Lemma: Estimates used by the causal certificate

Set

On every regular developed region, $F_s$ is spatially $\lambda$-Lipschitz, including the axial variable. For $s^{\prime }>s$,

The graphs are embedded, and their regular affine pieces are spacelike with induced dimensionless metric bounded below by $I/9$.

The prescribed transverse cellulation has one incidence-event parameter, $s=0$, where its singular transverse link changes from three incident faces to six.

Proof. The active gradients are

up to sector rotations. Minima and maxima preserve their common Lipschitz bound. The graph metric is $I-a{a}^{\mathsf{T}}$; its least eigenvalue is at least $1-{\lambda }^2=1/9$. Graph projection gives embeddedness. For fixed $x$, the term after the initial $s$ in (eq:family) is nondecreasing and 1-Lipschitz in $s$, proving (eq:time-order).

For the incidence, use

When $s\le 0$, the $h$-branch dominates and one face in each sector meets the apex. When $s>0$, in the half-sector $\xi \ge \eta$, the new boundary is $10\xi +\eta =15s$; its symmetric counterpart is $\xi +10\eta =15s$. Their vertices are

They define two distinct new apex triangles in each sector. Their outermost radial coordinate is $30s/11<3$, so they do not meet the persistent crest $\rho =3$. Within each open epoch these boundaries have fixed combinatorics, and all new coordinates vanish only at zero. Thus the normal transverse link changes exactly once from three to six.