Lemma 62 ([The maximal hypersurface as an orbit section]). Let Σ' be the maximal hypersurface of Proposition 60, and let B₀ be its bifurcation boundary. Projection along the Killing flow identifies intΣ' diffeomorphically with intΩ . This …

Each Killing orbit in the right wedge is a complete, nonzero causal curve. It is inextendible there: in the original product its spatial label is fixed and its z coordinate traverses all of ℝ . The Cauchy property of intΣ' therefore gives exactly one intersection with each orbit. A nonzero causal vector cannot be tangent to a spacelike hypersurface. Consequently the restriction of orbit projection is a local diffeomorphism, and its bijectivity makes it a global diffeomorphism. Flowing the section thus gives a diffeomorphism ℝΣ' ⟶ℝΩ. For a compact cut its two sections bound the compact chain swept out by the intervening finite orbit segments, which proves the homology assertion without a filling behind either boundary. For clarity, the identification of boundary labels does not assert that the two hypersurfaces have the same transverse defining function. In the attachment coordinates of Proposition 56, B_0= U=V=0 and the right wedge has U,V>0 . A spacelike hypersurface through B_0 is transverse to both null faces. If s≥0 is a smooth defining function on that hypersurface, then U=s\,a(s,y), V=s\,b(s,y), a(0,y),b(0,y)>0. Thus √UV=s√ab is a smooth defining function, and the restriction of y to B_0 is its original boundary label. This also proves that the graph approaches precisely that boundary and no other limiting orbit. The end assertion follows from the rest-chart graph estimates in Proposition 60.