Lipschitz causal height and level-set Jacobian bound

Put $R=R_0{q}^{-2\mathrm{/}3}$ in the physical comparison end, choosing $R_0$ large enough that this physical sphere and its compact original truncation are supplied. By [F29], let $C_{\mathrm{e}\mathrm{x}\mathrm{t}}={\mathrm{\Sigma }}_0\cap \mathcal{E}$ and $B_0=C_{\mathrm{e}\mathrm{x}\mathrm{t}}\cup S$, keeping its adapted topological chart separate. The section in [F8] is a connected smooth individually admissible section, with interior graph diffeomorphism $G:C_{\mathrm{e}\mathrm{x}\mathrm{t}}\to {\mathrm{\Sigma }}^{\circ }$; its physical normalized end is the one in [F27]. We first establish the needed end-metric convergence in the actual asymptotic chart, for this fixed $q>0$. Put $c_q=d_q\mathrm{/}(4q)$, $y=x_q=qx$ and $s=t_q=c_{q}t$, and let $G_{\mu \nu }$ denote the components of $g_q$ in $(s,y)$. The original metric obeys $g_{00}\to -1$, $g_{0i}=O(r^{-2})$ and $g_{ij}\to {\delta }_{ij}$ by regularity. By [F28], $D_q\to 4q^2>0$ uniformly in angle and $(W_q{)}_i=(1-q{)}^2(W_0{)}_i-4q(1-q)g_{0i}=O_q(r^{-2})$. Write $f=-g_{00}$. Substituting $W_q(T)=d_q-D_q$ in the tensor formula and rescaling coordinates gives $G_{00}=4q^2{g}_{00}\mathrm{/}{D}_q$, $G_{0i}=g_{0i}-f(W_q{)}_i\mathrm{/}{D}_q$, and $G_{ij}=(D_q{g}_{ij}+g_{0i}(W_q{)}_j+(W_q{)}_i{g}_{0j}-f{D}_q^{-1}(W_q{)}_i(W_q{)}_j)\mathrm{/}(4q^2)$. Thus $G_{00}\to -1$, $G_{0i}\to 0$ and $G_{ij}\to {\delta }_{ij}$ uniformly in angle. This uses the supplied smooth decaying primitive, without identifying it with the exact radial integral. The normalized graph slope tends to zero by [F8] and [F27]. Apply [F30] to the smooth graph parameterization $j(y)=({\tilde{u}}_q(y),y)$ and the coordinate scalings; its tangent vectors are ${\mathrm{\partial }}_i+({\mathrm{\partial }}_i{\tilde{u}}_q){\mathrm{\partial }}_s$. By the induced-metric definition in [F7], its components in $y$ are $G_{ij}+G_{0i}{\mathrm{\partial }}_j{\tilde{u}}_q+G_{0j}{\mathrm{\partial }}_i{\tilde{u}}_q+G_{00}{\mathrm{\partial }}_i{\tilde{u}}_q{\mathrm{\partial }}_j{\tilde{u}}_q$, so they tend to the Euclidean identity. Hence $h_0\ge \frac{1}{2}{\sum }_i(d{x}_q^i{)}^2$ sufficiently far out. Since $r=\mathrm{\mid }{x}_q\mathrm{\mid }\mathrm{/}q$ and $q>0$ is fixed, $\mathrm{\mid }dr{\mathrm{\mid }}_{h_0}\le \sqrt{2}\mathrm{/}q$ there. Apply the universal physical-core conclusion in [F26] to this same individually admissible section, using a lower positive parameter bound $q\mathrm{/}2$ if applying its constant-sequence formulation. Set $B=\mathrm{\Sigma }\setminus G(\{r_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}>R\})=S\cup G(B_{0,R}\setminus S)$ in the near-horizon definition. This core is compact and contains a neighborhood of $S$, has smooth outer boundary $\mathrm{\Gamma }=G(\{r_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=R\})$, and has compact positive-distance trims. Put $\mathrm{\Omega }=({\mathrm{int}}_{\mathrm{\Sigma }}B)\cap {\mathrm{\Sigma }}^{\circ }$. If $\mathrm{\Omega }$ had a separation, its two pieces would define a nonconstant continuous map to the two-point discrete space. At every point of the manifold boundary $S\bigsqcup \mathrm{\Gamma }$ of $B$, a sufficiently small boundary half-ball has connected interior by [F31], so that map extends uniquely and locally constantly to $B$. The extensions agree on overlaps because those interiors are dense. Its restriction to $\mathrm{\Gamma }$ is constant: the sphere parameterization pulls it back to a continuous map on connected $S^2$. Extend that constant to the physical graph tail, identified by the graph diffeomorphism. Boundary half-balls at $\mathrm{\Gamma }$ make this extension continuous on the whole connected section $\mathrm{\Sigma }$, contradicting [F31]. Hence $\mathrm{\Omega }$ is connected. Adding $S$ preserves connectedness: any separation of $\mathrm{\Omega }\cup S$ would restrict to a separation of $\mathrm{\Omega }$, and every relative neighborhood of a point of $S$ meets $\mathrm{\Omega }$. Thus ${\mathrm{int}}_{\mathrm{\Sigma }}B$ is connected. The same half-balls show $B={\overline{{\mathrm{int}}_{\mathrm{\Sigma }}B}}^{\mathrm{\Sigma }}$. Its outer sphere lies on exactly the physical comparison cylinder; hence the inside component containing $S$ in stationary projection is ${\mathbb{R}}_v\times \mathrm{\Omega }$, where $\mathrm{\Omega }=({\mathrm{int}}_{\mathrm{\Sigma }}B)\cap {\mathrm{\Sigma }}^{\circ }$. No equality of the physical radius and an adapted topological radius is asserted.

