The cone dictionary

Let $L_{\mu }=2\pi -\mu$. The spatial cone $C_{\mu }$ is the wedge $0\le \theta \le L_{\mu }$, with the two boundary rays identified and the origin completed to an apex. The regular metric is

The spacetime is

The upper boundary ray is mapped to the lower ray by $J(\mu )$, because

This is wedge gluing, not a quotient of the entire plane by the group generated by a rotation. We take this spinless conical dictionary as the admissibility model. A normalization such as $\mu =8\pi Gm$ would be additional dictionary input. The singular particle interpretation is not derived from an observation-boundary identity.

Regular charts are Minkowski charts, so their curvature and Einstein tensor vanish. The singular interpretation is the supplied conical dictionary, consistent with the classical setting of DJH1984,Kadar2005.