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\title{\textbf{I-B-VP-1: Prescribed apex finite parts\\
on a conical Enter spacetime}\\[5pt]
\large Wick square, stress tensor, and charged current:\\
strong extension, Ward identities, and classified contact freedom}
\author{Ngo Cheung}
\date{October 2026}

\begin{document}
\maketitle

\begin{abstract}
We construct distributional extensions across the prescribed apex
worldsheet of the massless Friedrichs ground-state one-point
observables computed in I-B-VP and I-B-VP-0. The construction fixes
a smooth structure on the topological completion, the conical
volume density, mixed stress-tensor components, and an explicit
extension of the regular radial Ward identities. These choices are
necessary: multiplication of apex-supported distributions by the
singular conical metric or its connection is not silently assumed.

A single transverse finite part of \(r^{-2}\) generates simultaneous
extensions of the Wick square, stress tensor, and circulating
current. The regulated distributions converge strongly, with
fixed-support seminorm estimates. Direct componentwise finite
parts are also constructed. Their stress divergence and trace
defects are evaluated exactly and repaired by explicit second
derivatives of the worldsheet Dirac distribution.

The principal result is a complete ambiguity theorem in the
declared stationary, boost-invariant, rotation-covariant,
charge-conjugation-compatible, almost-homogeneous class.
For a generic charged holonomy sector, the simultaneous
Ward-compatible extension has three real contact parameters;
the real neutral problem has two. Nonzero normal-dilation
residues cannot be removed in this class. A current contraction
identity exhibits explicitly why extending a formula and preserving
all its differential presentations are different requirements.

The contribution is this prescribed extension and compatibility
analysis, not new regular cosmic-string coefficients or a canonical
singular-metric renormalization. It partially resolves the
Friedrichs one-point extension question V1 in a stated distributional
structure. It does not extend the quantum state or field algebra to
the apex, select a defect action, construct backreaction, or derive
Einstein dynamics. Finite Enter reporting remains an interface for
installed descriptions, not acquisition of a continuum distribution.
\end{abstract}

\tableofcontents

%======================================================================
\section{Purpose, inputs, and the essential qualification}
\label{sec:introduction}
%======================================================================

Cheung's I-B prescribes a regular conical spacetime and a separate
finite operational interface \cite{CheungIB}. I-B-G constructs
intrinsic scalar Green operators after a spatial extension has
been supplied \cite{CheungIBG}. I-B-VP-v2 constructs the distinguished
massless Friedrichs spectral ground state, proves its local Hadamard
form, and evaluates its Wick square and stress tensor
\cite{CheungIBVP}. I-B-VP-0-v1 evaluates the charged circulating
current \cite{CheungIBVPZero}.

Those observables are smooth on the regular spacetime. Their
singular radial behaviours do not, without an additional
prescription, define distributions across the completed apex
worldsheet. The present paper supplies one such prescription and
classifies its finite freedom.

\paragraph{The qualification that cannot be omitted.}
The metric completion is not a smooth Lorentzian manifold carrying
the displayed conical metric at its apex. Before speaking of
tensor distributions there, one must specify the test space,
smooth structure, tensor bundle, density, and meaning of the
identities to be preserved. In particular, an expression such as
\(\Gamma(g)T\), with an apex-supported \(T\), is not automatically
a product of distributions.

We therefore construct distributions on a declared smooth
completion and preserve a declared smooth representative of the
regular invariant Ward equations. We do \emph{not} claim that this
defines the Levi-Civita derivative of arbitrary tensor distributions
for the singular metric. The distinction is consistent with the
general cautions about concentrated sources and distributional
geometry in Geroch--Traschen \cite{GerochTraschen}.

\paragraph{Enter and the non-extraction boundary.}
The philosophy of Enter distinguishes a destructive readout from
a quantum-return map, a record from a decoder, and an analogy from
an identification \cite{CheungPhilosophy}. I-B and En09-A implement
a finite destructive instrument returning a classical code and a
surviving residual, not the measured clock register
\cite{CheungIB,CheungEn09}. That instrument does not supply the
field, apex condition, state, continuum test space, or
renormalization prescription used below.

The cone, worldsheet, Friedrichs extension, spectral state, and
completion prescription remain declared inputs. No continuum
distribution is extracted from a finite conditional density matrix.
Conservation does not impose Einstein dynamics
\cite[\SourceLabel{prop:not-source}]{CheungIB}
\cite[\SourceLabel{prop:noeinstein}]{CheungXAST}.

\paragraph{Analytical context and contribution.}
Extension by Taylor subtraction, scaling degree, and distributions
supported on a submanifold are classical tools
\cite{BF,Dang,Hormander}. Almost-homogeneous extensions and their
dilation residues are likewise established concepts
\cite{NikolovStoraTodorov}. Our contribution is their explicit
application to the simultaneous one-point data, including the
stress Ward repair and the complete contact space in the
prescription fixed here.

VII-D's strong finite-part construction and VII-H's explicitly
restricted counterterm space are methodological comparisons
\cite{CheungVIID,CheungVIIH}. Their numerical kernels and contact
classes are not imported. Similarly, the relative-lifting and
normal-jet distinctions in XV-B motivate a compatibility check,
not an identification with its characteristic graph
\cite{CheungXVB}.

\paragraph{Proof dependencies.}
The displayed regular coefficients in Section~\ref{sec:data}
are the input identified with the Friedrichs state by the cited
companions. Every assertion about their distributional extension,
cutoff limit, Ward repair, scaling, and contact classification is
proved below. No missing interacting product or characteristic
composition is used.

%======================================================================
\section{The completion, regular data, and Ward prescription}
\label{sec:data}
%======================================================================

\subsection{Geometry and the additional smooth structure}

Fix
\[
 0<\alpha<1,\qquad
 \mu=2\pi(1-\alpha),\qquad
 0\leq\beta<1,\qquad \xi\in\RR.
\]
The regular host is
\[
 M=\RR_t\times(0,\infty)_r
       \times(\RR/2\pi\ZZ)_\varphi\times\RR_z,
\]
\[
 g=-\dd t^2+\dd r^2+\alpha^2r^2\dd\varphi^2+\dd z^2.
\]
Its metric completion adds
\[
 \WW=\RR_t\times\{o\}\times\RR_z,
\]
a worldsheet, not a worldline \cite{CheungIB,CheungIBG}.

\begin{assumption}[Smooth completion and density]
\label{ass:completion}
Identify the topological completion with
\[
 \overline M=\RR_y^{1+1}\times\RR_x^2,\qquad
 y=(t,z),\qquad
 x=(x^1,x^2)=(r\cos\varphi,r\sin\varphi).
\]
Equip it with this Cartesian smooth structure and the auxiliary
smooth metric
\[
 \sg=-\dd t^2+\dd z^2+(\dd x^1)^2+(\dd x^2)^2.
\]
The pairing density is
\begin{equation}
 \dd\Omega=\alpha\,\dd t\,\dd z\,\dd^2x.
 \label{eq:density}
\end{equation}
It equals the host volume density on \(M\).

All tensor distributions below are sections of the ordinary smooth
tensor bundles of \(\overline M\). Stress is extended as a mixed
tensor \(T^\mu{}_\nu\); the current is extended first as a
contravariant vector \(J^\mu\).
\end{assumption}

These \(x\)-coordinates are \emph{not} the developing coordinates
\((r\cos(\alpha\varphi),r\sin(\alpha\varphi))\).
Assumption~\ref{ass:completion} is additional data.
For a quotient angle \(\alpha=1/N\), it is not automatically the
orbifold test-function structure of the covering plane.

Write
\[
 \deltaW:\quad
 \pair{\deltaW}{f}
 =\int_{\RR^{1+1}}f(y,0)\,\dd t\,\dd z.
 \label{eq:delta}
\]
Thus, as a generalized function relative to \(\dd\Omega\),
\(\deltaW=\alpha^{-1}\delta^{(2)}_{\mathrm{Leb}}(x)\).
Derivatives are transposed with respect to the constant density
\eqref{eq:density}. Put
\[
 \Delta=\partial_1^2+\partial_2^2,\qquad
 E=x^i\partial_i,\qquad
 \salpha=2\pi\alpha,
\]
\[
 K_{ij}=\partial_i\partial_j-\delta_{ij}\Delta.
 \label{eq:K}
\]
Indices \(A,B\) range over \(t,z\), and \(i,j\) over \(1,2\).

