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\title{\textbf{I-B-VP-0: The holonomy-odd vacuum current\\
of the charged Friedrichs ground state}\\[5pt]
\large A local observable on the prescribed conical Enter spacetime}
\author{Ngo Cheung}
\date{October 2026}

\begin{document}
\maketitle

\begin{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
\(\Phi(\varphi+2\pi)=e^{2\pi\ii\beta}\Phi(\varphi)\), opening
parameter \(0<\alpha<1\), and the convention
\(J_\mu=\ii\{\Phi^\dagger D_\mu\Phi-(D_\mu\Phi)^\dagger\Phi\}\),
the only nonzero covariant component is
\[
 \langle J_\varphi\rangle_\ren
 =\frac{\beta(1-\beta)(2\beta-1)}
        {12\pi^2\alpha^2r^2},
 \qquad 0\leq\beta<1.
\]
The physical orthonormal component is
\(\langle J_{\hat\varphi}\rangle_\ren
 =\langle J_\varphi\rangle_\ren/(\alpha r)\).

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.
\end{abstract}

\tableofcontents

%======================================================================
\section{Purpose, background, and precise scope}
\label{sec:introduction}
%======================================================================

The expectation value of a charged scalar current around a
flux-carrying cosmic string is an established vacuum-polarization
effect. Explicit calculations include Sriramkumar
\cite{Sriramkumar2001}, Sitenko--Vlasii \cite{SitenkoVlasii2009},
and Bezerra de Mello et al.\ \cite{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 \cite{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.

\paragraph{Enter and its philosophy.}
Cheung's \emph{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 \cite{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.

\paragraph{The spacetime Enter.}
I-B pairs finite clock and transport operations with a
\emph{prescribed} regular conical product
\cite{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.

\paragraph{What is new in this companion.}
The mathematical contribution within the supplied companion
framework is:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a renormalized current derived from the full complex
regular covariance, rather than from its real part;
\item an all-angle near-diagonal justification, avoiding
unjustified differentiation of changing image sectors;
\item an independent finite-character image identity fixing
the current's sign and normalization;
\item a precise charge-conjugation discrimination statement
within the declared state family.
\end{enumerate}
The last item concerns identifiability in a specified mathematical
family. It is not a physical acquisition or universal reconstruction
theorem.

\paragraph{Symmetry being used.}
The current below is the Noether current of the internal
transformation \(\Phi\mapsto e^{\ii\epsilon}\Phi\).
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.

\paragraph{Source conventions.}
We cite the supplied I-B-G-v2 and I-B-VP-v2 by source labels rather
than unstable theorem numbers. The analytical ingredients needed
for the current are also stated and proved below.
Other companion comparisons retain their own domains and
non-transfer boundaries.

%======================================================================
\section{Prescribed geometry, bundle, and spectral state}
\label{sec:inputs}
%======================================================================

\begin{assumption}[Fixed background and charged field]
\label{ass:inputs}
Fix
\[
 0<\mu<2\pi,\qquad
 \alpha=1-\frac{\mu}{2\pi}\in(0,1).
\]
The regular spacetime is
\[
 M=\RR_t\times(0,\infty)_r
       \times(\RR/2\pi\ZZ)_\varphi\times\RR_z
\]
with
\begin{equation}
 g=-\dd t^2+\dd r^2+\alpha^2r^2\dd\varphi^2+\dd z^2.
 \label{eq:metric}
\end{equation}
The completed apex worldsheet \(\WW=\{r=0\}\) is not part of \(M\).
We use \(\hbar=c=1\).

For \(\beta\in[0,1)\), the charged complex scalar is represented
in a lifted parallel frame by
\begin{equation}
 \Phi(r,\varphi+2\pi,z)
 =e^{2\pi\ii\beta}\Phi(r,\varphi,z).
 \label{eq:equivariance}
\end{equation}
This equation fixes the meridian and character conventions.
The massless equation is
\[
 (-D^\mu D_\mu+\xi R)\Phi=0.
\]
On \(M\), the metric and bundle curvatures vanish, so in a parallel
frame the equation is \(\Box_g\Phi=0\).
No distributional \(\xi R\) interaction at \(\WW\) is modeled.

The spatial operator is the Friedrichs extension
\[
 B_\beta=A_{F,\beta}\geq0
\]
of the covariant expression \(-\Delta_q\), where
\[
 q=\dd r^2+\alpha^2r^2\dd\varphi^2+\dd z^2,\qquad
 \dd V=\alpha r\,\dd r\,\dd\varphi\,\dd z.
\]
\end{assumption}

Its Hilbert space is \(\HH_\beta=L^2_\beta(N,\dd V)\).
The angular functions and radial Bessel orders are
\begin{equation}
 \frac{e^{\ii(n+\beta)\varphi}}{\sqrt{2\pi\alpha}},
 \qquad
 \nu_n=\frac{|n+\beta|}{\alpha},
 \qquad n\in\ZZ.
 \label{eq:orders}
\end{equation}
The Friedrichs form is the closure of
\(\int_N|D u|^2\,\dd V\) on compact smooth sections.
The extension is an input, not a consequence of the regular metric
\cite{CheungIBG,KayStuder}.

\paragraph{Signature and developing frame.}
With
\[
 \mathrm{dev}(t,r,\varphi,z)
 =(t,r\cos(\alpha\varphi),r\sin(\alpha\varphi),z)
\]
and completion metric
\(\eta=\dd t^2-\dd X^2-\dd Y^2-\dd z^2\),
\[
 \mathrm{dev}^*\eta=-g,\qquad
 -\Box_g=\mathrm{dev}^*\Box_\eta.
\]
This is the equation-operator ledger of I-B-G and the ST sheets
\cite{CheungIBG,CheungXAST,CheungXBST}.
An inverse of an action Hessian \(-P\) is \(-G_\pm\), not \(G_\pm\).
The current is defined directly from its Noether expression below,
without importing a Hessian sign.

\subsection{Massless infrared finiteness}

\begin{lemma}[Heat kernel and compact-test covariance]
\label{lem:IR}
The spatial Friedrichs heat kernel, relative to \(\dd V\), is
\begin{align}
 K_\beta(s;x,x')
 ={}&
 \frac{e^{-\Delta z^2/(4s)}}{\sqrt{4\pi s}}\,
 \frac{e^{-(r^2+r'^2)/(4s)}}{4\pi\alpha s}
 \nonumber\\
 &\times\sum_{n\in\ZZ}
 e^{\ii(n+\beta)(\varphi-\varphi')}
 I_{\nu_n}\left(\frac{rr'}{2s}\right).
 \label{eq:heat}
\end{align}
On every compact regular spatial set \(K\),
\begin{equation}
 \sup_{x,x'\in K}|K_\beta(s;x,x')|
 \leq C_Ks^{-3/2-\nu_*},
 \qquad s\geq1,\qquad
 \nu_*=\frac{\min(\beta,1-\beta)}{\alpha}.
 \label{eq:heatbound}
\end{equation}
Every compact smooth spatial section \(v\) belongs to
\(\operatorname{dom}(B_\beta^{-1/4})\), continuously in the
fixed-support test topology.
\end{lemma}

\begin{proof}
The radial Friedrichs transform uses \(J_{\nu_n}(kr)\).
Weber's integral
\[
 \int_0^\infty e^{-sk^2}J_\nu(kr)J_\nu(kr')\,k\,\dd k
 =\frac{e^{-(r^2+r'^2)/(4s)}}{2s}
 I_\nu\left(\frac{rr'}{2s}\right)
\]
gives \eqref{eq:heat} after the angular normalization and
longitudinal heat kernel are included
\cite{Watson,CheungIBG}.
For bounded \(r,r'\) and \(s\geq1\), the Bessel power series gives
\[
 I_\nu\left(\frac{rr'}{2s}\right)
 \leq C_K\frac{(C_K/s)^\nu}{\Gamma(\nu+1)}.
\]
The orders grow linearly in \(|n|\), so their sum is bounded by a
constant times \(s^{-\nu_*}\). This proves \eqref{eq:heatbound}.

