Closed generator set and nonintersection before the cut

In the original stationary end chart, [F22] identifies $C_0$ with the Euclidean sphere $\mathrm{\mid }x\mathrm{\mid }=R$ in ${\mathbb{R}}^3$, where $R=R_0{q}^{-2\mathrm{/}3}>a>0$. Give $C_0$ the metric $d_0(c,c^{\mathrm{\prime }})=\mathrm{\mid }x(c)-x(c^{\mathrm{\prime }})\mathrm{\mid }$; the chart identifies this metric topology with its manifold subspace topology. This metric space is compact by [F23] and nonempty, since the coordinate point $x=(R,0,0)$ lies on it. The smooth function $v_q$ is continuous on $C_0$, so [F24] gives the finite attained minimum $m={\min }_{C_0}{v}_q$. The stationary translation in [F21] gives $v_q({\varphi }_t^T(c))=t+v_q(c)$ and hence ${\min }_{C_t}{v}_q=t+m$. Choose $t$ beyond the localization threshold in [F7] and with $t>-m$. Localization and the strict time ordering therefore hold simultaneously for every sufficiently late $t$, without any additional geometric assumption.

We first rule out two target initial points on one oriented maximal generator by a local end calculation. Follow the future outward affine geodesic from any $c\in C_t$, writing its affine parameter as $s=0$ at $c$, and put $p_{\mu }=g_{\mu \nu }{\dot{x}}^{\nu }$ in the stationary Cartesian chart $(\tau ,x)$. The geodesic equation and the Christoffel formula supplied by [F25], after differentiating this pairing, give ${\dot{p}}_{\mu }=\frac{1}{2}({\mathrm{\partial }}_{\mu }{g}_{\alpha \beta }){\dot{x}}^{\alpha }{\dot{x}}^{\beta }$. Stationarity therefore gives $p_{\tau }=-E=-d\mathrm{/}4$ throughout any end portion, by the normal normalization in [F11]. Differentiating the inverse-matrix identity then gives the equivalent equations ${\dot{x}}^i={\mathrm{\partial }}_{p_i}H$ and ${\dot{p}}_i=-{\mathrm{\partial }}_{x^i}H$ for $H=\frac{1}{2}{g}^{\mu \nu }{p}_{\mu }{p}_{\nu }=0$, with $p_{\tau }=-E$ held fixed. Write $p=(p_1,p_2,p_3)$, with Euclidean norm $\mathrm{\mid }p\mathrm{\mid }$. Since $q^2{r}^{-1}+q{r}^{-2}+r^{-3}\le a^2\mathrm{/}r$, [F26] gives, after scaling the time basis vector by $4a\mathrm{/}d$ and spatial basis vectors by $a^{-1}$, a metric matrix $\eta +O(r^{-1})$, where $\eta =\mathrm{diag}(-1,1,1,1)$. Its inverse is also $\eta +O(r^{-1})$: write its inverse difference as $-(\eta +e{)}^{-1}e\eta$ for the error matrix $e$. A spatial rotation differentiates $e$ to $O(r^{-1})$ by the differentiated estimates; differentiating the inverse identity preserves that bound. Thus, at fixed $p_{\tau }=-d\mathrm{/}4$, $H=H_0+\mathcal{R}$, where $H_0=\frac{1}{2}(-a^2+a^{-2}\mathrm{\mid }p{\mathrm{\mid }}^2)$ and $\mathrm{\mid }\mathcal{R}\mathrm{\mid }\le C{r}^{-1}(a^2+a^{-2}\mathrm{\mid }p{\mathrm{\mid }}^2)$. The quadratic coefficients also give $\mathrm{\mid }{\mathrm{\partial }}_p\mathcal{R}\mathrm{\mid }\le C{r}^{-1}(1+a^{-2}\mathrm{\mid }p\mathrm{\mid })$. For $i<j$, define ${\mathcal{Q}}_{ij}=x^i{\mathrm{\partial }}_{x^j}-x^j{\mathrm{\partial }}_{x^i}+p_i{\mathrm{\partial }}_{p_j}-p_j{\mathrm{\partial }}_{p_i}$. The same rotation bounds give $\mathrm{\mid }{\mathcal{Q}}_{ij}\mathcal{R}\mathrm{\mid }\le C{r}^{-1}(a^2+a^{-2}\mathrm{\mid }p{\mathrm{\mid }}^2)$. All constants here are independent of $q$, angle and sufficiently large $r$.