Let \(\epsilon\) be the positive quarter-turn matrix,
\[
 \epsilon=
 \begin{pmatrix}0&-1\\1&0\end{pmatrix},
 \qquad
 R^i=\epsilon^i{}_j x^j.
\]
The smooth vector \(R\) is \(\partial_\varphi\) on \(M\).

\subsection{Regular one-point data}

The charged field obeys
\[
 \Phi(\varphi+2\pi)=e^{2\pi\ii\beta}\Phi(\varphi)
\]
in a lifted parallel frame. Its spatial extension and state are the
massless Friedrichs operator and distinguished spectral ground state
of I-B-VP-v2
\cite[\SourceLabel{def:state},
      \SourceLabel{lem:infrared},
      \SourceLabel{thm:hadamard}]{CheungIBVP}.
For the real field take \(\beta=0\).

Introduce a multiplicity
\[
 n=
 \begin{cases}
 1,&\text{one real neutral scalar},\\
 2,&\text{one charged complex scalar}.
 \end{cases}
\]
Define
\begin{align}
 P_\beta&=\alpha^{-2}\bigl(1-6\beta(1-\beta)\bigr)-1,
 \nonumber\\
 C_\beta&=\alpha^{-4}\bigl(1-30\beta^2(1-\beta)^2\bigr)-1,
 \nonumber\\
 s&=\frac{P_\beta}{48\pi^2},\qquad
 h=\frac{nC_\beta}{1440\pi^2},\qquad
 q=\xi-\frac16,
 \nonumber\\
 k&=nqs,\qquad a=h+4k,\qquad b=h-2k.
 \label{eq:coefficients}
\end{align}
The regular Wick square is
\[
 F^\circ=\frac{s}{r^2},
\]
meaning \(\langle\phi^2\rangle_\ren\) for the real field and
\(\langle\Phi^\dagger\Phi\rangle_\ren\) for the complex field.
The effective field square in the stress trace is \(nF^\circ\).

In the smooth coordinates of Assumption~\ref{ass:completion},
the regular mixed stress is
\begin{align}
 (T^\circ)^A{}_B&=\frac{a}{r^4}\delta^A_B,
 &
 (T^\circ)^A{}_i&=(T^\circ)^i{}_A=0,
 \nonumber\\
 (T^\circ)^i{}_j
 &=\frac{b}{r^4}
       \left(4\frac{x^ix_j}{r^2}-3\delta^i_j\right).
 \label{eq:regularstress}
\end{align}
These are precisely the mixed components
\(r^{-4}\diag(a,b,-3b,a)\) in order \((t,r,\varphi,z)\)
\cite[\SourceLabel{thm:phi2},
      \SourceLabel{thm:T-real},
      \SourceLabel{thm:T-complex}]{CheungIBVP}.

For the charged field put
\begin{equation}
 j=\frac{\beta(1-\beta)(2\beta-1)}
          {12\pi^2\alpha^2}.
 \label{eq:j}
\end{equation}
With
\[
 J_\mu=\ii\left(
 \Phi^\dagger D_\mu\Phi-(D_\mu\Phi)^\dagger\Phi
 \right),
\]
I-B-VP-0-v1 gives
\begin{equation}
 (J^\circ)^i=\frac{j}{\alpha^2}
                 \frac{\epsilon^i{}_j x^j}{r^4},
 \qquad
 (J^\circ)^A=0,
 \qquad
 J^\circ_\varphi=\frac{j}{r^2}.
 \label{eq:regularcurrent}
\end{equation}
The current uses the complex first jet of the regular covariance,
not the real part used for stress
\cite[\SourceLabel{def:current},
      \SourceLabel{thm:current}]{CheungIBVPZero}.
Set \(j=0\) for the real field.

\subsection{Which Ward identities are extended?}

\begin{definition}[Prescribed Ward representative]
\label{def:Ward}
For the stationary distributions considered here, use
\begin{equation}
 \mathfrak D_\nu T:=\partial_\mu T^\mu{}_\nu,
 \qquad
 \mathfrak dJ:=\partial_\mu J^\mu,
 \qquad
 \mathfrak tT:=T^\mu{}_\mu.
 \label{eq:Ward}
\end{equation}
For a stationary scalar distribution \(G\), define the smooth
reference improvement
\begin{equation}
 \mathcal I[G]^\mu{}_\nu
 =\delta^\mu_\nu\Box_{\sg}G
       -\bar\partial^\mu\partial_\nu G.
 \label{eq:I}
\end{equation}
In the present translation-invariant sector,
\begin{equation}
 \mathcal I[G]^A{}_B=\delta^A_B\Delta G,
 \qquad
 \mathcal I[G]^i{}_j=-K_{ij}G,
 \qquad
 \mathcal I[G]^A{}_i=\mathcal I[G]^i{}_A=0.
 \label{eq:Iblocks}
\end{equation}
\end{definition}

\begin{lemma}[Agreement with the regular invariant equations]
\label{lem:regularWard}
For the regular tensors \eqref{eq:regularstress},
\[
 \nabla^g_\mu(T^\circ)^\mu{}_\nu
 =\mathfrak D_\nu T^\circ=0,
 \qquad
 \nabla^g_\mu(J^\circ)^\mu
 =\mathfrak dJ^\circ=0.
\]
For every smooth radial scalar \(G(r)\), the mixed improvement
\((g\Box_g-\nabla^g\nabla^g)^\mu{}_\nu G\) equals
\(\mathcal I[G]^\mu{}_\nu\). Consequently,
\begin{align}
 \mathfrak tT^\circ&=3q\,\Delta(nF^\circ),
 \label{eq:regulartrace}\\
 T^\circ(\xi)-T^\circ(1/6)
 &=nq\,\mathcal I[F^\circ].
 \label{eq:regularimprovement}
\end{align}
\end{lemma}

\begin{proof}
In polar coordinates the only nonzero difference between the
transverse Christoffel symbols of \(g\) and \(\sg\) is
\[
 \Gamma(g)^r_{\varphi\varphi}
       -\Gamma(\sg)^r_{\varphi\varphi}
 =(1-\alpha^2)r.
\]
Its contraction with the diagonal mixed tensor in
\eqref{eq:regularstress} vanishes in the mixed divergence.
The two volume densities differ by the constant \(\alpha\),
so vector divergences agree as well.

For a radial scalar, both mixed Hessians have radial entry \(G''\)
and angular entry \(G'/r\); both wave operators give
\(G''+G'/r\). This proves the improvement assertion.
Finally,
\[
 a-b=6k,\qquad \Delta r^{-2}=4r^{-4}
\]
on \(r>0\), which gives the trace and improvement identities.
\end{proof}

\begin{remark}[Not an unrestricted singular-metric derivative]
Definition~\ref{def:Ward} extends a regular, already-cancelled
invariant equation by a smooth differential operator.
It does not define products of the conical connection with delta
derivatives. Likewise, the trace is the smooth mixed trace;
we do not first lower or raise distributional indices with the
singular metric at \(\WW\).
All claims of Ward compatibility below refer to this definition.
\end{remark}

%======================================================================
\section{Strong radial finite parts}
\label{sec:finiteparts}
%======================================================================

\subsection{The scalar generating distribution}

Fix a length \(L>0\) and a smooth profile
\[
 \rho:[0,\infty)\longrightarrow[0,1],
 \qquad
 \rho(u)=0\ (u\leq1),\qquad
 \rho(u)=1\ (u\geq2).
\]
Put
\begin{equation}
 c_\rho=\int_1^2\frac{\rho(u)-1}{u}\,\dd u.
 \label{eq:crho}
\end{equation}

\begin{definition}[Generating finite part]
\label{def:U}
Define
\begin{equation}
 U_{L,\varepsilon}^{\rho}
 =\rho(r/\varepsilon)r^{-2}
  +\salpha\left(\log\frac{\varepsilon}{L}-c_\rho\right)\deltaW.
 \label{eq:Uepsilon}
\end{equation}
Its limit will be denoted by \(U_L\).
The first term is interpreted as a locally integrable function
with the density \(\dd\Omega\).
\end{definition}

\begin{theorem}[Existence, profile independence, and strong convergence]
\label{thm:U}
The limit
\[
 U_L=\lim_{\varepsilon\downarrow0}U_{L,\varepsilon}^{\rho}
\]
exists strongly in \(\D'(\overline M)\), is independent of \(\rho\),
and restricts to \(r^{-2}\) on \(M\). It is rotation invariant,
translation invariant in \(t,z\), and real.

