on those compact parameter sets and hence uniformly bounded, by [F8]. The number of pieces is bounded because only finitely many families were retained. End endpoints have only finitely many spatial positions in $U_f$.

The actual exterior section is maximal by the Statement, its complete stationary product is [F1], and the selected smooth stationary Harrison family is [F2]. Thus these are the revised transfer supplier's hypotheses in [F3], with no identification of the physical end chart and the hypersurface identification. Apply the transfer result to each smooth piece after positive affine reparametrization to $[0,1]$. Translate each subsequent transferred piece by the time correction accumulated on the preceding pieces. Stationarity preserves the integrands and future cones, so these pieces join to future $g_q$-timelike curves with the original spatial paths. Their total time corrections are uniformly bounded: each piece contributes $-d_q^{-1}\int W_q({\gamma }^{\mathrm{\prime }})$, and [F2] expresses its integrand through the bounded evaluations of $W_0$ and $T^{\mathrm{♭}}$ from step 5.1, with uniformly bounded coefficients. There are uniformly finitely many pieces, and $d_q^{-1}$ is uniformly bounded. The initial points are unchanged and both endpoint times are uniformly bounded.

At each of the finitely many end base points, let $y$ be its physical Cartesian coordinate, and use $u_q(y)$ for the graph function expressed in that chart. The endpoints belong to $U_f$, so the matched coordinates in [F4] and step 1.1 give $\mathrm{\mid }y\mathrm{\mid }>R_f$ and $r_q=q\mathrm{\mid }y\mathrm{\mid }\ge \delta \mathrm{\mid }y\mathrm{\mid }>\delta R_f>R$. The end estimate and $u_q(y)=(4q\mathrm{/}{d}_q){\tilde{u}}_q(qy)$ therefore bound $\mathrm{\mid }{u}_q(y)\mathrm{\mid }$ uniformly in $q$. For a transferred future curve from an end point $(t_{-},y_{-})$ to $(t_{+},x)$, step 1.2 gives $u_q(x)<u_q(y_{-})+t_{+}-t_{-}$. A transferred future curve from $(s_{-},x)$ to an end point $(s_{+},y_{+})$ similarly gives $u_q(x)>u_q(y_{+})+s_{-}-s_{+}$. The endpoint times and transfer corrections are uniformly bounded by steps 5.1 and 6.1. These two inequalities give a common bound for $\mathrm{\mid }{u}_q(x)\mathrm{\mid }$ for all $x\in K$ and $q\in [\delta ,1]$.