Proposition: All noncentral edges remain positive

The list in tab:edges is complete. Every actual noncentral edge has length greater than $1/2$ throughout $[-1,1]$. All eight faces remain two-dimensional, and the only incidence event in the patch is the central exchange.

Proof. For the incoming north-east arm, the displacement from the central endpoint to the inner corner is

Its squared Minkowski norm is

For the outgoing arm the displacement is $Y_s(q,q/p,1)$, with norm $Y_s\sqrt{1921}/39$. These give eq:arm-length; symmetry gives the other arms.

The remaining edges lie in the resting plane and have the Euclidean lengths listed in the table. The smallest such length over the window is $3/5>1/2$. The arms have length at least $3\sqrt{1921}/25>1/2$.

The central endpoints remain strictly inside ${\mathcal{R}}_s$. Each core face contains a strict-dominance neighbourhood just inside its inner side midpoint. Each collar polygon contains a nonempty open region because the margins are positive. The mixed arms and all peripheral incidences therefore persist. The central laws determine the only exchange.