Intrinsic angle defect at the exchange

The angle at a vertex of a spacelike planar face is computed using the positive definite restriction of $\eta$ to that face. At $s=0$, the two edge rays of the $E$-face may be represented by

Their squared norms are equal to

and their inner product is $1-p^2-(p/q{)}^2$. Thus

Reflection gives the same angle for $W$.

For the $N$-face, use

Then

and $S$ has the same angle. Both angles lie in $(0,\pi )$.

The event junction therefore has defect

Since

strict decrease of cosine on $[0,\pi ]$ gives ${\alpha }_N>\pi -{\alpha }_E$. Hence ${\kappa }_0<0$.

At the incoming northern triple junction, the central edge points in direction $d=(0,0,-1)$. Its angle ${\beta }_E$ with the north-east arm satisfies

Because ${\beta }_E\in (0,\pi )$,

The $W$-angle is the same, and the $N$-angle is ${\alpha }_N$. The triple-junction defect is consequently

Reflection gives the same value at the southern incoming junction.

At an outgoing triple junction the roles of $E$ and $N$ exchange: the two equal angles are ${\beta }_N=(\pi +{\alpha }_N)/2$, and the third is ${\alpha }_E$. Its defect is again ${\kappa }_0/2$. All relevant edge directions are constant within their respective epochs, so these defects are independent of $s$ there.

Thus the sum of the two central triple defects equals the single event defect on both sides. This is a statement about intrinsic geometry of the marked piecewise-flat slice. It is consistent with the identity ambient transition products proved in prop:products,prop:collar.