Proposition: Intrinsic compact-patch audit

For every regular slice, the total intrinsic interior angle defect of $\rho \le 4$ is $\mu$. The same holds for the common limiting slice, counting its active cells only.

Proof. The patch is a piecewise Euclidean metric disk. Triangulate the simple faces without changing their metrics. The triangle angle sum and Euler identity for a disk give

At each outer vertex $D_i$, the two resting sector corners have total interior angle $\pi -\alpha$. Hence boundary turning totals $3\alpha$, and

At $s=0$, the active incoming core triangles and collar form the same limiting disk. Collapsed outgoing slots contribute no extra vertices or faces.