Theorem: Unique particle-incidence event

The family has exactly one incidence-event parameter in $[-1,1]$, namely $s=0$. At that parameter:

1. the six vertices $B_i,C_i$ coalesce at $P_0$;

2. the twelve new edges $P{B}_i,P{C}_i,B_i{C}_i,C_i{B}_{i+1}$ collapse;

3. the six faces ${U_i}^{\pm }$ collapse;

4. the particle link changes from three incident faces to six.

There is no other collision, vanishing active finite edge or face, or contact of a new cell with the outer crest. At $s=0$, the regular cellulation is the common limiting three-face core; outgoing collapsed slots are germ metadata.

Proof. The exact dominance regions show constant incidence on each sign-preserving open epoch. For $s>0$, the explicit new coordinates are distinct, and A:lem:metric gives positive active lengths and areas. The bound $\rho (C_i)\le 30/11<3$ excludes crest contact. Coverage and nonintersection were established by A:lem:cells. The incoming finite cells are nondegenerate and have unchanged incidence.

All new coordinates are proportional to $s$, so their specified cells collapse only at $s=0$. The persistent $e_i,V_i$ have the incoming nondegenerate segments and triangles as their limits.

Incoming, the particle meets exactly the three $V_i$. Outgoing, it meets exactly the six ${U_i}^{\pm }$; the pentagons no longer touch it. Across each bisector edge $P{C}_i$, the gradients are $(3/5,0)$ and $(0,3/5)$. Across a ray edge $P{B}_i$, unfolding the adjacent sectors changes the sign of the transverse gradient. These are genuine creases. The six-face link is therefore not a mere subdivision of one plane or a relabelling of an unchanged star.