Abstract

We construct and evaluate the renormalized $U(1)$ current of the massless charged Friedrichs spectral ground state on the regular conical spacetime prescribed in I-B. For angular equivariance $\Phi (\phi +2\pi )=e^{2\pi \mathrm{i}\beta }\Phi (\phi )$, opening parameter $0<\alpha <1$, and the convention $J_{\mu }=\mathrm{i}\{{\Phi }^{†}{D}_{\mu }\Phi -(D_{\mu }\Phi {)}^{†}\Phi \}$, the only nonzero covariant component is

The physical orthonormal component is $⟨J_{\hat{\phi }}{⟩}_{\mathrm{ren}}=⟨J_{\phi }{⟩}_{\mathrm{ren}}/(\alpha r)$.

The principal contribution is a convention-complete, extension-dependent derivation that remains valid at every regular point, including cone angles at which an image boundary meets the diagonal. We retain the complex, holonomy-odd first jet of the Hadamard remainder; replacing it by the real part used for the stress tensor would incorrectly erase the current. A finite-character quotient calculation gives an independent exact sign and normalization check. The result agrees with the established cosmic-string current literature and is not claimed as a new cosmic-string coefficient.

Unlike the Wick square and stress tensor, the current distinguishes charge-conjugate holonomy sectors. Within the fixed-background Friedrichs family, the Wick square together with this current determines the fractional holonomy. This provides a useful orientation-sensitive diagnostic, not general state tomography. The current is conserved and independent of the curvature coupling on the regular host. No apex extension of the observable, electromagnetic backreaction, Einstein dynamics, interacting algebra, or downstream programme closure is asserted.