Localization supplies compact $D\subset {\mathrm{\Sigma }}^{\circ }\cap {\mathrm{\partial }}_{\mathcal{E}}{I}_{g_q}^{-}(C_t)$ inside the cylinder at level $v_q=0$; step 1.1 gives $0<{\min }_{C_t}{v}_q$. The completed Cauchy and temporality suppliers give a smooth globally hyperbolic $(\mathcal{E},g_q)$ and temporal $v_q$. The normalized outward initial null vector $L_{q,t}$ depends smoothly on $c\in C_t$ by its explicit smooth branch and nonzero normalization. Apply [F20] in $\mathcal{E}$ to these initial data. Differentiating in a coordinate chart and using the metric compatibility in [F25] gives $\frac{d}{ds}{g}_q({\dot{\gamma }}_c,{\dot{\gamma }}_c)=2g_q(D_s{\dot{\gamma }}_c,{\dot{\gamma }}_c)=0$; uniqueness excludes a zero tangent on this nonconstant geodesic, so it remains past null. The chosen family is therefore maximal in the exterior, exactly as required by the endpoint lemmas; a truncated supplied curve would only be a restriction and is not the family defining $A$ here. Thus every hypothesis of the normalized-null-flow and compact-diamond lemmas is met, relative to the exterior. They give unique hits and closed $A$. Their compactness argument keeps the lifted rays over $J^{+}(D)\cap J^{-}(C_t)$ and normalizes $d{v}_q(w)=1$; it never assumes affine completeness or an unproved uniform affine parameter bound.

Put $p_r=r^{-1}{\sum }_i{x}^i{p}_i$, ${\mathrm{\ell }}_{ij}=x^i{p}_j-x^j{p}_i$ for $i<j$, and $J=({\sum }_{i<j}{\mathrm{\ell }}_{ij}^2{)}^{1\mathrm{/}2}$. The algebraic identity $\mathrm{\mid }p{\mathrm{\mid }}^2=p_r^2+J^2\mathrm{/}{r}^2$ follows by expanding both sides. Nullity and step 1.2 imply $\mathrm{\mid }p{\mathrm{\mid }}^2\mathrm{/}{a}^4=1+O(r^{-1})$ by absorbing the error in $-a^2+a^{-2}\mathrm{\mid }p{\mathrm{\mid }}^2=O(r^{-1})(a^2+a^{-2}\mathrm{\mid }p{\mathrm{\mid }}^2)$. The Hamilton equations give $\dot{r}=a^{-2}{p}_r+O(r^{-1})$. They also give ${\dot{\mathrm{\ell }}}_{ij}=-{\mathcal{Q}}_{ij}H=-{\mathcal{Q}}_{ij}\mathcal{R}$, since the simultaneous rotation fixes $r$ and $\mathrm{\mid }p{\mathrm{\mid }}^2$, and hence $\mathrm{\mid }{\dot{\mathrm{\ell }}}_{ij}\mathrm{\mid }\le K{a}^2\mathrm{/}r$. Initially [F11] gives $p=(L_q{)}_{r}x\mathrm{/}r$ with $(L_q{)}_r=a^2(1+O(R^{-1}))>0$, so $J(0)=0$ and $p_r(0)>0$, where $R=R_0{q}^{-2\mathrm{/}3}$. On any exterior interval with $J\mathrm{/}(a^{2}r)\le 1\mathrm{/}2$, continuity keeps $p_r$ on its positive branch: for a common sufficiently large radius, $p_r^2\mathrm{/}{a}^4=1+O(r^{-1})-J^2\mathrm{/}(a^4{r}^2)\ge 2\mathrm{/}3$. Consequently $\dot{r}\ge 1\mathrm{/}2$ on that interval. This uses only finite portions of the existing geodesic.

Apply the transverse-hit lemma to obtain local smooth endpoint extensions, continuity, compact/Borel domain and countably many chart-Lipschitz pieces. This is the precise regularity needed by the endpoint-area interface, even if $D$ is nonsmooth or the derivative loses rank at a focal endpoint. The Jacobian is the nonnegative Gram two-Jacobian, and coincident endpoints are counted with their actual multiplicities. No image-smoothness or almost-everywhere injectivity is inferred from compactness.

