First cylinder contact, outward source branch and the focusing contradiction

We first derive the end bounds for the actual smooth primitive in [F10]. Stationarity and [F2] give $F=\frac{1}{2}d{T}^{\mathrm{♭}}=O_5(r^{-2})$. Cofactor expansion verifies that the inverse metric is the transposed cofactor matrix divided by its determinant, which tends to $-1$; the Hodge formula in [F10] uses these entries and the volume coefficient $\sqrt{-\det g}$ from [F33]. Repeated coordinate differentiation using [F30] therefore gives $\mathcal{F}=O_5(r^{-2})$ and $d\sigma =O_5(r^{-2})$. For each unit vector $\omega$, apply the fundamental theorem on $[r,b]$ to the real and imaginary parts of $\sigma (s\omega )$, bound their derivatives by a constant times $s^{-2}$ using monotonicity, and let $b\to \mathrm{\infty }$ using the supplied normalization. This gives $\sigma -1=O(r^{-1})$ uniformly in $\omega$; the bounds for $d\sigma$ give all spatial derivatives through order six, so $\sigma -1=O_6(r^{-1})$. For fixed $q>0$, [F29] then gives $D_q=4q^2+O_6(r^{-1})$, $(W_q{)}_i=O_5(r^{-2})$ and $(W_q{)}_0=d_q-D_q$. Put $c_q=d_q\mathrm{/}(4q)>0$, $t_q=c_{q}t$, $y=x_q=qx$, $r_q=\mathrm{\mid }y\mathrm{\mid }$ and let $G_{\mu \nu }$ be the normalized metric components. Direct substitution in [F29], with $f=-g_{00}$ and $W_i=(W_q{)}_i$, gives $G_{00}=4q^2{g}_{00}\mathrm{/}{D}_q$, $G_{0i}=g_{0i}-f{W}_i\mathrm{/}{D}_q$ and $G_{ij}=(D_q{g}_{ij}+g_{0i}{W}_j+W_i{g}_{0j}-f{W}_i{W}_j\mathrm{/}{D}_q)\mathrm{/}(4q^2)$. The denominators have positive limits. Iterating the same product, quotient and chain rules through order five yields $G_{00}=-1+O_5(r_q^{-1})$, $G_{0i}=O_5(r_q^{-2})$ and $G_{ij}={\delta }_{ij}+O_5(r_q^{-1})$. In this calculation ${\mathrm{\partial }}_y=q^{-1}{\mathrm{\partial }}_x$, with $q$ fixed; no parameter-uniform bounds or exact radial gauge are used. Finally $T=c_q{\mathrm{\partial }}_{t_q}$ is future timelike far out by [F9].

We verify that sufficiently distant outer regions have no hidden boundary in the exterior. Work first on the original section $C$ with its positive metric $h$ in [F31]. Choose an original end radius $a$ so large that $h\ge \frac{1}{2}\delta$ for $r\ge a$, and write $V_a=\{r>a\}\subset C$. Its coordinate cut $S_a=\{r=a\}$ is compact and closed in $C$ by [F32]. If $p\in {\overline{V_a}}^C\setminus S_a$, choose a relatively compact coordinate ball $V$ about $p$, convex in those local coordinates and disjoint from $S_a$. Take $z\in V\cap V_a$. A straight coordinate segment from $z$ to any point of $V$ has uniformly bounded $h$-speed, by the compact-coordinate comparison in [F32]. As long as the segment is in $V_a$, its end coordinate speed satisfies $\mathrm{\mid }dx\mathrm{/}ds\mathrm{\mid }\le \sqrt{2}\mathrm{\mid }\dot{\gamma }{\mathrm{\mid }}_h$. At a proposed first exit, this bound and the fundamental theorem in [F30] make its end coordinates Cauchy, hence convergent to a finite point of ${\mathbb{R}}^3$ by completeness in [F32]. That limit has radius at least $a$; radius $a$ would put the segment on the excluded $S_a$, and radius greater than $a$ continues it inside the open end chart. Thus no exit is possible: $V\subset V_a$, with end radius bounded on $V$ by $r(z)$ plus a fixed multiple of its coordinate diameter. It follows that ${\overline{V_a}}^C=V_a\cup S_a$ and ${\mathrm{\partial }}_C{V}_a=S_a$; the points of $S_a$ are boundary points by their end chart. Every point of $C$ has a neighborhood on whose intersection with $V_a$ the radius is bounded: use the end chart on $V_a\cup S_a$ and the complement of its closure elsewhere. A finite subcover shows that the radius on any compact subset of $C$ is bounded wherever that subset meets $V_a$. The projection $\pi :\mathcal{E}\to C$ is smooth by [F31], so the projection of any compact source is compact by [F32]. Choose a fixed larger normalized radius $R_1$ above the bound for that projection and above the far-end threshold in [F24]. The product agrees with the designated end because the section is original time zero there and both flows have velocity $T$. Consequently, for every $R\ge R_1$, the region $U_R=\{r_q>R\}\subset \mathcal{E}$ has closure $\{r_q\ge R\}$ and boundary the entire cylinder $\{r_q=R\}$, with the source disjoint from its closure. Its closed outer region is covered by a slightly larger end chart. This argument uses no compatibility between the global section chart and the end chart.