The spectral identity
\[
 \norm{B_\beta^{-1/4}v}^2
 =\frac1{\sqrt\pi}\int_0^\infty
 s^{-1/2}\langle v,e^{-sB_\beta}v\rangle\,\dd s
\]
is finite: on \(0<s<1\) use the contraction bound
\(\langle v,e^{-sB_\beta}v\rangle\leq\norm v^2\);
on \(s\geq1\) use \eqref{eq:heatbound} and \(\norm v_{L^1}^2\).
The large-\(s\) integrand is \(O(s^{-2-\nu_*})\).
These bounds also prove continuity.
There is no \(L^2\) zero eigenfunction, since zero form energy
would give a parallel section, which cannot be square integrable
on \(N\).
\end{proof}

\begin{definition}[Distinguished charged Friedrichs ground state]
\label{def:state}
The state \(\omega_F\) is the zero-mean, gauge-invariant quasifree
spectral Fock state with particle and antiparticle sectors on
\(L_\beta\) and its conjugate bundle. Its positive-frequency kernel is
\begin{equation}
 W_\beta(t-t';x,x')
 =\left[
 \frac{e^{-\ii(t-t')\sqrt{B_\beta}}}{2\sqrt{B_\beta}}
 \right](x,x')
 =\omega_F(\Phi(X)\Phi^\dagger(X')).
 \label{eq:W}
\end{equation}
\end{definition}

Indeed, on spacetime tests its positive form is the Gram form of
\[
 f\longmapsto
 \int_\RR e^{\ii t\sqrt{B_\beta}}
            (2\sqrt{B_\beta})^{-1/2}f(t)\,\dd t,
\]
which is finite by Lemma~\ref{lem:IR}. The standard Fock construction
has nonnegative one-particle Hamiltonian \(\sqrt{B_\beta}\)
\cite{Wald1994}. With the conventions of I-B-G,
\[
 [\Phi(X),\Phi^\dagger(X')]
 =-\ii E_\beta(X,X'),\qquad
 E_\beta(t)=\frac{\sin(t\sqrt{B_\beta})}{\sqrt{B_\beta}}.
\]
This is the state of
I-B-VP-v2 \SourceLabel{def:state}, not a claim of uniqueness among
all algebraic ground states \cite{CheungIBVP}.

\subsection{Euclidean kernel and analytic remainder}

For Euclidean points \(X=(\tau,r,\varphi,z)\), set
\begin{equation}
 \cosh\eta=
 \frac{r^2+r'^2+\Delta\tau^2+\Delta z^2}{2rr'},
 \qquad
 \vartheta=\alpha(\varphi-\varphi').
 \label{eq:eta}
\end{equation}

\begin{proposition}[Mode kernel and regular diagonal]
\label{prop:kernel}
For \(\eta>0\), the Euclidean covariance is
\begin{equation}
 G_{E,\beta}(X,X')
 =\frac1{8\pi^2\alpha rr'\sinh\eta}
 \sum_{n\in\ZZ}
 e^{-|n+\beta|\eta/\alpha}
 e^{\ii(n+\beta)(\varphi-\varphi')}.
 \label{eq:GE}
\end{equation}
On a sufficiently small developing neighbourhood of every
regular point, in a local parallel frame,
\begin{equation}
 H_{E,\beta}(X,X')
 =G_{E,\beta}(X,X')-\frac1{4\pi^2\sigma_E^2(X,X')},
 \label{eq:H}
\end{equation}
where
\[
 \sigma_E^2
 =r^2+r'^2-2rr'\cos\vartheta+\Delta\tau^2+\Delta z^2,
\]
is jointly real-analytic, including the diagonal.
It is a bisolution and is Hermitian under exchange of endpoints.
This holds for every \(0<\alpha<1\), including image-boundary
parameter values.
\end{proposition}

\begin{proof}
Multiply \eqref{eq:heat} by
\((4\pi s)^{-1/2}e^{-\Delta\tau^2/(4s)}\) and integrate in \(s\).
The large-\(s\) integral converges on compact regular sets by
\eqref{eq:heatbound}; the small-\(s\) integral defines a distribution.
Substitute \(\zeta=rr'/(2s)\) and use
\[
 \int_0^\infty e^{-\zeta\cosh\eta}I_\nu(\zeta)\,\dd\zeta
 =\frac{e^{-\nu\eta}}{\sinh\eta}.
\]
Absolute convergence for \(\eta>0\) gives \eqref{eq:GE}
\cite{Watson}. This also derives the mode kernel used in
I-B-VP-v2 \SourceLabel{prop:euclidean}
\cite{CheungIBVP}.

The heat-integrated kernel is a fundamental distribution for
\(-\Delta_E\) in each endpoint. In a parallel Cartesian chart,
the direct term in \eqref{eq:H} has the same delta source.
Their difference therefore satisfies both homogeneous equations
across the diagonal. It is annihilated by the elliptic operator
\[
 \Delta_{E,X}+\Delta_{E,X'}
\]
on the product chart. Elliptic and analytic elliptic regularity
make the difference jointly real-analytic
\cite{Hormander,MorreyNirenberg}. Hermiticity follows from the
self-adjoint heat kernel.

This argument applies to the total kernel, not separately to
moving image-sector indicators.
\end{proof}

The spectral continuation
\[
 \frac{e^{-a\sqrt{B_\beta}}}{2\sqrt{B_\beta}},
 \qquad a>0,\qquad a\longrightarrow\epsilon+\ii(t-t')
\]
gives \eqref{eq:W}. Continuing the analytic remainder in
Proposition~\ref{prop:kernel} gives, on a sufficiently small
Lorentzian neighbourhood,
\begin{equation}
 W_\beta(X,X')
 =\frac1{4\pi^2\bigl(
 |\mathbf X-\mathbf X'|^2-(t-t'-\ii0)^2\bigr)}
 +H_\beta(X,X'),
 \label{eq:localHadamard}
\end{equation}
with smooth \(H_\beta\). This is the local Hadamard property
proved in I-B-VP-v2 \SourceLabel{thm:hadamard}
\cite{CheungIBVP}. We use only this local statement.
A global microlocal-spectrum characterization and behaviour at
\(\WW\) are not established here.

%======================================================================
\section{Noether normalization and renormalized current}
\label{sec:definition}
%======================================================================

\subsection{The classical normalization}

Use the local charged action
\[
 S[\Phi]=-\int_M
 \left((D_\mu\Phi)^\dagger D^\mu\Phi+\xi R\Phi^\dagger\Phi\right)
 \dd\mathrm{vol}_g.
\]
The global phase transformation \(\delta\Phi=\ii\epsilon\Phi\)
has Noether current
\begin{equation}
 J_\mu
 =\ii\left(
 \Phi^\dagger D_\mu\Phi-(D_\mu\Phi)^\dagger\Phi
 \right).
 \label{eq:classicalcurrent}
\end{equation}
The normalization is unit \(U(1)\) charge. An electric current
with charge \(e\) is \(eJ_\mu\) when the same connection and
orientation conventions are retained.

Locally,
\[
 \Phi=\frac{\phi_1+\ii\phi_2}{\sqrt2},
 \qquad
 J_\mu=\phi_2\partial_\mu\phi_1-\phi_1\partial_\mu\phi_2
\]
in a parallel frame. This is compatible with
\(T^{\mathrm{complex}}=T[\phi_1]+T[\phi_2]\).
There is no additional factor of two to append to
\eqref{eq:classicalcurrent}.

