Indexed compact heights and uniform smooth horizon collars

If the indexed family is empty, all candidate estimates are vacuous; choose a smaller background reference collar from [F2]. An empty compact target set also has a vacuous height bound. For the path argument assume the family and the target set are nonempty. Let $Q⋐{\mathrm{\Sigma }}_1^{\circ }$ be compact, identified with $\{0\}\times Q$ using [F1]. Apply the original product result in [F1] to the actual maximal exterior section: its closure is a connected smooth spherical exterior with compact boundary $S$, its induced positive metric is smooth up to $S$, its physical end has $h_{ij}={\delta }_{ij}+O_6(r^{-1})$ and compact truncations, and its original lapse is positive. The regular data in [F6] give the ambient smooth spacelike extension across $S$, tangency of $T$ there, and the remaining stationary end bounds. Shrink only the coordinate tail past the compact-truncation threshold, retaining the same exterior product. These are the hypotheses of [F10]. Choose a fixed physical radius $R_{\ast }$ sufficiently large for that input and so that $r\ge R_{\ast }$ belongs to the common normalized end in [F1] for every $q\in [\delta ,1]$. For each $y\in Q$, [F10] supplies a future timelike path from $(0,y)$ to a point $b_y$ with $r>R_{\ast }$. Apply the same input with reversed time orientation, stationary coordinate $a=-t$, field $T^{\mathrm{\prime }}=-T$, and normal $n^{\mathrm{\prime }}=-n$. Its lapse is still $-g(T^{\mathrm{\prime }},n^{\mathrm{\prime }})=N>0$, its shift is $-X$, its second fundamental form is $-k$ and it remains maximal; $g_{00}$ and $g_{ij}$ are unchanged and the mixed end components change sign. All decay, compactness, smoothness and boundary-tangency hypotheses are therefore retained. Reversing the resulting curve gives a future original-metric timelike path from a point $a_y$ with $r>R_{\ast }$ to $(0,y)$. Both paths lie entirely in the original exterior and both endpoints lie in the common end-height region. No equality of chronological exteriors after end truncation or timelikeness on the entire originally designated end is assumed.

Subdivide each path into finitely many convex normal neighborhoods contained in $\mathcal{E}$ and replace each subarc by its smooth future timelike geodesic, using [F6]. Near $(0,y)$ keep the penultimate incoming and the first outgoing points fixed and vary only the joining geodesic to $(0,z)$. The exponential-cone characterization and chronological openness give a neighborhood of $y$ where both geodesics remain future timelike. Shrink to a neighborhood with compact closure inside that neighborhood. The varying geodesics and their velocities are smooth on this compact endpoint set times $[0,1]$, and all other pieces are fixed. A finite subcover of $Q$ therefore gives finitely many piecewise smooth path families ${\gamma }_z^{-}:a_i\to (0,z)$ and ${\gamma }_z^{+}:(0,z)\to b_i$, with fixed end endpoints, compact image and uniformly bounded coordinate velocities on their finitely many pieces. This construction, rather than openness alone, controls the integrals as endpoints vary.

For either path family put $A={\sup }_{\gamma }\int \mathrm{\mid }{W}_0({\gamma }^{\mathrm{\prime }})\mathrm{\mid }$ and $B={\sup }_{\gamma }\int \mathrm{\mid }{T}^{\mathrm{♭}}({\gamma }^{\mathrm{\prime }})\mathrm{\mid }$, integrating over all its pieces. These constants are finite: the forms are smooth on the fixed compact images and the coordinate velocities and number of pieces have common bounds from step 2.1. The explicit family in [F7] gives $\int \mathrm{\mid }{W}_q({\gamma }^{\mathrm{\prime }})\mathrm{\mid }\le A+4B$ for every $q\in [\delta ,1]$. Since $d_q\ge 1$, every partial correction ${\alpha }_q(s)=-d_q^{-1}{\int }_0^s{W}_q({\gamma }^{\mathrm{\prime }})$ satisfies $\mathrm{\mid }{\alpha }_q(s)\mathrm{\mid }\le A+4B$. Apply the transfer formula in [F1] on each smooth piece, retaining the accumulated correction at each join; a constant stationary translation preserves the metric and future direction. The finite concatenation remains future timelike by [F6]. This proves a uniform correction bound for the actual chosen family, without deducing parameter regularity from a separate fixed-$q$ choice.

