Lemma: Uniform estimates through the event

Let

For every $s,s^{\prime }\in [-1,1]$, $s^{\prime }\ge s$,

Every face is spacelike, with induced-metric lower bound $1-{{\lambda }_0}^2=1/9$. At $s=0$ the limiting graph remains embedded and uniformly spacelike in the stratified Lipschitz sense.

Proof. The nonconstant spatial gradients are

up to rotations. Their norms are at most ${\lambda }_0$. Maximum and minimum preserve the common Lipschitz bound. Integrate it along spatial paths and take their length infimum.

For fixed $x$, the minimum–maximum expression after the leading $s$ is nondecreasing and one-Lipschitz in $s$, proving the time estimates. The induced metric on an affine graph with gradient $a$ is $I-a{a}^{\mathsf{T}}$, bounded below by $(1-{{\lambda }_0}^2)I$. Projection of the graph to the cone is a homeomorphism. No vanished germ cell is counted as a positive-dimensional regular face at the event.