\subsection{Point splitting and the complex first jet}

\begin{definition}[Renormalized one-point current]
\label{def:current}
In a local parallel frame, define
\begin{equation}
 \mathcal J_\mu
 :=\langle J_\mu\rangle_\ren
 =\ii\left[
  \left(\partial_\mu-\partial_{\mu'}\right)H_\beta(X,X')
 \right],
 \label{eq:currentdefinition}
\end{equation}
where brackets mean coincidence and primed covectors are
parallel transported before identification.

Equivalently, one may use the locally regularized symmetrized
two-point function with coefficient \(\ii/2\).
For spatial components the same formula is obtained from
\(H_{E,\beta}\) at equal Euclidean times.
\end{definition}

At spacelike-separated endpoints the fields commute, so
\[
 \ii(\partial_\mu-\partial_{\mu'})
 \omega_F(\Phi(X)\Phi^\dagger(X'))
\]
has exactly the ordering in \eqref{eq:classicalcurrent} after
coincidence. The Hadamard subtraction extends this to the
local definition \eqref{eq:currentdefinition}.
The difference of endpoint derivatives is essential.

\begin{remark}[Do not replace the kernel by its real part]
\label{rem:imaginary}
The stress tensor in I-B-VP uses \(2\operatorname{Re}H_\beta\).
The current does not. Its information is in the imaginary,
endpoint-antisymmetric first jet of the \emph{complex} kernel.
Hermiticity gives
\[
 \left[
 (\partial_\mu-\partial_{\mu'})H_\beta
 \right]
 =2\ii\,\operatorname{Im}[\partial_\mu H_\beta],
\]
and consequently
\[
 \mathcal J_\mu=-2\operatorname{Im}[\partial_\mu H_\beta].
\]
Replacing \(H_\beta\) by its real part would incorrectly set this
current to zero.
Symmetrizing operator order is not averaging the two
charge-conjugate bundle sectors.
\end{remark}

The charged Hadamard current prescription and its finite
renormalization freedom are treated in
\cite{BalakumarWinstanley,KleinZahn}.

\begin{assumption}[Renormalization class]
\label{ass:ren}
The current has the fixed Noether normalization
\eqref{eq:classicalcurrent}. Renormalization is local and gauge
covariant, state independent, compatible with the standard
engineering dimension and charge conjugation, and normalized to
zero in the ordinary Minkowski vacuum with trivial connection.
Only the usual local background counterterms are allowed.

No preferred radial vector, angular Killing field, global Wilson
loop, or apex-supported distribution is admitted as an additional
local subtraction datum.
\end{assumption}

\begin{proposition}[Well-definedness, gauge invariance, and conservation]
\label{prop:well-defined}
Definition~\ref{def:current} gives a real, smooth, gauge-invariant
one-form on \(M\). In the class of
Assumption~\ref{ass:ren}, no nonzero local background ambiguity
survives on this regular flat host. Moreover,
\begin{equation}
 \nabla_\mu\mathcal J^\mu=0
 \label{eq:conservation}
\end{equation}
as an identity of smooth functions, and hence of distributions
on \(M\).
\end{proposition}

\begin{proof}
Smoothness and independence of the direction of coincidence
follow from Proposition~\ref{prop:kernel} and
\eqref{eq:localHadamard}. Hermiticity proves reality.
Using covariant derivatives in both endpoints makes the
coincidence expression gauge invariant.

The standard charged-current ambiguity includes a multiple of
\(\nabla^\nu F_{\nu\mu}\), which vanishes here
\cite{BalakumarWinstanley,KleinZahn}.
More directly, a locally covariant, state-independent background
covector on this host can be evaluated in a flat parallel chart.
The curvature jets vanish, and a flat connection is locally gauge
equivalent to zero. There is no nonzero Lorentz-invariant covector
built from these local data. Thus the allowed local ambiguity is
zero. This reasoning does not classify nonlocal additions or
distributions on the removed worldsheet.

Finally, in a parallel Cartesian chart, differentiation of a
coincidence limit acts as \(\partial+\partial'\). Therefore
\[
 \partial^\mu\mathcal J_\mu
 =\ii\left[
 (\partial^\mu+\partial^{\mu'})
 (\partial_\mu-\partial_{\mu'})H_\beta
 \right]
 =\ii[(\Box_X-\Box_{X'})H_\beta]=0.
\]
The mixed derivatives cancel, and the two remaining terms vanish
by the bisolution equations. The calculation is covariant under
the local identifications used above.
\end{proof}

\begin{remark}[A current-specific ambiguity statement]
The preceding argument is not inferred from the stress-tensor
ambiguity theorem. The two observables have different
renormalization freedoms. Analyticity makes the selected
point-split expression finite; the declared covariance class
controls its allowed finite changes.
Neither statement supplies apex counterterms or a contact
classification for a current--current product.
\end{remark}

\subsection{Parallel and periodic gauges}

Write the connection as \(D=\partial+\ii a\). In the lifted
parallel representation \(a=0\). The periodic field
\[
 v=e^{-\ii\beta\varphi}\Phi
\]
has
\[
 D_\varphi v=(\partial_\varphi+\ii\beta)v,\qquad
 H_{\mathrm{per}}(X,X')
 =e^{-\ii\beta(\varphi-\varphi')}H_{\mathrm{par}}(X,X').
\]
Thus
\begin{equation}
 \mathcal J_\varphi
 =\ii\left[
 \left(\partial_\varphi-\partial_{\varphi'}+2\ii\beta\right)
 H_{\mathrm{per}}
 \right]
 =\ii[(\partial_\varphi-\partial_{\varphi'})H_{\mathrm{par}}].
 \label{eq:gaugecheck}
\end{equation}
The endpoint phase must also be included in the subtracted
parametrix in periodic gauge.
Using ordinary derivatives on \(H_{\mathrm{per}}\) alone would
produce the spurious shift \(2\beta[H_{\mathrm{par}}]\).

Changing angular lifts multiplies a bundle kernel by the
corresponding character. The local fiber identification in the
coincidence operation cancels that change. The final current is
a global ordinary one-form.

%======================================================================
\section{Exact evaluation}
\label{sec:evaluation}
%======================================================================

Define the third Bernoulli polynomial
\begin{equation}
 \mathsf B_3(\beta)
 =\beta^3-\frac32\beta^2+\frac12\beta
 =\frac12\beta(1-\beta)(1-2\beta).
 \label{eq:B3}
\end{equation}

\begin{lemma}[The signed angular sum]
\label{lem:signed}
For \(a>0\),
\begin{equation}
 \mathcal A_\beta(a)
 :=\sum_{n\in\ZZ}(n+\beta)e^{-|n+\beta|a}
 =-\frac{\dd}{\dd a}
 \left\{
 \frac{\sinh((\frac12-\beta)a)}{\sinh(a/2)}
 \right\}.
 \label{eq:signedsum}
\end{equation}
As \(a\downarrow0\),
\begin{equation}
 \mathcal A_\beta(a)
 =\frac23\mathsf B_3(\beta)a+O(a^3).
 \label{eq:signedexpansion}
\end{equation}
\end{lemma}

\begin{proof}
Separate \(n\geq0\) from \(n\leq-1\). The difference of the
corresponding unweighted geometric sums is
\[
 \sum_{j=0}^\infty
 \left(e^{-(j+\beta)a}-e^{-(j+1-\beta)a}\right)
 =\frac{\sinh((\frac12-\beta)a)}{\sinh(a/2)}.
\]
Its negative derivative is \eqref{eq:signedsum}.
All series and derivatives converge absolutely for \(a>0\).