For a candidate graph let $F(t,x)=t-u(x)$. Extend its future normal $n$ and positive lapse $N=-g_q(T,n)$ to its stationary translates, taking $n^{\mathrm{♭}}=g_q(n,\cdot )$. Since $F(T)=1$ and its level sets are those spacelike translates, $dF=-n^{\mathrm{♭}}\mathrm{/}N$ throughout the product; thus $F$ increases along each transferred future timelike path. Write $a_i=(t_i^{-},z_i^{-})$ and $b_i=(t_i^{+},z_i^{+})$. The transferred incoming path ends at $({\alpha }_q^{-},z)$, giving $u(z)\le u(z_i^{-})+{\alpha }_q^{-}-t_i^{-}$; the transferred outgoing path starts at $(0,z)$ and ends at $(t_i^{+}+{\alpha }_q^{+},z_i^{+})$, giving $u(z)\ge u(z_i^{+})-t_i^{+}-{\alpha }_q^{+}$. The finitely many $t_i^{\pm }$ are fixed, step 3.1 bounds the corrections, and the common normalized end bound in [F1], with $u=4q\tilde{u}\mathrm{/}{d}_q$ from [F8] and $d_q\ge 1$ from [F7], bounds all endpoint heights. Hence $\mathrm{\mid }u\mathrm{\mid }$ is uniformly bounded on $Q$. Only existing indexed pairs were used.

Each candidate has its smooth individual K-height by the fixed-q result in [F2]. Shrink the reference collar so its closed collar has all its axial orbits in the common local symmetry neighborhood of [F9]: the smooth circle action preserves compact $S$, and continuity uniformly over its compact period interval gives such a smaller collar. The union $C$ of the outer reference boundaries over $q\in [\delta ,1]$ is compact and disjoint from $S$, by the smooth common boundary-distance maps in [F2]. Its axial saturation $A=\{{\varphi }_s^{Z}x:x\in C,0\le s\le 2\pi \}$ is compact and lies in that local neighborhood. It also lies in the exterior collar domain: on the shrunken collar the local fields agree with the complete extended fields of [F2], whose flows preserve their exterior domain. The stationary projection of $A$ is therefore compact in ${\mathrm{\Sigma }}_1^{\circ }$, and its stationary time coordinate is bounded. For $F(t,x)=t-u(x)$, commutation and $K=T+{\mathrm{\Omega }}_{H}Z$ give $F({\varphi }_s^{K}x)=s+F({\varphi }_{{\mathrm{\Omega }}_{H}s}^{Z}x)$. Periodicity and step 4.1 bound the second term uniformly, using that compact projection and bounded time coordinate. At the candidate intersection this expression is zero, so the outer K-height is uniformly bounded. Apply the strict barriers $c\mp M\rho$ of [F2] to the maximum of $u^K+M\rho$ and minimum of $u^K-M\rho$. The contact lemma includes S: equal boundary flow times give tangent planes because K vanishes there, and its strict curvature signs rule out an extremum beyond the outer bound. Thus $\mathrm{\mid }{u}^K\mathrm{\mid }\le C$ down to S for every indexed pair.

Use [F3] with this common height bound. The normal boost gives a common boundary angle; the Gaussian flow estimates give a strict cone and radial comparison. Projection is locally invertible, proper onto one fixed target collar and one-sheeted by its individual boundary germ, giving common zero-boundary normal graph domains. All these generic lemmas concern candidates actually supplied, not a requirement that every background parameter have a candidate.

The compact smooth background controls the local observer tensors, so the full-range logarithmic estimate in [F4] applies to each candidate. Its cutoff lemmas give a uniform normal-angle bound on a smaller common collar. The Gaussian graph identity then bounds the height and gradient in a compact spacelike jet cone. The coefficient extension in [F4] gives a common full uniformly elliptic jet box and bounded forcing, while the smooth fixed S supplies uniform boundary charts and the zero Dirichlet data. With the Hölder-norm convention in [F8], apply [F5] for a common C1-beta seed, then the seeded boundary bootstrap in [F4] for all finite higher estimates. Thus no uniform initial C4 norm or missing-parameter section has been assumed.