For every compact test support there is a constant \(C\) such that
\begin{equation}
 \left|\pair{U_{L,\varepsilon}^{\rho}-U_L}{f}\right|
 \leq C\varepsilon^2 p_2(f),
 \label{eq:Urate}
\end{equation}
where \(p_2\) is a fixed-support smooth seminorm.
In particular, all fixed distributional derivatives converge
strongly, with the corresponding extra derivatives in the seminorm.
\end{theorem}

\begin{proof}
Let
\[
 \overline f(y,r)=\frac1{2\pi}\int_0^{2\pi}
             f(y,r\cos\varphi,r\sin\varphi)\,\dd\varphi.
\]
Taylor's theorem and angular averaging give
\[
 \overline f(y,r)=f(y,0)+O(r^2p_2(f))
\]
uniformly on the relevant compact set.

For the sharp cutoff, the expression
\[
 \int_{r>\varepsilon}r^{-2}f\,\dd\Omega
 +\salpha\log(\varepsilon/L)\int f(y,0)\,\dd^2y
\]
has a limit: subtract \(f(y,0)\) in a fixed small radial
neighbourhood and integrate the remainder with measure
\(\salpha\,\dd r/r\).
The omitted remainder on \(0<r<\varepsilon\) is \(O(\varepsilon^2)\).

The smooth cutoff differs from the sharp cutoff only on
\(\varepsilon<r<2\varepsilon\). Its constant Taylor term contributes
\[
 \salpha c_\rho\int f(y,0)\,\dd^2y,
\]
which is exactly removed in \eqref{eq:Uepsilon}.
Its remaining contribution is \(O(\varepsilon^2p_2(f))\).
This proves existence, profile independence, and \eqref{eq:Urate}.

The estimates are uniform on bounded sets of tests with fixed
compact support, and hence give strong distributional convergence.
The symmetries and reality follow directly from the construction.
Derivative convergence follows by transposition.
\end{proof}

\subsection{Direct componentwise finite parts}

The following elementary extension lemma will also be used to
audit direct cutoff prescriptions.

For \(p\in\{2,3,4\}\) and a smooth angular coefficient \(A\), define
\[
 f^\circ_A(x)=r^{-p}A(x/r).
\]
For a test \(f\), put
\begin{equation}
 M_k^A[f]
 =\frac{\alpha}{k!}\int_{\RR^{1+1}}\int_0^{2\pi}
 A(\omega)(\omega\cdot\partial_x)^k f(y,0)
 \,\dd\varphi\,\dd^2y,
 \qquad
 \omega=(\cos\varphi,\sin\varphi).
 \label{eq:moments}
\end{equation}
When \(k<p-2\), set
\[
 m_{p,k}^{\rho}
 =\int_0^\infty \rho(u)u^{k+1-p}\,\dd u.
\]

\begin{lemma}[Finite-part formula for the component kernels]
\label{lem:raw}
The limit
\begin{align}
 \pair{\mathcal R_L[f_A^\circ]}{f}
 =\lim_{\varepsilon\downarrow0}\Bigg\{&
 \int \rho(r/\varepsilon)f_A^\circ f\,\dd\Omega
 \nonumber\\
 &-\sum_{k=0}^{p-3}
 \varepsilon^{k+2-p}m_{p,k}^{\rho}M_k^A[f]
 \nonumber\\
 &+\left(\log\frac{\varepsilon}{L}-c_\rho\right)
                  M_{p-2}^A[f]\Bigg\}
 \label{eq:rawfinitepart}
\end{align}
exists and is independent of the profile. The sum is empty when
\(p=2\). It defines a distribution of transverse scaling degree
at most \(p\), equal to \(f_A^\circ\) off \(\WW\).

For the scalar, stress, and vector kernels below, parity improves
the fixed-support convergence remainder to \(O(\varepsilon^2)\).
\end{lemma}

\begin{proof}
Expand the test transversely through degree \(p-2\). Radial
integration of the \(k\)-th term gives the power
\(\varepsilon^{k+2-p}\) when \(k<p-2\), and a logarithm when
\(k=p-2\). These are precisely the subtractions in
\eqref{eq:rawfinitepart}. The Taylor remainder is integrable.

Comparing two profiles on their transition annuli gives the
same moment contributions, so the normalized limit is
profile independent. Taylor remainder estimates prove continuity
and the stated scaling-degree bound.

For \(r^{-2}\), angular odd terms vanish.
For the stress kernels the angular coefficient is even, so the
cubic Taylor term vanishes.
For \(x_i/r^4\), the angular coefficient is odd, so the quadratic
Taylor term vanishes. In these cases the first unremoved angular
term yields an \(O(\varepsilon^2)\) remainder, bounded respectively
by test seminorms of orders \(2,4,3\).
\end{proof}

\begin{remark}[The actual divergence powers]
The codimension is two. Thus \(r^{-4}\) produces an
\(\varepsilon^{-2}\) divergence and a logarithmic second-jet
subtraction, not an \(\varepsilon^{-4}\) divergence.
For the circulating vector, the prospective
\(\varepsilon^{-1}\) term has zero angular moment; only a
logarithmic first-jet subtraction remains.
\end{remark}

We shall write
\begin{equation}
 V_L^{\mathrm c}=\mathcal R_L[r^{-4}],\qquad
 Q_{L,i}^{\mathrm c}=\mathcal R_L[x_i/r^4],
 \label{eq:rawVQ}
\end{equation}
\begin{equation}
 S_{L,ij}^{\mathrm c}
 =\mathcal R_L\left[
 r^{-4}\left(4\frac{x_ix_j}{r^2}-3\delta_{ij}\right)
 \right].
 \label{eq:rawS}
\end{equation}
The superscript \(\mathrm c\) denotes componentwise cutoff
finite parts, not a conserved prescription.

%======================================================================
\section{Simultaneous extensions and the exact Ward repair}
\label{sec:construction}
%======================================================================

\subsection{A derivative-compatible prescription}

Define
\begin{equation}
 \widehat F_L=sU_L,
 \label{eq:Fhat}
\end{equation}
\begin{align}
 (\widehat T_L)^A{}_B
 &=\frac a4\delta^A_B\Delta U_L,
 &
 (\widehat T_L)^A{}_i&=(\widehat T_L)^i{}_A=0,
 \nonumber\\
 (\widehat T_L)^i{}_j
 &=\frac b2 K_{ij}U_L,
 \label{eq:That}
\end{align}
and
\begin{equation}
 (\widehat J_L)^i
 =-\frac{j}{2\alpha^2}\epsilon^i{}_k\partial_kU_L,
 \qquad
 (\widehat J_L)^A=0.
 \label{eq:Jhat}
\end{equation}

\begin{theorem}[Simultaneous Ward-compatible extension]
\label{thm:extension}
The distributions \eqref{eq:Fhat}--\eqref{eq:Jhat} are real
extensions of the regular data
\eqref{eq:regularstress}--\eqref{eq:regularcurrent}.
They have transverse scaling degrees at most \(2,4,3\),
respectively. They satisfy
\begin{align}
 \mathfrak D_\nu\widehat T_L&=0,
 &
 \mathfrak d\widehat J_L&=0,
 \label{eq:extendedconservation}\\
 \mathfrak t\widehat T_L
 &=3q\,\Delta(n\widehat F_L),
 \label{eq:extendedtrace}\\
 \widehat T_L(\xi)-\widehat T_L(1/6)
 &=nq\,\mathcal I[\widehat F_L].
 \label{eq:extendedimprovement}
\end{align}
The reference-lowered stress
\(\sg_{\lambda\mu}(\widehat T_L)^\mu{}_\nu\) is symmetric.

