Axial boundary term in the mass of a cylindrical spacetime (${\mu }_{\mathrm{bulk}}=c^2[f^{\prime }(0^{+})-f^{\prime }(R^{-})]/(4G)$)

For $d{s}^2=-(d{x}^0{)}^2+d{r}^2+f(r{)}^{2}d{\varphi }^2+d{z}^2$, the smooth bulk mass density is $-c^2{f}^{\prime \prime }/(8\pi Gf)$. Integrate using proper transverse area $2\pi fdr$ to obtain the boundary term. With no shell at the core edge, $f^{\prime }(R^{-})$ equals the exterior conical slope. A regular axis requires $f(0)=0$ and $f^{\prime }(0)=1$ for angular period $2\pi$. A different printed inner derivative gives a different bulk integral; any singular-axis matter must be specified separately. This prevents silently applying the standard cosmic-string mass-deficit relation to nonregular cores.