Theorem: Cone dictionary

For every $\mu \in I$, the three sectors have total opening $2\pi -\mu$. Identifying the upper boundary with the lower boundary using $J(\mu )$ produces a spinless spatial cone $C_{\mu }$. The ray data, metric coefficients, seam, and packet vary continuously with $\mu$. The particle axis is fixed pointwise and the oriented rest-frame rotation is $\mu$.

Proof. Since $0<\alpha <\pi$, each sector is a convex wedge with independent boundary rays. Its opening is $\alpha$, so the total is $3\alpha =2\pi -\mu$. The upper ray has angle $3\alpha$; rotation by $\mu$ sends it to angle $2\pi$, the lower ray. All formulas are continuous, and $I\subset (0,2\pi )$, so Lemma lem:spinless applies.