Replacing \(U_L\) by \(U_{L,\varepsilon}^{\rho}\) gives
strongly convergent regulated extensions. The errors are bounded
by \(C\varepsilon^2p_2\), \(C\varepsilon^2p_4\), and
\(C\varepsilon^2p_3\) for field square, stress, and current.
\end{theorem}

\begin{proof}
On \(r>0\),
\[
 \frac14\Delta r^{-2}=r^{-4},\qquad
 -\frac12\partial_i r^{-2}=\frac{x_i}{r^4},
\]
\[
 \frac12K_{ij}r^{-2}
 =r^{-4}\left(4\frac{x_ix_j}{r^2}-3\delta_{ij}\right).
\]
This proves exact regular-part matching.

All derivatives in \eqref{eq:That}--\eqref{eq:Jhat} are derivatives
of a defined distribution. Since
\[
 \partial_iK_{ij}=0,\qquad
 \partial_i\epsilon^i{}_k\partial_k=0,
\]
conservation follows without a boundary-flux omission.
The mixed trace is
\[
 \mathfrak t\widehat T_L
 =\left(\frac a2-\frac b2\right)\Delta U_L
 =3k\Delta U_L
 =3q\,\Delta(n\widehat F_L).
\]
The changes of \(a/4\) and \(b/2\) under a change from conformal
coupling are \(nqs\) and \(-nqs\), respectively, giving
\eqref{eq:extendedimprovement}. Symmetry is immediate from the
block expressions. The scaling-degree and strong-convergence
claims follow from Theorem~\ref{thm:U}.
\end{proof}

This is a construction of c-number one-point distributions.
It is not an extension of the two-point function, quantum state,
or field algebra to \(\WW\).

\subsection{The direct cutoff prescription need not obey the Ward equations}

\begin{proposition}[Exact finite conversions]
\label{prop:conversion}
The direct finite parts satisfy
\begin{align}
 \frac14\Delta U_L
 &=V_L^{\mathrm c}-\frac{\salpha}{4}\Delta\deltaW,
 \label{eq:convertV}\\
 -\frac12\partial_iU_L
 &=Q_{L,i}^{\mathrm c}+\frac{\salpha}{4}\partial_i\deltaW,
 \label{eq:convertQ}\\
 \frac12K_{ij}U_L
 &=S_{L,ij}^{\mathrm c}
 +\salpha\left(
 \frac7{16}\delta_{ij}\Delta
 -\frac38\partial_i\partial_j
 \right)\deltaW.
 \label{eq:convertS}
\end{align}
In particular,
\begin{equation}
 \partial_iS_{L,ij}^{\mathrm c}
 =-\frac{\salpha}{16}\partial_j\Delta\deltaW,
 \qquad
 S_{L,ii}^{\mathrm c}=-2V_L^{\mathrm c}.
 \label{eq:rawdefects}
\end{equation}
\end{proposition}

\begin{proof}
Profile independence permits the sharp cutoff in this comparison.
Twice integrating by parts on \(r>\varepsilon\), and expanding
the boundary test jets, gives
\begin{align*}
 \partial_iU_L
 &=\mathcal R_L[\partial_i r^{-2}]
                    -\frac{\salpha}{2}\partial_i\deltaW,\\
 \partial_i\partial_jU_L
 &=\mathcal R_L[\partial_i\partial_j r^{-2}]
 -\frac{3\salpha}{4}\partial_i\partial_j\deltaW
 -\frac{\salpha}{8}\delta_{ij}\Delta\deltaW.
\end{align*}
The boundary computations are detailed in
Appendix~\ref{app:boundary}. Their trace gives
\eqref{eq:convertV}; the first identity gives
\eqref{eq:convertQ}; subtracting the traced second identity
gives \eqref{eq:convertS}. Taking a divergence of
\eqref{eq:convertS} proves the first identity in
\eqref{eq:rawdefects}. The second follows either by tracing
the regular angular kernel before applying the linear finite-part
operation, or directly from the moment subtractions.
\end{proof}

Define the preliminary componentwise lift by
\[
 (T_L^{\mathrm c})^A{}_B=a\delta^A_BV_L^{\mathrm c},
 \qquad
 (T_L^{\mathrm c})^i{}_j=bS_{L,ij}^{\mathrm c},
 \qquad
 (J_L^{\mathrm c})^i
 =\frac j{\alpha^2}\epsilon^i{}_kQ_{L,k}^{\mathrm c}.
\]

\begin{corollary}[Computed defects and an admissible correction]
\label{cor:repair}
The preliminary lift has
\begin{align}
 \mathfrak D_jT_L^{\mathrm c}
 &=-\frac{b\salpha}{16}\partial_j\Delta\deltaW,
 \label{eq:stressdefect}\\
 \mathfrak tT_L^{\mathrm c}
       -3q\Delta(n\widehat F_L)
 &=3k\salpha\Delta\deltaW,
 \label{eq:tracedefect}\\
 \mathfrak dJ_L^{\mathrm c}&=0.
 \label{eq:currentrawconserved}
\end{align}
The corrections
\begin{align}
 (\widehat T_L-T_L^{\mathrm c})^A{}_B
 &=-\frac{a\salpha}{4}\delta^A_B\Delta\deltaW,
 \nonumber\\
 (\widehat T_L-T_L^{\mathrm c})^i{}_j
 &=b\salpha\left(
 \frac7{16}\delta^i_j\Delta
 -\frac38\partial^i\partial_j
 \right)\deltaW,
 \label{eq:stressrepair}\\
 (\widehat J_L-J_L^{\mathrm c})^i
 &=\frac{j\salpha}{4\alpha^2}
              \epsilon^i{}_k\partial_k\deltaW
 \label{eq:currentrepair}
\end{align}
give exactly Theorem~\ref{thm:extension}.
\end{corollary}

\begin{proof}
The divergence follows from \eqref{eq:rawdefects}.
The raw trace is
\[
 2(a-b)V_L^{\mathrm c}=12kV_L^{\mathrm c}.
\]
Equation~\eqref{eq:convertV} makes
\(3q\Delta(n\widehat F_L)=12kV_L^{\mathrm c}
-3k\salpha\Delta\deltaW\), proving \eqref{eq:tracedefect}.
The vector in \eqref{eq:convertQ} differs from a gradient by a
delta derivative, so its rotation by \(\epsilon\) is conserved.
The corrections are the three conversions multiplied by their
regular coefficients.
\end{proof}

\begin{remark}[Subtraction, not algebra]
The raw finite part exists, but is not automatically a
Ward-compatible extension. Corollary~\ref{cor:repair} evaluates
the defect and supplies an actual correction in an explicit
supported space. This is the relevant compatibility calculation;
merely naming a relative-lifting criterion would not supply it
\cite{CheungVIID,CheungXVB,CheungEmOneA,CheungEmOneD}.
Nothing here defines a product of singular observables.
\end{remark}

%======================================================================
\section{The complete contact space in the declared class}
\label{sec:ambiguity}
%======================================================================

A finite-dimensional ambiguity requires more than a bound on
normal derivative order. We now fix the full class.

\begin{assumption}[Admissible one-point extensions]
\label{ass:class}
For each fixed \((\alpha,\beta,\xi)\), consider real distributional
extensions \((F,T,J)\) of the displayed regular data satisfying:
\begin{enumerate}[label=\textup{(A\arabic*)}]
\item Translation invariance along \(\WW\), covariance under
longitudinal Lorentz transformations, and transverse rotations.
The reference-lowered stress is symmetric.
\item Charge conjugation sends \(\beta\) to \(1-\beta\):
\(F,T\) are even and \(J\) is odd. Transverse reflection reverses
the meridian and therefore also sends \(\beta\) to \(1-\beta\).
These requirements apply to the family of sectors.
\item The componentwise transverse scaling degrees are at most
\(2,4,3\) for \(F,T,J^i\), respectively, and the extensions are
almost homogeneous of logarithmic order at most one:
\begin{equation}
 (E+2)^2F=0,\qquad
 (E+4)^2T^\mu{}_\nu=0,\qquad
 (E+3)^2J^\mu=0.
 \label{eq:AHclass}
\end{equation}
These are equations for the stated Cartesian component
distributions.
\item The Ward representative is Definition~\ref{def:Ward}, and
the required identities are
\begin{equation}
 \mathfrak DT=0,\qquad \mathfrak dJ=0,\qquad
 \mathfrak tT=3q\,\Delta(nF).
 \label{eq:classWard}
\end{equation}
\end{enumerate}
For a coupling-dependent family, \(F,J\) are required to be
independent of \(\xi\), and the improvement identity is required
with this same \(F\).
\end{assumption}

The almost-homogeneity condition excludes new power-law defect
scales such as \(L^{-2}\deltaW\) in the stress. It does not
require strict homogeneity, which will generally be impossible.
The class is a prescription, not a classification of every
possible singular-background renormalization.