If \(t=1-2\beta\), then
\[
 \frac{\sinh(ta/2)}{\sinh(a/2)}
 =t+\frac{t^3-t}{24}a^2+O(a^4)
 =(1-2\beta)-\frac{\mathsf B_3(\beta)}3a^2+O(a^4).
\]
Differentiate to obtain \eqref{eq:signedexpansion}.
\end{proof}

\begin{theorem}[Charged Friedrichs vacuum current]
\label{thm:current}
For Assumptions~\ref{ass:inputs} and \ref{ass:ren}, the current
of Definition~\ref{def:current} is
\begin{equation}
 \boxed{
 \mathcal J
 =-\frac{\mathsf B_3(\beta)}
         {6\pi^2\alpha^2r^2}\,\dd\varphi
 =\frac{\beta(1-\beta)(2\beta-1)}
         {12\pi^2\alpha^2r^2}\,\dd\varphi.}
 \label{eq:main}
\end{equation}
In particular,
\[
 \mathcal J_t=\mathcal J_r=\mathcal J_z=0.
\]
The formula is valid for every \(0<\alpha<1\) and every
\(0\leq\beta<1\), including image-boundary parameters.
\end{theorem}

\begin{proof}
Use a longitudinal spacelike split:
\[
 t=t',\qquad r=r',\qquad
 \varphi=\varphi',\qquad z-z'=s\ne0.
\]
Equivalently on the Euclidean section, \(\Delta\tau=0\) and
\[
 \cosh\eta=1+\frac{s^2}{2r^2}.
\]
This split requires no convention for a Euclidean time component
of the current.

At fixed \(s\ne0\), differentiate \eqref{eq:GE} in the angular
variables before setting their separation to zero. Absolute
convergence holds even after multiplication by \(n+\beta\).
The angular derivative of the direct flat term is zero at
\(\varphi=\varphi'\). Hence
\begin{align}
 &\ii(\partial_\varphi-\partial_{\varphi'})
 H_{E,\beta}\big|_{\Delta\varphi=0}
 \nonumber\\
 &\qquad
 =-\frac1{4\pi^2\alpha r^2\sinh\eta}
 \mathcal A_\beta(\eta/\alpha).
 \label{eq:regulatedcurrent}
\end{align}
By Lemma~\ref{lem:signed},
\[
 \frac{\mathcal A_\beta(\eta/\alpha)}{\sinh\eta}
 \longrightarrow \frac{2\mathsf B_3(\beta)}{3\alpha}.
\]
This proves \eqref{eq:main}. Analyticity of the regular remainder
makes the result the coincidence limit, independently of the
chosen split.

The Euclidean kernel is even in \(\Delta\tau\) and \(\Delta z\),
so its regular remainder has vanishing first difference
derivatives in those variables at coincidence.
It is also symmetric in \(r,r'\) at fixed angular difference,
so the radial difference derivative vanishes.
Continuation gives the stated Lorentzian components.
\end{proof}

\begin{remark}[The limits are taken in the required order]
The series
\(\sum_n(n+\beta)\) is not assigned a value by formal
coincidence. We first differentiate the convergent kernel for
\(s\ne0\), sum \eqref{eq:signedsum}, and only then take the
regular coincidence limit. This is ordinary point splitting,
not a replacement of the proof by a summability prescription
for a divergent numerical series.
\end{remark}

%======================================================================
\section{Properties and information carried by the current}
\label{sec:properties}
%======================================================================

\subsection{Components, scaling, and charge conjugation}

The positive angular orthonormal coframe is
\(e^{\hat\varphi}=\alpha r\,\dd\varphi\). Thus
\begin{align}
 \mathcal J_\varphi
 &=\frac{\beta(1-\beta)(2\beta-1)}
         {12\pi^2\alpha^2r^2},
 \nonumber\\
 \mathcal J_{\hat\varphi}
 &=\frac{\beta(1-\beta)(2\beta-1)}
         {12\pi^2\alpha^3r^3},
 \nonumber\\
 \mathcal J^\varphi
 &=\frac{\beta(1-\beta)(2\beta-1)}
         {12\pi^2\alpha^4r^4}.
 \label{eq:components}
\end{align}
The \(r^{-2}\) statement concerns the covariant component in a
dimensionless angular coordinate. The physical orthonormal
current has the expected \(r^{-3}\) scaling.

Under
\[
 \delta_\lambda(t,r,\varphi,z)
 =(\lambda t,\lambda r,\varphi,\lambda z),
\]
the one-form satisfies
\[
 \delta_\lambda^*\mathcal J=\lambda^{-2}\mathcal J.
\]
Directly,
\[
 \nabla_\mu\mathcal J^\mu
 =\frac1{\alpha r}\partial_\varphi
       (\alpha r\,\mathcal J^\varphi)=0,
\]
in agreement with Proposition~\ref{prop:well-defined}.

\begin{corollary}[Holonomy parity and zeros]
\label{cor:parity}
Extend the result periodically in \(\beta\). Then
\[
 \mathcal J_{\beta+k}=\mathcal J_\beta\quad(k\in\ZZ),
 \qquad
 \mathcal J_{1-\beta}=-\mathcal J_\beta.
\]
On \(0\leq\beta<1\), its only zeros are
\(\beta=0\) and \(\beta=1/2\).
It is negative in the positive covariant angular component for
\(0<\beta<1/2\), and positive for \(1/2<\beta<1\).
\end{corollary}

\begin{proof}
Integer shifts reindex the angular modes.
Equation~\eqref{eq:B3} gives
\(\mathsf B_3(1-\beta)=-\mathsf B_3(\beta)\).
The factorization in \eqref{eq:main} gives the remaining claims.
\end{proof}

The relevant ``symmetry under \(\beta\leftrightarrow1-\beta\)''
is therefore \emph{oddness}, not evenness.
Reversing the meridian or the current's charge convention reverses
the corresponding sign.

\subsection{Flat-space Aharonov--Bohm limit}

At fixed \(r>0,\beta\),
\begin{equation}
 \mathcal J_\varphi
 \longrightarrow
 \frac{\beta(1-\beta)(2\beta-1)}{12\pi^2r^2}
 \qquad(\alpha\uparrow1).
 \label{eq:flatlimit}
\end{equation}
This is the punctured flat-space flux result, not the ordinary
untwisted Minkowski vacuum. For \(\beta=0\), the current vanishes
for every cone angle. The geometric deficit alone does not supply
an independent charged-scalar character.

\subsection{Compatibility with the even observables}

For reference, set
\[
 P_\beta=\alpha^{-2}(1-6\beta(1-\beta))-1,\qquad
 C_\beta=\alpha^{-4}(1-30\beta^2(1-\beta)^2)-1.
\]
With the same complex-field normalization, I-B-VP-v2 proves
\cite[\SourceLabel{thm:phi2},
      \SourceLabel{thm:T-complex}]{CheungIBVP}
\begin{align}
 F_\beta:=\langle\Phi^\dagger\Phi\rangle_\ren
 &=\frac{P_\beta}{48\pi^2r^2},
 \label{eq:wick}\\
 \langle T^\mu{}_\nu\rangle_\ren
 &=\frac{C_\beta}{720\pi^2r^4}\diag(1,1,-3,1)
 \nonumber\\
 &\quad+
 2\left(\xi-\frac16\right)
 \frac{P_\beta}{48\pi^2r^4}\diag(4,-2,6,4).
 \label{eq:stress}
\end{align}
Their mixed-component order is \((t,r,\varphi,z)\).
Both are even under \(\beta\leftrightarrow1-\beta\).
The current is odd. Its \(\xi\)-independence is consistent with
\eqref{eq:wick}--\eqref{eq:stress}: on the regular host, changing
\(\xi\) changes the stress-tensor improvement but neither the
chosen equation, the spectral state, nor the internal Noether
current.

