= Solution
The relevant <Penrose singularity theorem> states: a <connected> time-oriented four-dimensional <globally hyperbolic spacetime> with a noncompact <Cauchy hypersurface>, a nonempty <compact> orientable boundaryless <trapped surface>, and the <null convergence condition> is \b[future null-geodesically incomplete]. Under the <Einstein field equations>, the <null energy condition> supplies the curvature hypothesis. A <trapped surface> here is spacelike and has both future normal <null expansions> strictly negative. The conclusion is an incomplete <null geodesic>, not necessarily a divergent curvature invariant at an identifiable point.
Suppose, for contradiction, that every future <null geodesic> is complete. Write $S$ for the <trapped surface> and
$$
E^+(S)=J^+(S)\setminus I^+(S)=\partial I^+(S).
$$
The equality follows because the <causal future> of a <compact> set is <closed> in a <globally hyperbolic spacetime>. This <future horismos> is a closed <achronal boundary>. It is nonempty: the restriction of a Cauchy time <function> to <compact> $S$ has a minimum, and a point at that minimum cannot be chronologically preceded by another point of $S$.
Normalize the two future null normal directions along $S$ against a smooth future timelike field, fixing the affine scale continuously. Their initial <null expansions> are continuous and strictly negative. <Compactness> of the normalized normal bundle gives a uniform $c>0$ with $\theta_\pm\leq-c$ everywhere on $S$. The preceding <null focusing theorem> forces a <conjugate point to a spacelike surface> along each normal generator within affine length $L=2/c$.
By the supplied boundary-generator result, every point of $E^+(S)$ is reached by an orthogonal future <null geodesic> which has no earlier conjugate point. No such boundary generator can remain on the boundary beyond its first focal point. Every point of $E^+(S)$ therefore lies in
$$
K=\left\{\exp_p(\lambda U):p\in S,\ U\text{ a normalized future null normal},\ 0\leq\lambda\leq L\right\}.
$$
The parameter set is <compact>. Future null completeness makes its <geodesic> flow defined throughout this common finite interval, and smooth dependence on initial data makes its image $K$ <compact>. Since $E^+(S)$ is closed and contained in $K$, it is <compact>. This is the <compactness of the future horismos of a trapped surface> step; a merely pointwise finite bound would not suffice without <compactness> and uniform normalization.
Choose a smooth complete timelike vector field, obtained if necessary by positive rescaling against a complete auxiliary <Riemannian metric>. Its inextendible <integral curves of a vector field> meet a chosen smooth <Cauchy hypersurface> $C$ exactly once. Projection along those curves defines a <continuous map> $\pi_C:E^+(S)\to C$. The <achronal set> property makes this map an <injective function>: two boundary points on the same timelike <integral curve of a vector field> would be timelike related.
An <achronal boundary> is a topological <hypersurface> without boundary, even at nonsmooth generator junctions; the submanifold property allowed in the question supplies this fact. Thus domain and codomain are both three-dimensional <topological manifolds> without boundary. <Invariance of domain> makes $\pi_C(E^+(S))$ <open> in $C$. <Compactness> makes it <closed> in the <Hausdorff space> $C$, and it is nonempty. <Connectedness> of the <spacetime> gives <connectedness> of $C$, so this image is all of $C$. That would make $C$ <compact>, contradicting the noncompact <Cauchy hypersurface> hypothesis. The completeness assumption is false, proving the theorem.
The noncompactness hypothesis and strict trapping are essential to this version. <Global hyperbolicity> cannot simply be omitted in the <compactness> and projection steps, and nonpositive initial expansion with zeros does not supply the uniform focusing bound used here.
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