Proposition: Total-angle audit

For every slice in the family, summing over all interior vertices of the compact patch gives

At $s=0$, only regular cells of the limiting slice enter the sum.

Proof. Triangulate the polygonal faces without changing their metric. The Euclidean angle sum in each triangle, together with Euler's formula for a disk, gives

Artificial triangulation edges do not change the summed angles.

The boundary lies in the resting collar. At each of the three boundary vertices $D_i$, the total interior angle is $\pi /2$, so the boundary turning sum is $3\pi /2$. Therefore the interior defect sum is $\pi /2$.