Define the conserved, traceless mixed contact tensor
\begin{align}
 \mathcal H^A{}_B&=\delta^A_B\Delta\deltaW,
 &
 \mathcal H^i{}_j&=2K_{ij}\deltaW,
 &
 \mathcal H^A{}_i&=\mathcal H^i{}_A=0.
 \label{eq:Hcontact}
\end{align}
Indeed, \(\mathfrak D\mathcal H=0\) and
\(\mathfrak t\mathcal H=2\Delta\deltaW-2\Delta\deltaW=0\).

\begin{theorem}[Complete ambiguity and its dimension]
\label{thm:classification}
Every extension in Assumption~\ref{ass:class} is uniquely of
the form
\begin{align}
 F&=\widehat F_L+c\,\deltaW,
 \label{eq:allF}\\
 T^\mu{}_\nu
 &=(\widehat T_L)^\mu{}_\nu
       +\tau\mathcal H^\mu{}_\nu
       +nq\,c\,\mathcal I[\deltaW]^\mu{}_\nu,
 \label{eq:allT}\\
 J^i&=(\widehat J_L)^i
       +w\,\epsilon^i{}_k\partial_k\deltaW,
 &
 J^A&=0,
 \label{eq:allJ}
\end{align}
with real \(c,\tau,w\).

Conversely, these formulas give all the required extensions.
For coupling-dependent families, \(c,\tau,w\) can be taken
independent of \(\xi\), and then the improvement identity holds.

The coefficients \(c,\tau\) are even in \(\beta\), and \(w\)
is odd. Consequently:
\begin{enumerate}[label=\textup{(\roman*)}]
\item for a generic charged sector
\(\beta\notin\{0,1/2\}\), the contact space has dimension \(3\);
\item at a self-conjugate charged sector
\(\beta=0\) or \(1/2\), \(w=0\), and the dimension is \(2\);
\item for the real neutral field, there is no charged current
and the dimension is \(2\).
\end{enumerate}
The dimension is for fixed background and field parameters,
not for arbitrary functions of those parameters.
\end{theorem}

\begin{proof}
The difference of two extensions is supported on \(\WW\).
The structure theorem for distributions supported on a
submanifold expresses it as a finite sum of transverse
delta derivatives with coefficient distributions on \(\WW\)
\cite[Section~8]{Dang}
\cite[Theorem~2.3.5]{Hormander}.
Translation invariance forces those coefficient distributions
to be constants times the longitudinal Lebesgue distribution.

A transverse derivative of order \(m\) obeys
\[
 E(\partial^\gamma\deltaW)
 =-(2+m)\partial^\gamma\deltaW,\qquad |\gamma|=m.
\]
The scaling-degree bounds and \eqref{eq:AHclass} therefore
allow only order \(0\) for the field-square difference,
order \(2\) for stress, and order \(1\) for current.
For example, a stress term proportional to \(\deltaW\)
has \((E+4)^2\deltaW=4\deltaW\), and is excluded.

Rotation covariance and the reflection/charge-conjugation
conditions give
\[
 \Delta F=c\deltaW,
\]
\[
 \Delta T^A{}_B=u\,\delta^A_B\Delta\deltaW,\qquad
 \Delta T^i{}_j=v\,\delta^i_j\Delta\deltaW
                         +z\,\partial^i\partial_j\deltaW,
\]
with no mixed blocks.
The parity-odd symmetric transverse tensor is excluded by the
even charge-conjugation behaviour of stress combined with
meridian reflection.

For the current, longitudinal Lorentz covariance excludes a
constant longitudinal vector. Rotation covariance allows
\(\partial_i\deltaW\) and
\(\epsilon_i{}^k\partial_k\deltaW\).
The radial term is even under meridian reflection, whereas
the current must reverse under charge conjugation.
The simultaneous requirements exclude the radial term.
Thus the current difference is precisely the term in
\eqref{eq:allJ}.

Stress conservation gives \(v+z=0\), so write the transverse
block as \(v'K_{ij}\deltaW\). Its trace is
\[
 (2u-v')\Delta\deltaW.
\]
The trace equation in \eqref{eq:classWard} requires
\[
 2u-v'=3nq\,c.
\]
Put \(\tau=u-nq\,c\). Then \(v'=2\tau-nq\,c\), which is
exactly \eqref{eq:allT} by
\eqref{eq:Iblocks} and \eqref{eq:Hcontact}.
The parameters are unique because the listed contact
generators are linearly independent.

Conversely, the contact tensors in
\eqref{eq:allF}--\eqref{eq:allJ} satisfy all the symmetries,
scaling conditions, and Ward equations.
The improvement identity follows by subtracting the same formula
at \(q=0\).
At a self-conjugate sector, oddness forces \(w=0\).
\end{proof}

\begin{remark}[What makes the dimension finite]
Without longitudinal translation invariance, the coefficients
of the normal delta derivatives could themselves be distributions
on \(\WW\), giving an infinite-dimensional ambiguity.
Without \eqref{eq:AHclass}, the scaling-degree bound also permits
lower-order contacts, including a longitudinal stress proportional
to \(L^{-2}\deltaW\).
Without the reflection and charge-conjugation requirements,
additional oriented tensor structures can occur.
The dimension in Theorem~\ref{thm:classification} is not a
universal dimension for every apex prescription.
\end{remark}

\begin{remark}[No physical normalization is selected]
The Friedrichs condition fixes the spatial wave operator on
the regular host. It does not fix \(c,\tau,w\).
A specified defect action, core model, variational prescription,
or independent apex normalization could impose additional
conditions. None is introduced here.
\end{remark}

%======================================================================
\section{Dilation residues and compatibility of presentations}
\label{sec:scaling}
%======================================================================

\subsection{The unavoidable scalar residue}

\begin{proposition}[Euler identity and scale dependence]
\label{prop:Euler}
The generating finite part obeys
\begin{equation}
 (E+2)U_L=\salpha\deltaW,
 \qquad
 L\frac{\partial U_L}{\partial L}=-\salpha\deltaW,
 \label{eq:EulerU}
\end{equation}
and hence
\begin{equation}
 U_{L'}-U_L=-\salpha\log(L'/L)\deltaW.
 \label{eq:scaleshift}
\end{equation}
For the transverse dilation \(x\mapsto\lambda x\),
\[
 U_L(\lambda x)
 =\lambda^{-2}\left(
 U_L(x)+\salpha\log\lambda\,\deltaW
 \right)
\]
as a pullback identity of generalized functions.
No extension of \(r^{-2}\) differing from \(U_L\) by a
multiple of \(\deltaW\) can be strictly homogeneous.
\end{proposition}

\begin{proof}
For a test \(f\),
\[
 \pair{(E+2)U_L}{f}=-\pair{U_L}{Ef}.
\]
The subtraction term vanishes because \(Ef(y,0)=0\).
Radial integration gives
\[
 \lim_{\varepsilon\downarrow0}
 \alpha\int_{r>\varepsilon}r^{-2}Ef\,\dd^2x
 =-\salpha f(y,0)
\]
after the longitudinal integration. This proves the first
identity. Differentiating the defining subtraction proves
the second, and integration in \(L\) gives
\eqref{eq:scaleshift}. Integrating the Euler equation along
dilations gives the pullback formula.
Since \((E+2)\deltaW=0\), adding a scalar contact cannot
remove the residue.
\end{proof}

\begin{corollary}[Simultaneous residues]
\label{cor:residues}
For the canonical prescription,
\begin{align}
 (E+2)\widehat F_L
 &=s\salpha\deltaW,
 \nonumber\\
 (E+4)(\widehat T_L)^A{}_B
 &=\frac{a\salpha}{4}\delta^A_B\Delta\deltaW,
 \nonumber\\
 (E+4)(\widehat T_L)^i{}_j
 &=\frac{b\salpha}{2}K_{ij}\deltaW,
 \nonumber\\
 (E+3)(\widehat J_L)^i
 &=-\frac{j\salpha}{2\alpha^2}
                         \epsilon^i{}_k\partial_k\deltaW.
 \label{eq:allresidues}
\end{align}
These residues are unchanged by the contacts in
Theorem~\ref{thm:classification}.
The \(L\)-derivative of each canonical distribution is the
negative of its displayed residue.
\end{corollary}

\begin{proof}
Use \([E,\partial_i]=-\partial_i\) in
\eqref{eq:EulerU}, then apply the defining differential
operators. Each permitted contact is exactly homogeneous
at the relevant degree and is therefore killed by the
corresponding Euler operator.
\end{proof}

