Lemma: Unique incidence-event parameter

For the declared cellulation, the only incidence-event parameter in $[-1,1]$ is $s=0$. At that parameter, six outgoing vertex slots, six particle-star radial edge slots, six perimeter edge slots, and six inner face slots collapse at $P$. The particle link changes from three to six faces.

Proof. Lemma lem:decomposition gives a constant pattern for each sign of $s$. For $s>0$, the three $B_i$ lie on distinct cone rays and the three $C_i$ lie strictly in their sectors. Their coordinates are positive multiples of $s$, so they coalesce only at $s=0$.

All particle-star edges are genuine creases. Across $P{C}_i$, the gradients are those of $b\xi$ and $b\zeta$, which are distinct. Across a cone ray, unfolding the adjacent sectors gives inner transverse gradients of opposite nonzero signs. Incoming core gradients also have opposite nonzero transverse components after unfolding: their magnitude is $a\tan (\alpha /2)>0$.

No inner region meets the cap, by Lemma lem:bounds. Persistent radial lengths remain positive, all born edges and inner areas are positive for $s>0$, the outgoing core area is bounded below by a positive coefficient, and the collar never degenerates, by Lemma lem:lengths. The exterior is resting with fixed incidence. Thus no other collision, edge collapse, face collapse, or crease loss occurs in the window.

For $s<0$ the three $V_i$ meet $P$. For $s>0$ the six ${U_i}^{\pm }$ meet $P$, proving the link change.