Theorem: Unique isolated particle-incidence event

The family (eq: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 edges $u_i,{v_i}^{\mathrm{edge}},{w_i}^{\pm }$ vanish;

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

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

There is no other collision, zero-length active edge, zero-area active face, or contact with the outer crest in the window.

Proof. The cellulation is independent of the sign-preserving choice of $s$. For $s<0$, all its finite lengths and areas are constant and positive.

For $s>0$, the vertex formulas show that the six new vertices are distinct and lie strictly inside $\rho =3$. Propositions prop:lengths and prop:areas give positivity of every active finite edge and face. Lemma lem:decomposition excludes self-intersection and omitted regions. The strict bound $\rho (C_i)\le 30/11<3$ excludes contact with the crest.

At $s=0$, the stated vertices, edges, and faces collapse according to their explicit formulas, while $e_i$ and $V_i$ have the incoming edge and face as their nondegenerate limits. The particle-link orders displayed above differ in valence, so the event cannot be a relabelling of an unchanged star.