The residues are extension scaling defects. They must not be
identified without further analysis with a curvature-dependent
quantum trace anomaly. For example, at \(\xi=1/6\) the mixed
trace of the constructed stress is exactly zero, even when
its normal-dilation residue is nonzero.
Changing \(L\) changes a contact convention and does not change
any regular-part value.

\subsection{The angular current is a smooth contraction, not singular lowering}

The smooth contraction defining an angular scalar extension is
\begin{equation}
 J_\varphi^{\mathrm{ext}}
 :=\alpha^2R_iJ^i,
 \qquad R_i=\delta_{ij}R^j.
 \label{eq:angularcontraction}
\end{equation}
On \(M\), this is the actual covariant angular component,
because \(g_{ij}R^i=\alpha^2R_j\).
Equation~\eqref{eq:angularcontraction} uses a smooth polynomial
multiplier on the completion; it does not multiply by the
singular metric there.

\begin{proposition}[A computed contraction defect]
\label{prop:contraction}
For the derivative-compatible current,
\begin{equation}
 \alpha^2R_i(\widehat J_L)^i
 =jU_L-\frac{j\salpha}{2}\deltaW.
 \label{eq:contractiondefect}
\end{equation}
For the general current \eqref{eq:allJ},
\begin{equation}
 J_\varphi^{\mathrm{ext}}
 =jU_L-\left(\frac{j\salpha}{2}+2\alpha^2w\right)\deltaW.
 \label{eq:generalangular}
\end{equation}
In contrast, the direct componentwise current obeys
\[
 \alpha^2R_i(J_L^{\mathrm c})^i=jU_L.
\]
Thus a contact can preserve this latter contraction identity,
but not simultaneously the unmodified derivative identity
\eqref{eq:Jhat} when \(j\ne0\).
\end{proposition}

\begin{proof}
Since \(R_i\epsilon^i{}_k=x_k\),
\[
 \alpha^2R_i(\widehat J_L)^i
 =-\frac j2 EU_L.
\]
Equation~\eqref{eq:EulerU} gives
\eqref{eq:contractiondefect}.
Also \(x^i\partial_i\deltaW=-2\deltaW\), proving
\eqref{eq:generalangular}.
Equation~\eqref{eq:currentrepair} then gives the direct
finite-part contraction. Choosing
\(w=-j\salpha/(4\alpha^2)\) removes the contraction defect,
but changes \eqref{eq:Jhat} by that nonzero contact.
\end{proof}

This is an explicit presentation-compatibility calculation.
It does not rely on checking finitely many null relations
and then assuming coverage of all relations.
The distinction between a distributional extension and a
descending realization of several presentations is also
emphasized in the supplied XV-B material
\cite{CheungXVB}.

%======================================================================
\section{What this completion does and does not provide}
\label{sec:scope}
%======================================================================

\subsection{Regular-part agreement and supported laboratories}

Every test supported in \(\{r\geq r_0>0\}\) sees exactly the
regular one-point functions, for all \(c,\tau,w,L\).
No extension parameter alters the Friedrichs wave operator,
the regular spectral state, or its local Hadamard subtraction.

The supported-transfer theorem of I-B-G concerns scalar
retarded and advanced operators, with both the additional-arrival
window and the independent retained-shadow reserve
\cite[\SourceLabel{thm:transfer},
      \SourceLabel{ass:transfer-contract}]{CheungIBG}.
The X-A-ST and X-B-ST supported-chart contracts remain
separate \cite{CheungXAST,CheungXBST}.
Neither Green-operator agreement nor a local developing chart
transfers a vacuum covariance or selects apex contacts.
No comparison with a flat-completion state is used in the
proofs above.

\subsection{Energy and positivity are additional questions}

For a static observer on \(M\), the measured energy density is
\(-a/r^4\). Its cutoff transverse integral per unit longitudinal
length is
\begin{equation}
 \int_{\varepsilon<r<R}-\frac a{r^4}
       \,\alpha r\,\dd r\,\dd\varphi
 =-\frac{\salpha a}{2}
       \left(\varepsilon^{-2}-R^{-2}\right).
 \label{eq:energy}
\end{equation}
It generally diverges. Likewise,
\[
 \int_{\varepsilon<r<R}F^\circ\,\alpha r\,\dd r\,\dd\varphi
 =\salpha s\log(R/\varepsilon).
\]
A finite distributional action on smooth compact tests is not
a proof of a finite unrenormalized energy per unit length.

If \(s>0\), no positive distribution on the completion can
restrict to \(s/r^2\) with this density: a positive distribution
is a locally finite measure, whereas every neighbourhood of the
apex has infinite regular \(s/r^2\) mass.
The constructed finite part is therefore not a positive measure.
This does not contradict positivity of the ground-state
two-point function on regular tests.

\subsection{No nonlinear singular geometry is manufactured}

The conical metric in the chosen smooth coordinates is
\[
 g_{ij}=\alpha^2\delta_{ij}
      +(1-\alpha^2)\frac{x_ix_j}{r^2}
 \qquad(r>0).
\]
It has no direction-independent value at \(x=0\) when
\(\alpha\ne1\). Products such as \(g_{ij}\partial_k\deltaW\),
a distributional Einstein tensor built from unspecified products,
or a singular-connection divergence of arbitrary contacts
are not defined by the finite parts above.

Our mixed tensors are ordinary tensor distributions on the
\emph{auxiliary smooth completion}. Their regular restrictions
are the physical mixed tensors, and their extended Ward
identities are those fixed in Definition~\ref{def:Ward}.
A core regularization or nonlinear generalized-function
framework would require a separate construction and comparison
\cite{GerochTraschen}. No such comparison is asserted.

%======================================================================
\section{Finite reporting, coverage, and evidence}
\label{sec:reporting}
%======================================================================

The finite supplied-report instrument can install words containing
\[
 (\alpha,\beta,\xi,L,c,\tau,w,\text{formula identifiers},
       \text{test descriptions},\text{evidence status}).
\]
For a calibrated clock basis \(\{\ket{\tau_j}\}\), set
\[
 D_j=\bra{\tau_j}_C\otimes I_A.
\]
The recorded instrument is
\[
 \mathcal M(X)=\sum_j
       \proj{j,w_j}_R\otimes D_jXD_j^\dagger.
\]
Its Kraus completeness and discard identities are
\[
 \sum_jD_j^\dagger D_j=I_{CA},
 \qquad
 \Tr_R\mathcal M(X)=\Tr_CX,
\]
including every untouched-reference extension
\cite[\SourceLabel{thm:instrument}]{CheungIB}
\cite{CheungEn09,CheungOOneST}.
It returns the surviving \(A\), not a quantum \(C\)-register.
The words are installed descriptions, not measurements acquiring
an unknown continuum distribution.

\paragraph{Coverage is class relative.}
Every regular-part test annihilates all contacts in
Theorem~\ref{thm:classification}. Thus even the collection of
\emph{all} tests supported away from \(\WW\) cannot select
\(c,\tau,w\).