\begin{corollary}[Conditional identification of fractional holonomy]
\label{cor:identification}
Fix \(\alpha,r\), the orientation, and the state family of
Definition~\ref{def:state}. The pair
\[
 \bigl(F_\beta,\mathcal J_{\varphi,\beta}\bigr)
\]
determines \(\beta\in[0,1)\).
In contrast, \(F_\beta\) and
\(\langle T^\mu{}_\nu\rangle_\ren\) together do not distinguish
\(\beta\) from \(1-\beta\) when these are distinct.
\end{corollary}

\begin{proof}
Equation~\eqref{eq:wick} determines
\[
 u=\beta(1-\beta)
 =\frac{1-\alpha^2(1+48\pi^2r^2F_\beta)}6.
\]
For \(0<u<1/4\), the two possible values are
\[
 \beta_\pm=\frac{1\pm\sqrt{1-4u}}2.
\]
The sign of \(\mathcal J_\varphi\) selects the upper or lower
root by Corollary~\ref{cor:parity}. If \(u=0\), the representative
is \(\beta=0\); if \(u=1/4\), it is \(\beta=1/2\).
The evenness of \eqref{eq:wick}--\eqref{eq:stress} proves the
last assertion.
\end{proof}

This corollary assumes the entire fixed-background,
fixed-extension spectral family in advance.
It does not determine an arbitrary state, select the extension,
or provide a finite measurement procedure acquiring these
continuum expectation values.

%======================================================================
\section{An independent exact quotient check}
\label{sec:quotient}
%======================================================================

For
\[
 \alpha=\frac1N,\qquad
 \beta=\frac pN,\qquad
 N\geq2,\quad p\in\{0,\ldots,N-1\},
\]
the character is compatible with the cyclic quotient. The
Friedrichs Euclidean kernel is the finite character-weighted
image kernel
\cite[\SourceLabel{cor:structure}]{CheungIBG}.
In a developing parallel frame, its regular part is
\begin{equation}
 H_{E,p/N}(X,X')
 =\sum_{k=1}^{N-1}
 \frac{e^{2\pi\ii pk/N}}
 {4\pi^2|\mathbf X_E-\mathcal R_k\mathbf X'_E|^2},
 \label{eq:quotientkernel}
\end{equation}
where \(\mathcal R_k\) rotates the transverse plane by
\(2\pi k/N\) and fixes \(\tau,z\).
The sum contains the full quotient images, also for even \(N\).

\begin{lemma}[Finite character identity]
\label{lem:finiteidentity}
For \(p=0,\ldots,N\),
\begin{equation}
 \sum_{k=1}^{N-1}
 \sin\left(\frac{2\pi pk}{N}\right)
 \cot\left(\frac{\pi k}{N}\right)
 \csc^2\left(\frac{\pi k}{N}\right)
 =\frac23p(N-p)(N-2p).
 \label{eq:finiteidentity}
\end{equation}
\end{lemma}

\begin{proof}
Call the left side \(S_p\), and put \(x_k=\pi k/N\).
Then \(S_0=S_N=0\), while
\[
 S_{p+1}-2S_p+S_{p-1}
 =-4\sum_{k=1}^{N-1}\sin(2px_k)\cot x_k.
\]
For \(1\leq p\leq N-1\),
\[
 \sin(2px)\cot x
 =1+\cos(2px)+2\sum_{j=1}^{p-1}\cos(2jx).
\]
Summing over \(k\), and using the finite root-of-unity sums,
gives
\[
 \sum_{k=1}^{N-1}\sin(2px_k)\cot x_k=N-2p.
\]
Thus the second difference of \(S_p\) is \(8p-4N\).
The polynomial
\[
 f_p=\frac23p(N-p)(N-2p)
\]
has the same second differences and endpoint values.
Their difference has zero second difference and zero values at
\(0,N\), so it vanishes identically.
\end{proof}

\begin{proposition}[Finite-image current and sign]
\label{prop:quotient}
Direct differentiation of \eqref{eq:quotientkernel} gives
\begin{equation}
 \mathcal J_\varphi
 =-\frac1{8\pi^2Nr^2}
 \sum_{k=1}^{N-1}
 \sin\left(\frac{2\pi pk}{N}\right)
 \cot\left(\frac{\pi k}{N}\right)
 \csc^2\left(\frac{\pi k}{N}\right).
 \label{eq:quotientcurrent}
\end{equation}
It equals Theorem~\ref{thm:current} at
\((\alpha,\beta)=(1/N,p/N)\).
\end{proposition}

\begin{proof}
The image denominator is
\[
 D_k^2=r^2+r'^2
 -2rr'\cos\left(\frac{\varphi-\varphi'}N-\frac{2\pi k}N\right)
 +\Delta\tau^2+\Delta z^2.
\]
At coincidence \(D_k^2=4r^2\sin^2(\pi k/N)\).
Applying \(\ii(\partial_\varphi-\partial_{\varphi'})\)
to \eqref{eq:quotientkernel}, and pairing conjugate images,
gives \eqref{eq:quotientcurrent}.
Lemma~\ref{lem:finiteidentity} then yields
\[
 \mathcal J_\varphi
 =\frac{p(N-p)(2p-N)}{12\pi^2Nr^2},
\]
which is precisely \eqref{eq:main}.
This calculation does not use the small-separation
Bernoulli expansion.
\end{proof}

For example, the following are exact rational identities, not
numerical execution reports:
\begin{center}
\begin{tabular}{@{}cccc@{}}
\toprule
\(N\) & \(p\) & \(\pi^2r^2\mathcal J_\varphi\) & Evidence\\
\midrule
2 & 1 & \(0\) & \ANALYTIC\\
3 & 1 & \(-1/18\) & \ANALYTIC\\
3 & 2 & \(1/18\) & \ANALYTIC\\
4 & 1 & \(-1/8\) & \ANALYTIC\\
4 & 3 & \(1/8\) & \ANALYTIC\\
\bottomrule
\end{tabular}
\end{center}

A strict geometric image set can omit boundary representatives
when \(N\) is even. Equation~\eqref{eq:quotientkernel} avoids that
ambiguity by using the full quotient sum. One must not use a
boundary omission to redefine the underlying covariance.

%======================================================================
\section{Literature comparison}
\label{sec:literature}
%======================================================================

A direct comparison is available in Sriramkumar
\cite{Sriramkumar2001}, equations~(2.10), (2.12), and (2.13).
Its covariant angular component is
\[
 \langle \widehat J_\varphi\rangle
 =-\frac{e\,\gamma(1-\gamma)(1-2\gamma)}
          {12\pi^2\alpha^2\rho^2}.
\]
Set \(\rho=r\), \(\gamma=\beta\), and divide by \(e\).
Its periodic-gauge convention
\(D_\varphi=\partial_\varphi+\ii eA_\varphi\), with
\(eA_\varphi=\gamma\), is \eqref{eq:gaugecheck}.
Its symmetrized covariance uses coefficient \(\ii e/2\);
our positive-frequency convention uses \(\ii\).
The formulas therefore agree, including normalization and sign.
Its opposite metric signature does not change this comparison
of the explicitly defined lower-index component.

This checks an established coefficient, not the wider
fluctuation calculations of that paper.
The current literature also includes massive and finite-core
models \cite{SitenkoVlasii2009,Bezerra2015}.
We import neither their core physics nor their Maxwell response.
A comprehensive audit of all conventions and extensions in that
literature is not claimed.

