Proposition: Intrinsic particle defect is not the packet deficit

The incoming and outgoing transverse particle angle defects are

and

They need not equal $\mu$. Nevertheless the total intrinsic interior angle defect of the compact transverse patch $\rho \le 4$ equals $\mu$.

Proof. On an incoming face the induced oblique metric is $G_{\vartheta }-(a,a{)}^{\mathsf{T}}(a,a)$. Its two ray tangents give the displayed cosine. On $U_i^{-}$, the ray and bisector tangents are $(1,0)$ and $(1,1)$. Their metric determinant and inner product give

There are three incoming or six outgoing corners.

For the total, triangulate the simple faces without changing their intrinsic metrics. The triangle angle sum and Euler identity for a disk give

At each outer vertex the two resting corners have total angle $\pi -\vartheta$, so boundary turning is $3\vartheta$. The interior sum is $2\pi -3\vartheta =\mu$. At zero count only active limiting cells.