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The relevant implications in the causal hierarchy areThe first follows from strict cone widening: failure of strong causality would give causal curves leaving and re-entering arbitrarily small neighborhoods, and the wider metric turns the limiting recurrence into a closed timelike curve. For the second, if distinct $p,q$ had $I^{+}(p)=I^{+}(q)$, arbitrarily short timelike segments near either point and this equality would produce a causal curve leaving and re-entering a sufficiently small causally convex neighborhood. ThereforeThe stated null geodesic completeness is stronger than needed: stable causality already gives the conclusion.

Apply the time-reversed Penrose singularity theorem. The two past-directed null congruences orthogonal to the compact anti-trapped surface have negative expansion. The null energy condition, through the Einstein field equations, supplies the null convergence condition, and the null focusing theorem forces each generator to acquire a conjugate point within finite affine length if it can be extended that far.

If every past-directed null generator were complete, the boundary of the causal past of the surface would therefore be generated only for a bounded affine interval. Compactness of the initial surface and continuous dependence of geodesics on initial data would make that achronal boundary compact. A globally hyperbolic spacetime provides a Cauchy hypersurface and a timelike flow projecting the boundary onto it. The standard Penrose argument then makes its image both open and closed, forcing the connected Cauchy hypersurface to be compact. This contradicts its stipulated topology ${\mathbb{R}}^3$.

Hence at least one past-directed null generator ends after finite affine parameter: the universe is null-geodesically incomplete to the past. Global hyperbolicity controls the causal boundary, the noncompact spatial topology supplies the contradiction, and the energy condition supplies focusing.