Within the declared three-dimensional contact space, a finite
family of appropriately chosen apex-jet tests can be injective:
the three generators are independent distributions, and linear
algebra selects separating test actions. That conclusion uses
the proved classification. It does not hold for an unrestricted
space of worldsheet-supported distributions.
This is the finite-coverage distinction of Oai282-A
\cite{CheungOaiTwoEightTwo}.

\paragraph{Evidence classes.}
We retain the reporting discipline used in XVI-O1-ST
\cite{CheungOOneST}:
\begin{description}[style=nextline,leftmargin=2.1cm]
\item[\ANALYTIC]
A statement established by the displayed argument and its
identified analytical inputs.
\item[\SUPPLIED]
An inherited computational report, not independently rerun.
\item[\UNCERTIFIED]
A numerical enclosure or verification claim without the artifacts
and total error bounds required for that interpretation.
\end{description}
The results of this paper are analytical. No numerical apex
runs or raw logs were supplied or are claimed.
The numerical residuals in the regular-part companion papers
are not relabeled as checks of these extension identities.

A future regulated numerical report must retain the test,
density, smooth structure, component convention, cutoff profile,
input encodings, raw subtraction terms, precision, residuals,
and remainder bounds. Input error, cutoff error, and quadrature
error are separate. A successful regression that retains a
known discrepancy does not remove it.
No successful \texttt{pdflatex} run, numerical execution,
proof-assistant verification, or independent rebuild is asserted.

%======================================================================
\section{Ledger update and conclusion}
\label{sec:conclusion}
%======================================================================

\begingroup
\small
\renewcommand{\arraystretch}{1.12}
\begin{longtable}{@{}L{.27\linewidth}L{.23\linewidth}L{.43\linewidth}@{}}
\toprule
Object or problem & Status here & Exact boundary\\
\midrule
\endfirsthead
\toprule
Object or problem & Status here & Exact boundary\\
\midrule
\endhead
\bottomrule
\endfoot

Friedrichs regular one-point data
&
Retained
&
The cited Wick square, stress, and current are unchanged on
every regular-part test.
\\

Apex finite parts
&
Constructed
&
Strong distributions on the smooth completion of
Assumption~\ref{ass:completion}.
\\

Simultaneous Ward identities
&
Constructed
&
Conservation, mixed trace, and improvement for the prescribed
smooth Ward representative, not an unrestricted
singular-connection derivative.
\\

Contact ambiguity
&
Classified
&
Dimension \(3\) for generic charged sectors, \(2\) for the real
neutral problem and self-conjugate charged sectors, under
Assumption~\ref{ass:class}.
\\

Strict normal homogeneity
&
Generally obstructed in this class
&
The nonzero residues in \eqref{eq:allresidues} cannot be removed
by its finite contacts.
\\

Canonical physical apex renormalization
&
\textbf{Open}
&
No core model, defect action, or intrinsic singular-metric
normalization is selected.
\\

State or field algebra at the apex
&
Not constructed
&
Extending one-point c-number distributions is not extending the
quantum theory to the completion.
\\

Non-Friedrichs, massive, thermal, spinor, interacting cases
&
\textbf{Open here}
&
No such extension/state calculation is supplied.
\\
\end{longtable}
\endgroup

\paragraph{The precise V1 advance.}
The Friedrichs one-point extension question is partially resolved:
a concrete completion prescription exists, its direct-cutoff
defects are repaired, and its residual contact space is fully
classified. The original demand for a uniquely normalized or
intrinsically defined apex observable is not thereby resolved.
V2, V3, and V5--V8 of I-B-VP retain their stated scopes;
I-B-VP-0's current evaluation remains its separate regular-part
achievement \cite{CheungIBVP,CheungIBVPZero}.

\paragraph{Non-sourcing and unchanged programme boundaries.}
A prescribed distributional extension is not a field equation.
No Maxwell equation, semiclassical Einstein equation,
backreaction solution, spacetime emergence, or interacting
Ward theorem is constructed. The non-sourcing distinctions of
I-B and X-A-ST remain unchanged
\cite{CheungIB,CheungXAST}.

No external OpenAI catalogue entry is a premise. No
characteristic graph coefficient is constructed by this
one-point extension. In particular, the existence of technical
XV-B manuscripts is not the opening of its distinct downstream
quantum-admission exit \cite{CheungXVB,CheungEn09}.
O.1 remains \NOTCLOSED{} wherever it was not independently
closed; XVI-N is not discharged; \bsr{} is not promoted
\cite{CheungOOneST,CheungEn09}.
These restrictions do not deny the free-field distributions
proved here.

\paragraph{Conclusion.}
A declared smooth completion and Ward representative turn the
regular \(r^{-2}\), \(r^{-4}\), and circulating-current formulas
into a precise extension problem. One generating finite part
solves it simultaneously. The raw finite-part defects, their
repair, the remaining contact parameters, and the nonremovable
dilation residues are explicit. The construction is therefore
more than a proposal, but deliberately less than a canonical
singular-background quantum theory or a derived source law.

%======================================================================
\appendix
\section{Boundary calculations for the finite conversions}
\label{app:boundary}
%======================================================================

We give the local transverse calculation; longitudinal integration
is passive. Write \(\langle\cdot\rangle_\varphi\) for an unnormalized
angular integral, and use the constant density factor \(\alpha\).
All remainder estimates are uniform in the required compact
longitudinal set.

For a smooth test,
\[
 \frac1{2\pi}\langle f(\varepsilon\omega)\rangle_\varphi
 =f(0)+\frac{\varepsilon^2}{4}\Delta f(0)
        +O(\varepsilon^4p_4(f)).
\]
The angular identities used below are
\[
 \int_0^{2\pi}\omega_i\omega_j\,\dd\varphi
 =\pi\delta_{ij},
\]
\[
 \int_0^{2\pi}\omega_i\omega_j\omega_k\omega_l\,\dd\varphi
 =\frac{\pi}{4}
   \left(\delta_{ij}\delta_{kl}
        +\delta_{ik}\delta_{jl}
        +\delta_{il}\delta_{jk}\right).
\]

\subsection{One derivative}

For \(u=r^{-2}\), integration by parts on \(r>\varepsilon\)
uses the inner outward normal \(-\omega\):
\[
 \alpha\int_{r>\varepsilon}u\,\partial_i f\,\dd^2x
 =-\alpha\int_{r>\varepsilon}\partial_i u\,f\,\dd^2x
  -\alpha\varepsilon^{-1}
                 \int_0^{2\pi}\omega_i f(\varepsilon\omega)
                      \,\dd\varphi.
\]
The last expression equals
\[
 -\frac{\salpha}{2}\partial_i f(0)+O(\varepsilon^2p_3(f)).
\]
After including the logarithmic subtraction in \(U_L\),
this gives
\[
 \partial_iU_L
 =\mathcal R_L[\partial_i u]
                  -\frac{\salpha}{2}\partial_i\deltaW.
\]
Since \(\partial_i u=-2x_i/r^4\), this is
\eqref{eq:convertQ}.

\subsection{Two derivatives}

Two integrations by parts give
\begin{align*}
 &\alpha\int_{r>\varepsilon}u\,\partial_i\partial_j f\,\dd^2x
 -\alpha\int_{r>\varepsilon}\partial_i\partial_j u\,f\,\dd^2x\\
 &\qquad
 =-\alpha\left\{
 \varepsilon^{-1}\int\omega_i\partial_jf(\varepsilon\omega)
                         \,\dd\varphi
 +2\varepsilon^{-2}\int\omega_i\omega_jf(\varepsilon\omega)
                         \,\dd\varphi
 \right\}.
\end{align*}
Taylor expansion gives the boundary expression
\[
 -\salpha\varepsilon^{-2}\delta_{ij}f(0)
 -\frac{3\salpha}{4}\partial_i\partial_jf(0)
 -\frac{\salpha}{8}\delta_{ij}\Delta f(0)
 +O(\varepsilon^2p_4(f)).
\]
The power and logarithmic terms are exactly those in the
direct finite part of \(\partial_i\partial_j u\). Hence
\[
 \partial_i\partial_jU_L
 =\mathcal R_L[\partial_i\partial_j u]
 -\frac{3\salpha}{4}\partial_i\partial_j\deltaW
 -\frac{\salpha}{8}\delta_{ij}\Delta\deltaW.
\]
Taking a trace yields
\[
 \Delta U_L=4V_L^{\mathrm c}-\salpha\Delta\deltaW.
\]
Subtracting this trace from the preceding tensor identity and
dividing by two gives
\[
 \frac12K_{ij}U_L
 =S_{L,ij}^{\mathrm c}
 +\salpha\left(
 \frac7{16}\delta_{ij}\Delta
 -\frac38\partial_i\partial_j
 \right)\deltaW.
\]
This verifies every coefficient in
Proposition~\ref{prop:conversion} without numerical execution.