The candidate section and the preserved horizon data in [F7] give a future/outward normal frame on compact S. The future outgoing ray is tangent to the local future branch: this agrees with the original future branch, and preservation of its null direction and the future orientation in [F7]–[F9] keep the label unchanged; replacing the reference spacelike section by the chosen one changes its future/outward frame by a boost. Since T is tangent to the spacelike S, choose the local tube with compact closure so small that $g_q(T,T)>-c_q^2\mathrm{/}4$ there, with $c_q$ from step 1.1. Apply [F13]–[F15] in this tube, restricting their common short affine interval if necessary. Their local defining pair and initial-data calculation show that every sufficiently small outward graph is in the exterior and that its inward null normal reaches the local future branch before leaving this fixed tube. The graph family in [F3] is trapped for all sufficiently small epsilon, so choose one epsilon meeting both requirements and then fix its delta*. Use step 2.1 for this single compact graph A to fix $R_1$ once and for all.

Step 1.1 gives $g_q(T,T)=c_q^2{G}_{00}<-c_q^2\mathrm{/}2$ far out, uniformly in stationary time. Thus every sufficiently distant target cylinder is disjoint from the compact tube chosen in step 3.1. By [F16], an exterior-contained source-normal segment to such a target cannot have the inward source ray. This proves the precise local branch exclusion needed here, without asserting a global transformed event-horizon theorem.

Fix one sufficiently large normalized R, above $R_1$, and set the free comparison constant to $R_0=R{q}^{-1\mathrm{/}3}$. Its original comparison radius $R_0{q}^{-2\mathrm{/}3}=R\mathrm{/}q$ is exactly the desired radius. At fixed q, taking R sufficiently large meets the comparison-range threshold in [F18]. The cuts in [F11] are therefore $C_s=\{r_q=R,t_q=c_{q}s\}$, with the same original stationary flow parameter $s$ used by the section time. Step 1.1 and [F24] make them compact smooth spacelike cuts, with a timelike cylinder and temporal $t_q$ on a neighborhood of its closed outer region. The source lies inside in the precise sense of step 2.1. The actual original maximal section in [F31] has the positive lapse, smooth compact-boundary spherical exterior, physical compact truncations and complete original stationary product specified in [F34]. Its ambient boundary extension, tangent field and end bounds are the regular data [F2]. Thus the far-tail lemma in [F34] applies in the original metric and proves outgoing access for every point of $\mathcal{E}$ to every sufficiently far original tail. Choose the present free target radius beyond that threshold as well. This verifies the additional outgoing-access hypothesis of [F18] before applying it. Apply [F18] to the compact graph $A\subset {\mathrm{\Sigma }}_q^{\circ }$: it supplies finite chronological access to this cylinder through the original-path transfer in [F4], without assuming that smooth-path transfer applies to every absolutely continuous curve. The flow parameter map is smooth and defined for all real times by [F31]. The section time in [F5]–[F6] satisfies $\tau {\mathrm{\mid }}_A=0$ and $\tau ({\varphi }_s^{T}x)=s+\tau (x)$, and the exterior is globally hyperbolic. All hypotheses of [F19] now hold, so it supplies an earliest contact $p_R$. Compactness here is of A, the target cut and a bounded-time cylinder slab, rather than of the full cylinder.

By [F20], $p_R$ lies on the future chronological and causal boundary of A and has strictly positive section time; no sign of the stationary cut parameter is needed. Apply [F21], which reverses time and uses compact target A with temporal function minus the section time. It supplies a null segment from A to $p_R$ with source orthogonality and no interior focal point measured from A. The local target variation in [F22] makes the same segment orthogonal to the target cut. Its additional outward-ray hypotheses hold by steps 2.1 and 4.2: the cylinder bounds $U_R$, A is outside its closure, $t_q$ is temporal there, $T=c_q{\mathrm{\partial }}_{t_q}$, and the very same cuts have $t_q=c_{q}s$. Thus the final ray is outward. The segment remains wholly in the open exterior. Step 4.1 excludes its inward source ray, so [F17] fixes its affine initial tangent to be exactly n plus nu.

The metric is stationary by [F9], so [F23] conserves the Killing energy along these affinely normalized segments, including any part in an ergoregion. The smooth outgoing source field on fixed compact A gives $0<E\le E_{\max }$ for every segment arriving at the end, by [F25]. The algebraic null-cone estimate [F24] applies to the actual normalized end of step 1.1. For the fixed radius $R_1$ chosen in step 3.1, step 2.1 verifies both additional last-passage hypotheses: $A\cap {\overline{U_{R_1}}}^{\mathcal{E}}=\mathrm{\varnothing }$ and ${\mathrm{\partial }}_{\mathcal{E}}{U}_{R_1}=\{r_q=R_1\}$. Therefore [F26] gives ${\mathrm{\ell }}_R\ge (R-R_1)\mathrm{/}(C{E}_{\max })$, with $R_1$, C and $E_{\max }$ independent of the target radius and target translation.

Increase R so that the displayed bound exceeds two divided by delta*. The segment itself now exists past that bound by the first-contact construction. The normal-geodesic flow gives a local congruence along its finite interval as explained in [F27]. Use null convergence [F12], the fixed outgoing expansion bound from [F3] and the exact normalization in step 5.1. Raychaudhuri forces a focal point strictly before the target, whereas [F21] excludes it. This contradiction rules out the original negative-eigenvalue graph alternative. No extension of an unconstructed geodesic to infinity is used.