%======================================================================
\section{Companion interfaces and reporting discipline}
\label{sec:interfaces}
%======================================================================

\subsection{Supported Green replacement does not replace the state}

In an angularly small chart
\[
 U\subset\{r>r_U,\ \varphi\in I\},
 \qquad r_U>0,\qquad |I|<\pi,
\]
I-B-G defines an additional-arrival threshold
\(T_{\mathrm{arr}}(x,x')\) and proves
\[
 T_{\mathrm{arr}}(x,x')
 \geq2r_U\sin(\pi\alpha/2).
\]
Replacing an intrinsic retarded or advanced scalar operation by
a completion operation requires its signed pair window,
a local parallel frame, and the independent retained-shadow
and compact-reserve conditions
\cite[\SourceLabel{thm:transfer},
      \SourceLabel{ass:transfer-contract}]{CheungIBG}.
The ST scalar sheets retain these support obligations
\cite{CheungXAST,CheungXBST,CheungXCST}.

This theorem does not transfer a symmetric covariance or a
vacuum state. In particular, it does not imply that the current
must vanish in a short laboratory window.
The current is a local observable of a globally specified
spectral state, not a retarded signal with a newly assigned
emission event.

XI-E's named polynomial Hadamard domain
\cite[\SourceLabel{def:domain},
      \SourceLabel{thm:star}]{CheungXIE}
has its own flat-background multiplet and product.
It does not select the present positive state or construct a
conical interacting algebra. Conversely, evaluating the
one-point current does not transfer that multiplet or establish
interacting gauge closure.

\subsection{Finite reports are not continuum acquisition}

A finite report may contain the inputs
\((\alpha,\beta,r)\), the current convention, a formula identifier,
an approximation, and its error status.
The I-B/En09-A destructive supplied-report interface is
\[
 \mathcal M(X)
 =\sum_j\proj{j,w_j}_R\otimes D_jXD_j^\dagger,
 \qquad
 D_j=\bra{\tau_j}_C\otimes I_A.
\]
It is completely positive and trace preserving because
\[
 \sum_jD_j^\dagger D_j=I_{CA}.
\]
Expanding \(X\) in the \(\{\tau_j\}\) basis gives
\[
 \Tr_R\mathcal M(X)=\Tr_CX,
\]
also after an untouched reference is adjoined
\cite{CheungIB,CheungEn09,CheungO1ST}.
The output contains the surviving \(A\), not the consumed
quantum clock \(C\). It reports installed words; it does not
evaluate an unknown vacuum current by measurement.

The finite-test coverage distinction of Oai282-A
\cite{CheungOai282A} also applies at the logical level.
Finitely many linear evaluations cannot determine a general
distribution in an infinite-dimensional class. For currents,
an explicit infinite-dimensional test class consists of
\[
 j^\mu=\partial_\nu A^{\nu\mu},
 \qquad A^{\nu\mu}=-A^{\mu\nu}\in C_c^\infty
\]
in a regular flat chart. These are conserved, and disjoint
compact supports give arbitrarily large linearly independent
families. Finite evaluation maps have nontrivial kernels there.
This example is not asserted to be a difference of two
Friedrichs-state currents. It shows why finite tests alone
cannot prove a general continuum identity.

\subsection{Evidence gate}

We use the following evidence labels:
\begin{description}[style=nextline,leftmargin=2.1cm]
\item[\ANALYTIC]
Established by the displayed proof, with cited analytical
inputs identified.
\item[\SUPPLIED]
An inherited computational report, not independently executed.
\item[\UNCERTIFIED]
A computational enclosure or verification claim whose required
artifacts or total error bounds have not been supplied.
\end{description}

No numerical current runs, raw current data, or execution logs
were supplied for this companion, and none are claimed.
In particular, the numerical discrepancies reported for the
Wick square and stress tensor in I-B-VP-v2 are not relabeled
as checks of the current.

Following XVI-O1-ST's residual-retention discipline
\cite{CheungO1ST}, a future \texttt{VERIFY} report must state
its actual scope and retain at least:
\[
 \begin{gathered}
 (\text{source/version},\text{current sign and component},\\
 \text{parameter encodings},\text{frame and lift},
 \text{raw paired values},\\
 \text{precision},\text{cutoffs},\text{tail and input bounds},
 \text{residual},\text{status}).
 \end{gathered}
\]
Agreement at shared rounded inputs is not an input-error bound.
For the closed form, input propagation can use the exact derivatives
\[
 \partial_\alpha\mathcal J_\varphi
 =-\frac2\alpha\mathcal J_\varphi,\qquad
 \partial_r\mathcal J_\varphi
 =-\frac2r\mathcal J_\varphi,
\]
\[
 \partial_\beta\mathcal J_\varphi
 =\frac{-6\beta^2+6\beta-1}{12\pi^2\alpha^2r^2}.
\]
Quadrature or mode-tail errors require separate certificates.
At a zero of the current, relative residuals may be inapplicable.

A successful test that preserves a known discrepancy does not
make the discrepancy vanish. No numerical execution, independent
rebuild, formal verification, or successful compiler run is
asserted in this manuscript.

%======================================================================
\section{Ledger update and remaining questions}
\label{sec:ledger}
%======================================================================

\begingroup
\small
\renewcommand{\arraystretch}{1.1}
\begin{longtable}{@{}L{.29\linewidth}L{.22\linewidth}L{.42\linewidth}@{}}
\toprule
Extension/state & Current status & Boundary\\
\midrule
\endfirsthead
\toprule
Extension/state & Current status & Boundary\\
\midrule
\endhead
\bottomrule
\endfoot

Friedrichs, charged distinguished spectral ground state
&
Constructed and evaluated
&
Theorem~\ref{thm:current}; all \(0<\alpha<1\), \(0\leq\beta<1\),
on \(r>0\).
\\

Friedrichs, untwisted complex spectral ground state
&
Zero current
&
The covariance, Wick square, and stress tensor need not vanish.
\\

Twisted non-Friedrichs \(\mathsf E_0,\mathsf E_\lambda\)
&
\textbf{Open}
&
No state or current evaluation is supplied here.
Nonnegativity alone does not settle infrared finiteness.
\\

Massless untwisted logarithmic \(\mathsf L_L\)
&
Outside the nonnegative square-root prescription
&
Its transverse negative eigenvalue generates a negative
longitudinal spectral band.
\\

Mass-shifted logarithmic or negative-\(\lambda\) families
&
\textbf{Open}
&
The positivity and threshold infrared qualifications of
I-B-G/I-B-VP remain.
\\

Thermal states, spinors, constrained gauge systems
&
\textbf{Open}
&
No such state or constrained apex domain is constructed.
\\
\end{longtable}
\endgroup

The update to I-B-VP-v2 is restricted to the current-evaluation
part of V4. A direct comparison with one current reference has
also been supplied in Section~\ref{sec:literature}; a broader
literature audit is still outside the result.
The remaining scopes are unchanged:
\begin{enumerate}[label=\textbf{V\arabic*.}]
\item \textbf{Open:} prescribed distributional extensions of
the local observables across \(\WW\), including apex-supported
counterterms. No such extension of the current is selected here.
\item \textbf{Open:} a complete global microlocal-spectrum
analysis on the regular non-globally-hyperbolic host.
\item \textbf{Open:} the non-Friedrichs state and local-observable
families, including massive logarithmic thresholds.
\item \textbf{Evaluated here:} the charged Friedrichs spectral
ground-state current. A comprehensive current-literature audit
remains a separate task.
\item \textbf{Open:} an apex curvature-coupling model selecting
or modifying the extension.
\item \textbf{Open:} the corresponding massive-field companion
construction and observable calculations.
\item \textbf{Open:} spinors and constrained gauge systems.
\item \textbf{Open:} thermal and other non-ground states.
\end{enumerate}

Here \textbf{Open} means unresolved in this manuscript or the
stated companion scope, not unknown throughout the literature.

