Divergence comparison between S and the translated causal cut

The maximal-section supplier gives the smooth spacelike section with included compact boundary $S$. Its induced metric is smooth and positive definite up to that boundary; the transformed horizon/metric supplier ensures the ambient smooth geometry there. Apply the first two new lemmas to this fixed $h$: the inward unit-normal exponential produces a compact Gaussian collar on which its coordinate equals intrinsic distance from $S$, even among paths allowed to leave the collar. Thus $\rho$ is smooth there and $\mathrm{\mid }{\mathrm{\nabla }}_h\rho \mathrm{\mid }=1$. No global regularity of boundary distance or extension of stationary graph height is assumed.

The third new lemma gives fixed constants $C_q,K_q$ on that collar with $\mathrm{\mid }{\mathrm{\Delta }}_h\rho \mathrm{\mid }\le C_q$ and product-volume density at most $K_q$. They depend on the fixed section and parameter, and remain valid when $\eta$ shrinks. Choose $\eta$ below this collar width. Use the localization and smooth-cut construction to fix a sufficiently late target $C_t$ and its localized data, independently of $ϵ$. For each $ϵ>0$, choose directly from the coarea-smoothing supplier a smooth compact sublevel region ${\mathrm{\Omega }}_{ϵ}=\{u_i\le b\}$ inside that collar, with boundary $S\bigsqcup D_{ϵ}$ and $\mathrm{\mid }{D}_{ϵ}\mathrm{\mid }\le \mathrm{\mid }{C}_t\mathrm{\mid }+ϵ$. The same supplier preserves $u_i=-2\delta$ on a neighborhood of $S$ and gives $b\in (-\delta ,-\delta \mathrm{/}2)$. Since $-2\delta <b$, that neighborhood lies in ${\mathrm{\Omega }}_{ϵ}$, so the region contains the inward side of $S$, as required by the divergence input.

Apply the fourth new lemma, which matches all the divergence and density hypotheses established in steps 1.1 and 2.1. The outward unit normal at $S$ is $-\mathrm{\nabla }\rho$, so the exact flux identity has the displayed minus sign on its volume integral. The outer flux is at most $\mathrm{\mid }{D}_{ϵ}\mathrm{\mid }$, and its volume error is bounded by $C_q{K}_q\eta \mathrm{\mid }S\mathrm{\mid }$. Substituting the selected coarea-smoothing area bound proves $\mathrm{\mid }S\mathrm{\mid }\le \mathrm{\mid }{C}_t\mathrm{\mid }+ϵ+C_q{K}_q\eta \mathrm{\mid }S\mathrm{\mid }$. This bound is uniform in $ϵ$, the possible disconnectedness of $D_{ϵ}$, and the late target.

The Harrison algebra supplier gives ${\mathcal{L}}_T{g}_q=0$. Apply [F21] to the smooth tensor $g_q$ and vector field $T$ on the stationary end: on each common flow domain containing $C_q$, $({\varphi }_t^T{)}^{\ast }{g}_q=g_q$. The coordinate-sphere convention supplies the product-end flow ${\varphi }_t^T(s,x)=(s+t,x)$ for every real $t$, so the comparison sphere is in such a domain for every selected $t$ and $C_t={\varphi }_t^T(C_q)$. Thus these translations satisfy the ambient isometry condition of the fifth new lemma. The fifth new lemma therefore gives $\mathrm{\mid }{C}_t{\mathrm{\mid }}_{g_q}=\mathrm{\mid }{C}_q{\mathrm{\mid }}_{g_q}$ exactly, for every selected $t$, including targets that move as $\eta$ shrinks. The collar and $C_q$ remain fixed at the chosen $q$. Let $ϵ\downarrow 0$ at each admissible $\eta$, then $\eta \downarrow 0$ with the fixed constant $C_q{K}_q\mathrm{\mid }S\mathrm{\mid }$. The same lemma gives $\mathrm{\mid }S{\mathrm{\mid }}_{g_q}\le \mathrm{\mid }{C}_q{\mathrm{\mid }}_{g_q}$ as asserted. No uniform estimate as $q\downarrow 0$ is needed for this fixed-$q$ conclusion.