A sufficiently long focal-free boundary segment contradicts negative outgoing expansion

Let $a=\gamma (0)$ and use the smooth outgoing null normal field $L_{ϵ}$ on a surface neighborhood of $a$ in $A$. The existing affine segment and its endpoint in the smooth open spacetime belong to a maximal geodesic defined on an open interval containing $[0,\mathrm{\ell }]$. By [F4], openness of the flow domain and compactness of this parameter interval give a common surface neighborhood $U$ of $a$ and an open interval $J\supset [0,\mathrm{\ell }]$ on which $F(y,\lambda )={\gamma }_{y,L_{ϵ}(y)}(\lambda )$ is smooth. Its central generator is the given $\gamma$ by uniqueness. Every parameter curve is affine, and metric compatibility gives ${\mathrm{\partial }}_{\lambda }g(L,L)=2g({\mathrm{\nabla }}_{L}L,L)=0$, where $L={\mathrm{\partial }}_{\lambda }F$. Its initially nonzero future null tangent therefore remains future null; it cannot vanish at a later time by geodesic uniqueness.

For a source-coordinate variation let $V={\mathrm{\partial }}_{s}F$. The parameter derivatives commute. In any ambient chart, the mixed second derivatives of $F$ agree and the symmetric connection coefficients in [F6] give $D_{\lambda }V=D_{s}L$ along the map, including at rank loss. Since $D_{\lambda }L=0$, metric compatibility yields ${\mathrm{\partial }}_{\lambda }g(L,V)=g(L,D_{\lambda }V)=g(L,D_{s}L)=\frac{1}{2}{\mathrm{\partial }}_{s}g(L,L)=0$. The initial value is zero because $L_{ϵ}$ is normal to $A$, so every variation field is orthogonal to $L$. Before screen rank loss, the two source variation fields are independent modulo $L$; together with the nonzero $L$ they span the three-dimensional space $L^{\perp }$. Thus the family is locally an immersed null hypersurface congruence on the regular interval, as required by [F3]. Its initial optical expansion is precisely ${\theta }_{L_{ϵ}}$ by [F2] and [F5].

Apply [F3] with nonnegative null Ricci curvature and $\theta (0)={\theta }_{L_{ϵ}}(a)\le -{\delta }_{\ast }<0$. It forces a focal instant no later than $2\mathrm{/}\mathrm{\mid }\theta (0)\mathrm{\mid }\le 2\mathrm{/}{\delta }_{\ast }<\mathrm{\ell }$. This is strictly before the endpoint, contradicting [F1]. The smooth normal geodesic family is defined through that bound by step 1.1 and has the required null hypersurface structure wherever its screen map is regular by step 2.1; no existence past the bound was inferred from the focusing inequality.