\paragraph{Behaviour at the apex.}
The current is a smooth one-form and therefore a distribution
on \(M\). Its orthonormal component behaves as \(r^{-3}\) when
the coefficient is nonzero. It does not define an ordinary
locally integrable vector field across the completed transverse
origin. This does not preclude distributional extensions; it
means an extension prescription and its ambiguities have not
been chosen. Regular-part conservation alone does not choose
them.

%======================================================================
\section{Non-identifications and conclusion}
\label{sec:conclusion}
%======================================================================

The geometry, character, extension, and state prescription are
inputs. Conservation of this current does not derive the cone
or impose a metric field equation.
The stress-tensor non-sourcing separations in
I-B \SourceLabel{prop:not-source} and
X-A-ST \SourceLabel{prop:noeinstein}
remain applicable in their stated scopes
\cite{CheungIB,CheungXAST}.

A nonzero current could enter a separately declared semiclassical
Maxwell model, but no Maxwell equation is imposed or solved here.
In particular, the flat background connection is not asserted
to solve a self-consistent sourced Maxwell equation with the
computed expectation value. There is no electromagnetic
backreaction or Einstein backreaction claim.

The result is a free-field one-point expectation value, not a
current--current product, a dynamical response hierarchy,
an interacting Ward theorem, or a conical BRST construction.
No external OpenAI catalogue family or documentary separation
is used as a mathematical premise. No corpus-wide nonexistence
claim is made.

No standing downstream status is updated. O.1 remains
\NOTCLOSED{} wherever it was not independently closed; XVI-N
is not discharged, XV-B's quantum-admission exit is not opened,
and \bsr{} is not promoted by this calculation
\cite{CheungO1ST,CheungEn09}.
These programme qualifications do not deny the explicit
free-field current proved here.

\paragraph{Conclusion.}
The complex regular covariance contains an odd first angular
jet that the Wick square and stress tensor do not retain.
Its point-split evaluation gives the exact current
\eqref{eq:main}. Analytic regularity controls coincidence for
all cone angles, and finite-character images independently
check the sign. Together with I-B-VP's even observables, this
completes the three specified local one-point functions for
the charged Friedrichs spectral ground state. It does not
complete the state's higher products, its global microlocal
analysis, or an apex or dynamical completion.

%======================================================================
\appendix
\section{A differentiated image--diffraction identity}
\label{app:contour}
%======================================================================

This appendix records a second representation, with its
differentiation restriction explicit. It is not needed for
Theorem~\ref{thm:current}.

For a chosen angular lift define
\[
 \mathcal K(\vartheta)
 =\{k\in\ZZ:|\vartheta-2\pi\alpha k|<\pi\},
\]
\[
 R_k^2=r^2+r'^2
 -2rr'\cos(\vartheta-2\pi\alpha k)+\Delta z^2,
 \qquad
 \rho(u)^2=r^2+r'^2+2rr'\cosh u+\Delta z^2.
\]
The diffraction weight is
\[
 \Phi_\beta(u,\vartheta)
 =\sum_{n\in\ZZ}
 \sin(\pi\nu_n)e^{-\nu_nu}
 e^{\ii(n+\beta)\vartheta/\alpha}.
\]
I-B-G's angular summation and the Euclidean heat integral give
\cite[\SourceLabel{thm:closed}]{CheungIBG}
\cite[\SourceLabel{prop:contour}]{CheungIBVP}
\begin{align}
 G_{E,\beta}
 ={}&
 \sum_{k\in\mathcal K(\vartheta)}
 \frac{e^{2\pi\ii\beta k}}{4\pi^2(R_k^2+\Delta\tau^2)}
 \nonumber\\
 &-\frac1{\pi\alpha}\int_0^\infty
 \frac{\Phi_\beta(u,\vartheta)}
 {4\pi^2(\rho(u)^2+\Delta\tau^2)}\,\dd u.
 \label{eq:contour}
\end{align}

Assume here that
\(\frac1{2\alpha}\notin\NN\), so zero angular separation is
not an image boundary. Define the real weight
\[
 \Xi_\beta(u)
 =-\ii\alpha\,\partial_\vartheta\Phi_\beta(u,0)
 =\sum_n(n+\beta)\sin(\pi\nu_n)e^{-\nu_nu},
 \qquad u>0.
\]
Then
\begin{align}
 \mathcal J_\varphi
 ={}&
 -\frac{\alpha}{4\pi^2r^2}
 \sum_{0<k<1/(2\alpha)}
 \frac{\sin(2\pi\beta k)\cos(\pi\alpha k)}
      {\sin^3(\pi\alpha k)}
 \nonumber\\
 &+\frac1{8\pi^3\alpha r^2}
 \int_0^\infty
 \frac{\Xi_\beta(u)}{\cosh^2(u/2)}\,\dd u.
 \label{eq:current-contour}
\end{align}
Its value is the polynomial in \eqref{eq:main}.

To justify \eqref{eq:current-contour}, choose a small angular
neighbourhood of zero containing no image boundary. The image
set is constant there. The closed geometric-series expression
for \(\Phi_\beta\) in I-B-G
\SourceLabel{lem:Phi} has denominators bounded away from zero
at \(u=0\), also after one angular derivative. At infinity,
the differentiated series decays exponentially.
The diffraction denominator grows as \(e^u\), uniformly on
compact regular spatial sets. Differentiation under the integral
is therefore justified on this neighbourhood.
Apply \(2\ii\alpha\partial_\vartheta\), set coincidence, pair
the \(k,-k\) image terms, and use
\(\rho(u)^2=4r^2\cosh^2(u/2)\).

When \(\alpha>1/2\), the image sum is empty, so the full current
is represented by the differentiated diffraction term.
This is a representation of a stationary spectral observable,
not a claim about an emitted retarded signal.

\paragraph{Why joint integrability alone is insufficient.}
At an image boundary, local integrability of
\(\Phi_\beta(u,\vartheta)\) does not by itself justify pointwise
differentiation under its integral. Angular derivatives can
worsen the boundary singularity, and changing image indicators
also contribute distributionally.
For such parameters, use the differentiated \emph{total}
distribution in \eqref{eq:contour} and its analytic remainder
from Proposition~\ref{prop:kernel}, or use the mode proof.
No separate pointwise boundary prescription is introduced.

%======================================================================
\begin{thebibliography}{99}
\addcontentsline{toc}{section}{References}
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\bibitem{Sriramkumar2001}
L.~Sriramkumar,
Fluctuations in the current and energy densities around a
magnetic-flux-carrying cosmic string,
\emph{Classical and Quantum Gravity}
\textbf{18} (2001), 1015--1025.
\newblock DOI: \nolinkurl{10.1088/0264-9381/18/6/304}.
\newblock \url{https://arxiv.org/abs/gr-qc/0011074}.
\newblock Used for the explicit current comparison:
equations~(2.10), (2.12), (2.13), and (A2).

\bibitem{SitenkoVlasii2009}
Yu.~A.~Sitenko and N.~D.~Vlasii,
Induced vacuum current and magnetic field in the background
of a cosmic string,
\emph{Classical and Quantum Gravity}
\textbf{26} (2009), 195009.
\newblock DOI: \nolinkurl{10.1088/0264-9381/26/19/195009}.
\newblock \url{https://arxiv.org/abs/0909.0405}.
\newblock Historical and analytical context; its Maxwell
response is not imported.

\bibitem{Bezerra2015}
E.~R.~Bezerra de Mello, V.~B.~Bezerra,
A.~A.~Saharian, and H.~H.~Harutyunyan,
Vacuum currents induced by a magnetic flux around a cosmic
string with finite core,
\emph{Physical Review D} \textbf{91} (2015), 064034.
\newblock DOI: \nolinkurl{10.1103/PhysRevD.91.064034}.
\newblock \url{https://arxiv.org/abs/1411.1258}.
\newblock Cited for the distinction between idealized and
finite-core current problems.

