Prescribed conical product and two different supports

Fix a length unit $\ell >0$, speed $c>0$, and time origin $T_{\ast }$. Write

The regular physical spacetime is

where $r>0$ and $\varphi$ has period $2\pi$. Its stipulated conical completion $\overline{M}$ has

No smooth field equation is imposed on $\mathcal{W}$.

Use a cut transverse wedge of opening $3\pi /2$, with developed directions

Sector $i$, $i=0,1,2$, has dimensionless coordinates

Physical spatial coordinates in that sector are $\ell x$. Cone-ray labels are cyclic modulo three, but $e_0$ and $e_3$ are distinct cut-space copies.

The upper seam is identified with the lower by the affine map

in dimensionless coordinates $(t,X,Y,\zeta )$. It sends $(t,0,-r,\zeta )$ to $(t,r,0,\zeta )$. This is wedge gluing, not a quotient of the entire plane by the finite rotation group.