%======================================================================
\section{The unconstrained supported extension space}
\label{app:unrestricted}
%======================================================================

The finite contact dimension should not obscure the larger
extension problem.

A translation-invariant scalar difference supported on \(\WW\)
and with transverse scaling degree at most \(2\) is
\(c\deltaW\). Without translation invariance its coefficient
can be an arbitrary distribution in \(t,z\).

For stress, scaling degree at most \(4\) permits transverse
delta derivatives through order \(2\). The almost-homogeneity
condition reduces these to order \(2\), and the symmetry and
Ward equations then give Theorem~\ref{thm:classification}.
If almost homogeneity is dropped but the other stated
symmetries and conservation are kept, one can additionally
add
\[
 T^A{}_B=\lambda L^{-2}\delta^A_B\deltaW,\qquad
 T^i{}_j=0.
\]
This is conserved for the reference representative, but it
violates the declared massless almost-homogeneous class and,
unless accompanied by a changed trace prescription, its trace
condition.

For a current, a normal-order bound alone likewise does not
select circulation or remove possible longitudinal structures
if longitudinal Lorentz covariance is dropped.
A statement that there are ``only three apex counterterms''
without the assumptions of
Theorem~\ref{thm:classification} would therefore be false.

These observations are elementary instances of the distinction
between a finite normal-jet module and a finite-dimensional
space of normalization parameters
\cite{Dang,CheungXVB}. They also explain why no counterterm
class from a different companion observable is identified
with the class constructed here.

%======================================================================
\begin{thebibliography}{99}
\addcontentsline{toc}{section}{References}
%======================================================================

\bibitem{BF}
R.~Brunetti and K.~Fredenhagen,
Microlocal analysis and interacting quantum field theories:
Renormalization on physical backgrounds,
\emph{Communications in Mathematical Physics}
\textbf{208} (2000), 623--661.
\newblock \url{https://arxiv.org/abs/math-ph/9903028}.
\newblock Used for the scaling-degree extension framework,
especially Theorems~5.2--5.3; no interacting theory on the
singular completion is imported.

\bibitem{Dang}
N.~V.~Dang,
The extension of distributions on manifolds, a microlocal approach,
\emph{Annales Henri Poincar\'e}
\textbf{17} (2016), 819--859.
\newblock \url{https://arxiv.org/abs/1412.2808}.
\newblock Used for supported normal-jet ambiguities and their
coefficient spaces, especially Section~8.

\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 distributions supported on submanifolds
and ordinary distributional differentiation.

\bibitem{NikolovStoraTodorov}
N.~M.~Nikolov, R.~Stora, and I.~Todorov,
\emph{Renormalization of Massless Feynman Amplitudes
in Configuration Space},
preprint, 2013.
\newblock \url{https://arxiv.org/abs/1307.6854}.
\newblock Cited for almost-homogeneous extensions and
renormalization-invariant dilation residues, not for a new
graph construction in this manuscript.

\bibitem{GerochTraschen}
R.~Geroch and J.~Traschen,
Strings and other distributional sources in general relativity,
\emph{Physical Review D} \textbf{36} (1987), 1017--1031.
\newblock DOI: \nolinkurl{10.1103/PhysRevD.36.1017}.
\newblock Cited for the need to control singular-metric operations;
no distributional curvature model is selected here.

\medskip
\noindent\textit{Cheung companion records.}
The following manuscripts are cited by supplied version,
identifier, and source label. Unpublished source records and
quoted excerpts are distinguished from journal articles.
Their citation does not assert verification of an entire corpus,
a successful build, or a formal proof certificate.
\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: prescribed geometry and worldsheet,
\SourceLabel{thm:instrument},
\SourceLabel{prop:not-source}, and the Einstein-comparator
non-extraction distinction.
\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{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{def:state},
\SourceLabel{lem:infrared},
\SourceLabel{thm:analytic},
\SourceLabel{thm:hadamard},
\SourceLabel{thm:phi2},
\SourceLabel{thm:T-real},
\SourceLabel{thm:T-complex}, and the V1--V8 ledger.

\bibitem{CheungIBVPZero}
N.~Cheung,
\emph{I-B-VP-0: The holonomy-odd vacuum current of the charged
Friedrichs ground state:
A local observable on the prescribed conical Enter spacetime},
version~1, October 2026.
\newblock Supplied source \SourceLabel{I-B-VP-0-v1.tex},
Enter Calculus Companion.
\newblock Used:
\SourceLabel{def:current},
\SourceLabel{thm:current},
\SourceLabel{rem:imaginary}, and
\SourceLabel{eq:gaugecheck}.
\newblock Its regular-current calculation does not select the
apex contacts constructed in this paper.

\bibitem{CheungXAST}
N.~Cheung,
\emph{X-A-ST}, version~1, Enter Calculus Companion.
\newblock Used:
\SourceLabel{prop:noeinstein} and the supported-completion
signature and scope distinctions.
\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 Cited for the retained-shadow and supported-chart
contract, not for an apex state or counterterm.
\newblock \url{https://enter.grok.me/browse/X-B-ST}.

\bibitem{CheungVIID}
N.~Cheung,
\emph{VII-D}, version~1,
\SourceLabel{VII-D.tex}, Enter Calculus Companion.
\newblock Supplied excerpt:
\SourceLabel{thm:main}, finite part for the prescribed VII-C family.
\newblock Methodological comparison of strong subtraction and
a stated counterterm class; its reflected kernel is not used.

\bibitem{CheungVIIH}
N.~Cheung,
\emph{VII-H}, version~1,
\SourceLabel{VII-H.tex}, Enter Calculus Companion.
\newblock Supplied release excerpt concerning a restricted
free-field response and its declared ambiguity.
\newblock No contact classification for the present apex
observables is imported.

\bibitem{CheungXVB}
N.~Cheung,
\emph{XV-B}, version~2,
\SourceLabel{XV-B.tex}, Enter Calculus Companion.
\newblock Supplied excerpts:
\SourceLabel{thm:lifting},
\SourceLabel{thm:jetnull},
\SourceLabel{thm:pointcorrection}, and
\SourceLabel{thm:release}.
\newblock Used for the distinction between an extension,
presentation descent, a specified correction class, and quantum
admission. Its graph-specific missing map is not constructed here.

\bibitem{CheungEmOneA}
N.~Cheung,
\emph{Em01-A}, version~1,
\SourceLabel{Em01-A.tex}, Enter Calculus Companion.
\newblock Supplied comparison excerpts distinguish smooth
representatives, equation-space pairings, and unconstructed
characteristic compositions. No such smoothing theorem is used
to define the present apex distributions.

\bibitem{CheungEmOneD}
N.~Cheung,
\emph{Em01-D}, version~2,
\SourceLabel{Em01-D-v2.tex}, Enter Calculus Companion.
\newblock Supplied scope excerpt distinguishes a named fixed-kernel
family from a map on every presentation.
\newblock Its origin-germ hypotheses and asymptotic orders are
not inputs here.

\bibitem{CheungOOneST}
N.~Cheung,
\emph{XVI-O1-ST}, version~5,
\SourceLabel{XVI-O1-ST-v5.tex}, Enter Calculus Companion.
\newblock Used:
\SourceLabel{thm:instrument} and the supplied
residual-retention, evidence, and execution-scope discipline.
\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 excerpts concerning destructive finite
reporting, record--residual separation, and unchanged
downstream boundaries.
\newblock \url{https://enter.grok.me/browse/En09-A}.

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

\end{thebibliography}

\end{document}