For every $p\in D$, the completed past-boundary generator theorem gives a future null segment to $C_t$ ending on the prescribed normalized outward ray and without an interior focal point. Reverse and positively affinely rescale that segment to give initial tangent $-L_{q,t}(c)$. By [F20] it is a restriction of the chosen maximal ${\gamma }_c$, including its endpoint in $D$; hence $c\in A$ and $F(c)=p$, proving $F(A)=D$. Conversely any selected segment from a hit in $D$ to $C_t$ stays on the past boundary by the cut-coverage lemma; the completed universal boundary focal-exclusion supplier rules out any interior focal point on it.

For $r\ge R$, define $B_q(r)=q^2\log (r\mathrm{/}R)+2q(R^{-1}-r^{-1})+\frac{1}{2}(R^{-2}-r^{-2})$. Then $B_q(R)=0$, $B_q\ge 0$, and $B_q^{\mathrm{\prime }}(r)=a^2\mathrm{/}r$, by [F27]. While the bound in step 2.2 holds, $d{B}_q(r(s))\mathrm{/}ds\ge a^2\mathrm{/}(2r)$. Therefore the derivatives of $2K{B}_q(r(s))\pm {\mathrm{\ell }}_{ij}(s)$ are nonnegative, and the mean value theorem gives $\mathrm{\mid }{\mathrm{\ell }}_{ij}(s)\mathrm{\mid }\le 2K{B}_q(r(s))$. In particular $J\le 2\sqrt{3}K{B}_q$. Since $a^{2}r=q^{2}r+2q+r^{-1}\ge 4q$ and also $a^{2}r\ge q^{2}r$, the estimate $\log z\le z$ for $z\ge 1$ gives $B_q(r)\mathrm{/}(a^{2}r)\le 3\mathrm{/}(2R)+1\mathrm{/}(8q{R}^2)$. The logarithmic estimate itself follows by differentiating $z-\log z$ and using [F27]. Hence $J\mathrm{/}(a^{2}r)\le C(R^{-1}+(q{R}^2{)}^{-1})=C(q^{2\mathrm{/}3}\mathrm{/}{R}_0+q^{1\mathrm{/}3}\mathrm{/}{R}_0^2)$. Choose the already sufficiently small positive $q$ so that this bound is less than $1\mathrm{/}4$, and $R$ exceeds the preceding common end thresholds. This choice is independent of $t$. Starting with $J=0$, a first failure of $J\mathrm{/}(a^{2}r)<1\mathrm{/}2$ on any finite exterior segment is impossible, because the improved bound is less than $1\mathrm{/}4$ and the fields are continuous. Thus $\dot{r}\ge 1\mathrm{/}2$ persists. The ray cannot return to $r=R$: before any putative first return it stays in this exterior chart with strictly increasing $r$. No assertion of affine completeness or continuation to infinite affine parameter is used. In particular an outward ray through one point of $C_t$ cannot later pass through another point of $C_t$.

Apply the independent causal-curve rigidity input [F19] to the smooth time-oriented exterior. As in the noncrossing lemma [F18], a meeting with nonparallel future tangents at a point interior to both boundary segments would produce a causal corner from $D$ to $C_t$, contradicting rigidity and boundary membership from step 3.2. A proportional-tangent meeting identifies the oriented maximal geodesics by [F20]. Their distinct initial points on $C_t$ would then be ordered along that geodesic by temporal $v_q$; the earlier one's future outward continuation reaches the later one, contrary to step 3.3. If a common point outside $D$ is the $C_t$ endpoint of just one selected segment and interior to the other, concatenate the first segment's past leg from $D$ with the other's future leg to its target in $C_t$. Rigidity again forces matching tangent directions there, so the same uniqueness and no-return argument excludes this case. A point in $D$ cannot be interior to either segment because $v_q$ strictly increases from zero to its positive target value. Thus selected segments from distinct initial points are disjoint away from $D$, covering the full stated nonintersection claim. Meetings at $D$ remain possible; the area interface retains their actual multiplicities.