The maximal-section and positive-lapse suppliers give a smooth spacelike section up to compact $S$, $N>0$ inside, $N=0$ at $S$, and $T{\mathrm{\mid }}_S\in TS$. The causal escape supplier constructs a smooth intrinsic boundary-distance collar. Choose a sufficiently small width $\eta$ and put $\mathcal{R}=\{0\le \rho \le \eta \}$. The compact normal-flow lemma supplies one common smooth flow $V=Nn=T-X$ on $\mathcal{R}$, fixing $S$, with $v({\mathrm{\Psi }}_{z}x)=z$ for interior $x$ and uniform inverse-flow two-dilation $1+ϵ(\delta )$, $ϵ(\delta )\to 0$. Smoothness of the ambient metric and embedded spacelike section up to $S$ is supplied by the section and transformed-geometry interfaces; no extension of the exterior product across the horizon is assumed.

First shrink $\eta$ so that $\mathcal{R}\subset {\mathrm{int}}_{\mathrm{\Sigma }}B$, using step 1.1 and compactness of $S$; the flow and dilation bounds of step 1.2 restrict to this smaller collar. Choose $0<b<\eta \mathrm{/}4$ sufficiently small for the compact-core supplier. By step 1.1, $B\cap \{\rho \ge b\}$ is compact in ${\mathrm{\Sigma }}^{\circ }$. The actual original maximal section $C_{\mathrm{e}\mathrm{x}\mathrm{t}}$ and product in [F29], together with the regular boundary-extension and physical-end data in [F2], meet every original-metric hypothesis of [F32]. The far-tail lemma therefore gives outgoing access from every original exterior point to each sufficiently far selected original stationary tail. Choose the free comparison constant beyond this background threshold before fixing the target. Thus the outgoing-access premise of [F11] is satisfied independently of any transformed chronological-access conclusion. The eventual-access supplier places the compact trim $B\cap \{\rho \ge b\}$ entirely in $I_{\mathcal{E}}^{-}(C_t)$ for all sufficiently late $t$; choose $t$ to satisfy the zero-time localization supplier as well. The target is a nonempty compact smooth spacelike sphere on the comparison cylinder. That cylinder is timelike, and its stationary $T$ flow is future timelike by the eventual-access supplier. Since $Tv=1$, increasing $t$ makes ${\min }_{C_t}v=t+{\min }_{C_0}v>0$. The causal escape supplier gives $\mathrm{\mid }d\rho \mathrm{/}dv\mathrm{\mid }\le K_{\rho }\rho$ for a fixed constant $K_{\rho }\ge 0$ in a stationary collar $0<\rho <a_0$; choose $a_0$ small enough that the target projects outside this collar. Thus the compact regular closed base $B$, its smooth outer boundary $\mathrm{\Gamma }$, the collar inclusion, compact-trim entry, temporal $v$ and causal collar estimate meet the hypotheses of the translated-cut localization lemma with $Q=C=C_t$. It supplies a common time interval on which all positive- and negative-time cuts $D_z$ are compact and lie in ${\mathrm{\Psi }}_z$ of a fixed compact interior annulus of $\mathcal{R}$, with fixed inner neighborhoods outside the chronological past and outer neighborhoods inside it. Fix this late target and shrink $\delta$ so that the common time bound holds and ${\min }_{C}v>2\delta$. This supplies the nonzero-time localization without inferring it from the zero-time cut.

The chronological past is a past set, and each interior normal flow line is future timelike. Apply the truncated-height lemma to obtain the Lipschitz threshold, its two constant boundary values, and the exact level identity for every interior slab time. Its local cone argument is carried out only on a compact annulus where $V$ is timelike; near $S$ the height is constant. Thus degeneration $V{\mathrm{\mid }}_S=0$ does not invalidate the Lipschitz argument or any flow chart.

Global hyperbolicity, temporal $v$, compact $C$ and each compact $D_z$ now satisfy the normalized-null-flow endpoint lemmas with $v-z$. Use the same maximal past null-normal geodesic family supplied by the zero-time endpoint result. The past-boundary generator theorem applies at every $p\in D_z$ because it is inside the cylinder and $v(p)=z<{\min }_{C}v$; its final tangent is precisely the prescribed outward normal. The selected-endpoint lemma therefore gives compact/Borel $A_z$, coverage of $D_z$ and local smooth transverse $F_z$ for every small $z$. At time zero the localization and endpoint suppliers identify these data with $D,A,F$ of the near-horizon definition. For negative $z$, the selected-endpoint lemma's boundary-cylinder argument proves $A_z\subset A$ and identifies the intermediate zero-time hit with $F$. These data meet the optical Jacobian supplier's hypotheses, which give $0\le J_2{F}_z\le J_{2}F\le 1$ on $A_z$.

Compose each endpoint map with the inverse normal flow and apply the multiplicity-aware area-transfer lemma. Its image is exactly the level set from step 3.1, and its area formula retains all endpoint multiplicities. The uniform two-dilation factor from step 1.2 gives the first inequality; the pointwise optical comparison and $A_z\subset A$ give the second. Set $o_{\delta }(1)=ϵ(\delta )$ to obtain the stated limits for fixed target, collar and parameter. The argument applies to every $z\in (-\delta ,-\delta \mathrm{/}2)$, with no unjustified regular-value or injectivity assumption.