Raychaudhuri expansion and generator Jacobians

In the original stationary end, regularity in [A1] gives $g(T,T)=g_{00}\to -1$ uniformly in angle. Choose a fixed radius beyond which $g(T,T)<0$, and choose the free comparison constant $R_0$ beyond that radius, retaining the large-radius requirements of the comparison-sphere supplier. Since $0<q\le 1$, the comparison cylinder has radius $R_0{q}^{-2\mathrm{/}3}\ge R_0$. For the same selected primitive and pole, [F6] gives a smooth stationary $W_q$, its normalization $W_q(T)=d_q-D_q$, and a positive denominator on the whole exterior. In the notation of the global construction, $d{W}_0=4((f-1)F+\psi {\star }_{g}F)$ and $d{T}^{\mathrm{♭}}=2F$, so direct differentiation gives $d{W}_q=4(1-q)(((1-q)f-(1+q))F+(1-q)\psi {\star }_{g}F)$, the required Harrison curl identity. Thus the supplied metric and one-form meet the algebra theorem's hypotheses. Its exact algebraic identity gives $g_q(T,T)=d_q^{2}g(T,T)\mathrm{/}(4D_q)<0$ on and outside the cylinder. The temporal section time satisfies $dv(T)=1$ by [F1], so this timelike field is future-directed: a past-directed timelike vector has negative pairing with the temporal covector. The complete stationary product and the invariance of $g_q$ preserve this conclusion along every cylinder orbit. This establishes future timelike cylinder orbits for the supplied smooth Harrison family. To match the expansion input as well, [F6] gives $(W_0{)}_i=O_5(r^{-3})$ in the unchanged original chart. Use the same mass-one end, supplied original maximal section with its stipulated ADM mass-one limit in $x^i$, normalized potential and pole in [F8]. The stationarity, curl and contraction identities just checked are the component supplier's Harrison one-form conditions, and its positivity domain is supplied by [F6]. Its remaining spatial-decay condition is exactly the bound in [F6], so the denominator and component estimates apply to this same $W_q$ and $g_q$, not to an auxiliary radial choice. The coordinate-null-normal and sphere-area-expansion suppliers then apply with those same data. Their uniform expansion estimate is ${\theta }_{L_q}=2q\mathrm{/}(1+qr)+O(r^{-2})$. At $r=R_0{q}^{-2\mathrm{/}3}$, the ratio of the leading term to $r^{-2}$ is at least $2R_0^2\mathrm{/}(1+R_0)$ for $0<q\le 1$; enlarge the same free $R_0$ so this exceeds the uniform error constant, retaining every other end threshold. This is precisely the positive-expansion application of the comparison-sphere supplier in [F1]. Stationary translation carries that sign to $C=C_t$. No equality between the smooth primitive and the radial integral is used.

Consider either an arbitrary $c\in A$ with endpoint $p=F(c)$, or an arbitrary $c\in A_z$ with endpoint $p=F_z(c)$. In both cases [F2] gives $p\in \mathrm{\partial }{I}^{-}(C)$ and $v(p)<{\min }_{C}v$. By [F3] and [F7], the entire selected null segment is on that boundary and has no focal point strictly between $C$ and $p$. Choose the affine parameter $\lambda =0$ at $C$, with future-directed tangent $L={\mathrm{\partial }}_{\lambda }$; its endpoint parameter ${\lambda }_p$ is negative. The stationary isometry carries the positive future outward expansion to $C$, and positive affine rescaling preserves its sign. Null convergence holds along the segment. The screen Jacobi map is initially the identity and is invertible on $({\lambda }_p,0]$ by focal exclusion. Applying the optical comparison on every closed regular subinterval ending at $0$ gives $\theta (\lambda )>0$ and a positive screen Jacobian $j$ satisfying $j({\lambda }_1)\le j({\lambda }_2)\le j(0)=1$ for ${\lambda }_p<{\lambda }_1\le {\lambda }_2\le 0$. This treats all of $A$ directly, without an exhaustion by the $A_z$.

Fix $c\in A_z$ and let $p_z=F_z(c)$. The chosen ray segment from $p_z$ to $c$ is causal, and $p_z$ lies on the relative past boundary of $C$, with $v(p_z)=z<{\min }_{C}v$. The compact-target hypotheses hold by [F7], so [F3] puts the whole segment on that boundary. Since $z<0<{\min }_{C}v$, temporal monotonicity gives a unique crossing $p_0$ of $v=0$ between $p_z$ and $c$. If $p_0$ were outside the cylinder, continuity from the inside point $p_z$ would give a cylinder crossing $y$ at $v(y)\le 0$. At its spatial coordinate $x$, $v(y)=t(y)-u_q(x)$ and the point of $C$ on the same stationary orbit has $v=t-u_q(x)\ge {\min }_{C}v>0$. Thus $t(y)<t$, and the future timelike stationary orbit from $y$ reaches $C$, placing $y$ in $I^{-}(C)$, contrary to boundary membership. The same orbit argument excludes $p_0$ on the cylinder. Hence $p_0$ is inside it, and localization identifies $p_0$ with a point of $D$. Uniqueness of the temporal hit gives $c\in A$ and $F(c)=p_0$.

At either type of endpoint use a smooth local normalized null-normal field on $C$. Smooth geodesic dependence gives a smooth evaluation map $G(x,\lambda )$ on an open neighborhood of its finite included hit. For $b=0$ or $b=z$, the scalar hitting equation is $H(x,\lambda )=v(G(x,\lambda ))-b=0$. Its solved-variable derivative at the hit is ${\mathrm{\partial }}_{\lambda }H=dv(L)>0$, since $L$ is nonzero future null and $v$ is temporal. In local coordinates on $C$, apply [F5]'s implicit function theorem with the two coordinates as the free variables and $\lambda$ as the one solved variable (and no additional parameter block). It supplies a unique smooth local hitting parameter ${\lambda }_b(x)$ and the extension $x\mapsto G(x,{\lambda }_b(x))$ into the smooth spacelike level $v=b$; temporal uniqueness identifies its restriction with the selected endpoint map. This construction remains valid at a focal endpoint because it uses $dv(L)$, not the rank of the screen map. Differentiating this extension adds a multiple of $L$ to each fixed-affine-parameter variation vector $V$. The normal congruence has $g(L,V)=0$ and $g(L,L)=0$, so these additions change neither its Gram matrix nor its determinant. The Gram two-Jacobian of each hitting map is therefore the screen Jacobian at the corresponding hit, also when the determinant vanishes.

For any $c\in A$, apply step 1.2 with $p=F(c)$. The screen Jacobi fields and their nonnegative Gram Jacobian are continuous through the finite endpoint by smooth geodesic dependence, including when that endpoint is focal. Taking the limit $\lambda \downarrow {\lambda }_p$ of $0<j(\lambda )\le 1$ and using step 2.2 gives $0\le J(F)(c)\le 1$. This proves the zero-cut estimate on every point of $A$ and positivity of future expansion on each regular part before its endpoint.

For $c\in A_z$, step 2.1 gives the intermediate point $p_0=F(c)$ and parameters ${\lambda }_z<{\lambda }_0<0$. Step 1.2 with endpoint $p_z$ makes $p_0$ an interior regular point. Its regular comparison gives $j(\lambda )\le j({\lambda }_0)\le 1$ for ${\lambda }_z<\lambda \le {\lambda }_0$. Continuity through $p_z$ and step 2.2 therefore give $0\le J(F_z)(c)\le J(F)(c)\le 1$, allowing a focal endpoint at $p_z$. The argument is pointwise and does not use injectivity or an image-area identity.