Theorem: Unique transverse three-to-six incidence

For every fixed $\mu \in I$, the declared transverse cellulation has exactly one incidence-event parameter, $s=0$. Six outgoing vertex slots, twelve outgoing edge slots, and six inner face slots collapse there. The particle link changes from three incident faces to six.

In the $3+1$ product this is a transverse statement along $\{\tau =0,x=o\}\subset {\mathcal{W}}_{\mu }$, not an isolated point event of the full spacetime.

Proof. For each sign of $s$, Lemma lem:decomposition fixes the cell pattern. The new vertices lie on distinct rays or strictly inside distinct sectors and are positive multiples of $s$. They coalesce only at zero. Proposition prop:metric-data excludes other collapses, and the cap separation excludes an extra intersection.

The outgoing bisector edges are genuine creases because the gradients of $b\xi$ and $b\eta$ are distinct. Across a cone ray, unfolding the neighbouring sectors gives opposite nonzero transverse gradients. The incoming ray crease has transverse magnitude $a\tan (\vartheta /2)>0$. Thus the link change is not an artificial subdivision of one flat face. Three $V_i$ meet $P$ before the event; six ${U_i}^{\pm }$ meet it afterwards.

At zero, the active limiting cells are the incoming core triangles and collar. Collapsed outgoing labels are one-sided germ metadata, not extra active zero-size cells. Taking the axial product proves the final statement.