\bibitem{BalakumarWinstanley}
V.~Balakumar and E.~Winstanley,
Hadamard renormalization for a charged scalar field,
\emph{Classical and Quantum Gravity}
\textbf{37} (2020), 065004.
\newblock DOI: \nolinkurl{10.1088/1361-6382/ab6b6e}.
\newblock \url{https://arxiv.org/abs/1910.03666}.
\newblock Used for charged point splitting and the current
renormalization ambiguity, especially equation~(5.6).

\bibitem{KleinZahn}
C.~Klein and J.~Zahn,
Renormalized charged scalar current in the
Reissner--Nordstr\"om--de Sitter spacetime,
\emph{Physical Review D} \textbf{104} (2021), 025009.
\newblock DOI: \nolinkurl{10.1103/PhysRevD.104.025009}.
\newblock \url{https://arxiv.org/abs/2104.06005}.
\newblock Cited for local gauge-covariant current point splitting
and its conserved-current ambiguity, not for its black-hole state.

\bibitem{KayStuder}
B.~S.~Kay and U.~M.~Studer,
Boundary conditions for quantum mechanics on cones and
fields around cosmic strings,
\emph{Communications in Mathematical Physics}
\textbf{139} (1991), 103--139.
\newblock DOI: \nolinkurl{10.1007/BF02102731}.

\bibitem{Watson}
G.~N.~Watson,
\emph{A Treatise on the Theory of Bessel Functions},
2nd edition, Cambridge University Press, 1944.

\bibitem{Hormander}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential Operators I:
Distribution Theory and Fourier Analysis},
2nd edition, Springer, 1990.
\newblock Used for elliptic regularity of distributional solutions.

\bibitem{MorreyNirenberg}
C.~B.~Morrey, Jr., and L.~Nirenberg,
On the analyticity of the solutions of linear elliptic systems
of partial differential equations,
\emph{Communications on Pure and Applied Mathematics}
\textbf{10} (1957), 271--290.
\newblock DOI: \nolinkurl{10.1002/cpa.3160100204}.

\bibitem{Wald1994}
R.~M.~Wald,
\emph{Quantum Field Theory in Curved Spacetime and
Black Hole Thermodynamics},
University of Chicago Press, 1994.
\newblock Cited for the spectral one-particle and quasifree
construction; no global-hyperbolicity theorem is applied to
the entire regular cone.

\medskip
\noindent\textit{Cheung companion records.}
The following unpublished manuscripts and supplied excerpts are
cited by identifier, version, and source label. Documentary use
does not assert independent verification of an entire record,
publication, compiler execution, or formal proof checking.
\medskip

\bibitem{CheungPhilosophy}
N.~Cheung,
\emph{The Philosophy of Enter:
Destructive Readout, Quantum Recovery, and the Refusal of Identification},
version~2, \SourceLabel{EntPhil-v2.tex},
Enter Calculus Companion.
\newblock Introductory philosophical map, not a theorem premise.
\newblock \url{https://enter.grok.me/enter}.

\bibitem{CheungIB}
N.~Cheung,
\emph{I-B-v1: A Finite Enter Spacetime:
Calibrated Destructive Time Readout on a Prescribed Conical Product},
version~1, Enter Calculus Companion.
\newblock Used: the prescribed host,
\SourceLabel{thm:instrument}, and
\SourceLabel{prop:not-source}.
\newblock \url{https://enter.grok.me/browse/I-B}.

\bibitem{CheungIBG}
N.~Cheung,
\emph{I-B-G: Intrinsic and holonomy-twisted Green operators
on a prescribed conical spacetime:
Self-adjoint extensions, diffraction, finite propagation,
and supported transfer from flat-completion laboratories},
version~2, October 2026.
\newblock Supplied source \SourceLabel{I-B-G-v2.tex},
Enter Calculus Companion.
\newblock Used:
\SourceLabel{ass:host},
\SourceLabel{ass:product},
\SourceLabel{thm:positivity},
\SourceLabel{thm:closed},
\SourceLabel{lem:Phi},
\SourceLabel{cor:structure},
\SourceLabel{thm:transfer}, and
\SourceLabel{ass:transfer-contract}.

\bibitem{CheungIBVP}
N.~Cheung,
\emph{I-B-VP: Vacuum polarization of the Friedrichs ground
state on a prescribed conical spacetime:
Local Hadamard subtraction, holonomy, and the renormalized
stress tensor},
version~2, October 2026.
\newblock Supplied source \SourceLabel{I-B-VP-v2.tex},
Enter Calculus Companion.
\newblock Used:
\SourceLabel{lem:infrared},
\SourceLabel{prop:euclidean},
\SourceLabel{def:state},
\SourceLabel{thm:analytic},
\SourceLabel{thm:hadamard},
\SourceLabel{thm:phi2},
\SourceLabel{thm:T-complex}, and
\SourceLabel{prop:contour}.
\newblock Its current-evaluation item V4 is the specific
successor target addressed here.

\bibitem{CheungXAST}
N.~Cheung,
\emph{X-A-ST}, version~1, Enter Calculus Companion.
\newblock Used:
\SourceLabel{prop:signature-ledger},
\SourceLabel{prop:noeinstein}, and the supported-completion scope.
\newblock \url{https://enter.grok.me/browse/X-A-ST}.

\bibitem{CheungXBST}
N.~Cheung,
\emph{X-B-ST: One scalar retarded map on the X-A-ST host},
version~1, Enter Calculus Companion.
\newblock Used:
\SourceLabel{ass:host},
\SourceLabel{def:supported}, and the equation/Hessian
normalization and retained-shadow distinctions.
\newblock \url{https://enter.grok.me/browse/X-B-ST}.

\bibitem{CheungXCST}
N.~Cheung,
\emph{X-C-ST: Second-order scalar completion on the inherited
X-A-ST/X-B-ST host},
version~1, Enter Calculus Companion.
\newblock Cited for compact intermediate-source support
obligations and unchanged product boundaries.
\newblock \url{https://enter.grok.me/browse/X-C-ST}.

\bibitem{CheungXIE}
N.~Cheung,
\emph{XI-E}, version~1, Enter Calculus Companion.
\newblock Supplied excerpts:
\SourceLabel{def:domain} and \SourceLabel{thm:star}.
\newblock Its declared polynomial Hadamard product domain is
not a positive-state selection or a conical interacting construction.
\newblock \url{https://enter.grok.me/browse/XI-E}.

\bibitem{CheungO1ST}
N.~Cheung,
\emph{XVI-O1-ST}, version~5,
\SourceLabel{XVI-O1-ST-v5.tex},
Enter Calculus Companion.
\newblock Used for the complete supplied-report interface and
the integrity-aware distinction between proof, retained discrepancy,
and execution evidence.
\newblock No unrelated closure status is imported.
\newblock \url{https://enter.grok.me/browse/XVI-O1-ST}.

\bibitem{CheungEn09}
N.~Cheung,
\emph{En09-A}, version~1, Enter Calculus Companion.
\newblock Supplied source excerpts concerning destructive
finite reporting, record--residual separation, and standing
non-identifications.
\newblock \url{https://enter.grok.me/browse/En09-A}.

\bibitem{CheungOai282A}
N.~Cheung,
\emph{Oai282-A}, version~1, Enter Calculus Companion.
\newblock Used for the finite-test/coverage distinction
identified by
\SourceLabel{prop:coverage} and \SourceLabel{prop:invisible}
in the supplied companion material.
\newblock No external catalogue theorem is used as a premise.
\newblock \url{https://enter.grok.me/browse/Oai282-A}.

\end{thebibliography}

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