Purpose, background, and precise scope

The expectation value of a charged scalar current around a flux-carrying cosmic string is an established vacuum-polarization effect. Explicit calculations include Sriramkumar Sriramkumar2001, Sitenko–Vlasii SitenkoVlasii2009, and Bezerra de Mello et al.\ Bezerra2015. Our purpose is not to claim priority for that effect or its massless coefficient. It is to complete one specified observable calculation for the extension and state already fixed in Cheung's I-B-G and I-B-VP, with all signs, bundle conventions, and regularity qualifications explicit CheungIBG,CheungIBVP.

I-B-G constructs intrinsic scalar wave operators after an apex extension has been supplied. I-B-VP-v2 constructs the distinguished Friedrichs spectral ground state and evaluates its Wick square and stress tensor. Its item V4 leaves the twisted current unevaluated. The present manuscript evaluates that current and supplies an exact finite-image cross-check. It does not resolve the other extension, state, apex, or interacting questions listed there.

and its philosophy. Cheung's The Philosophy of Enter distinguishes destructive readout from quantum recovery, a classical record from a decoder, a channel from a thermal machine, and an analogy from an identification CheungPhilosophy. These distinctions constrain the interpretation here: a finite report of a current coefficient is not a continuum field, a state-reconstruction map, or a source law. The essay is an introductory philosophical map, not a premise of the analysis.

spacetime Enter. I-B pairs finite clock and transport operations with a prescribed regular conical product CheungIB. Its finite destructive instrument does not select a scalar field, an apex extension, or a vacuum. Those are separate inputs in this paper. In particular, our expectation value is not extracted from a conditional finite-register density matrix.

is new in this companion. The mathematical contribution within the supplied companion framework is:

1. a renormalized current derived from the full complex regular covariance, rather than from its real part;

2. an all-angle near-diagonal justification, avoiding unjustified differentiation of changing image sectors;

3. an independent finite-character image identity fixing the current's sign and normalization;

4. a precise charge-conjugation discrimination statement within the declared state family.

The last item concerns identifiability in a specified mathematical family. It is not a physical acquisition or universal reconstruction theorem.

being used. The current below is the Noether current of the internal transformation $\Phi \mapsto e^{\mathrm{i}ϵ}\Phi$. It is not the angular-momentum current associated with spacetime rotations. Changing the holonomy parameter changes the background bundle data; it is not itself a continuous symmetry operation within one fixed bundle.