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\title{\textbf{Landmark VII-C}\\[0.4em]
A mixed-Wick obstruction to a normal-topology boundary extension\\
for sharp Cauchy harmonic insertions}
\author{}
\date{}

\usepackage{tikz}
\begin{document}
\maketitle

\begin{abstract}
We retain the sharp zero-Cauchy harmonic dressing and its exact
off-shell classical functional, and examine a boundary enlargement of
the free polynomial Wick domain. The outcome is a specified
continuity obstruction, not an algebra construction.
For the fixed Minkowski de Donder two-point matrix, the full tensor
calculation gives \(H_{ww}=0\), while \(H_{wh}\) contains a
Cauchy-reflected frequency branch. Linear products, the insertion
self-product, and a mixed quadratic test admit compact distributional
polynomial representatives. Nevertheless, the mandatory ordered word
\(B_f\star W_g\star B_{f'}\), with
\(W_g=\int g\,h_{00}^{\,2}\), has a fully contracted coefficient whose
regular-kernel approximations grow logarithmically.
We compute the nonzero coefficient explicitly as a product of
reflected light-ray transforms. The approximating insertion kernels
converge in a specified normal topology, so this divergence rules out
a sequentially continuous extension of that coefficient. Consequently,
it rules out a hypocontinuous Wick algebra extension satisfying the
stated admission and coefficient-continuity requirements.
The logarithmic coefficient can be nonzero for three pairwise
disjoint observation supports.
No free equation is imposed on configurations, no additional
coordinate field is introduced, and no retarded product is constructed.
\end{abstract}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Inherited sharp dressing};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {New insertion domain};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {A new field equation};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {Changing the domain does not replace the field equation.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{VII-C changes the insertion domain only. The three boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:scope-VII-C}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (a) at (0,0) {Inherited sharp dressing};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {A new field equation};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {Changing the domain does not replace the field equation.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{VII-C changes the insertion domain only. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:split-VII-C}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Inherited sharp dressing};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {New insertion domain};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {A new field equation};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {Changing the domain does not replace the field equation.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (c) -- (s);
\end{tikzpicture}
\caption{Reading downward is not a transfer. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:chain-VII-C}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (a) at (0,0) {Inherited sharp dressing};
\node[align=center,font=\small] (m) at (5.2,0) {not the same object};
\node[draw,rounded corners,align=center,text width=4.2cm,font=\small,inner sep=3pt] (b) at (10.4,0) {A new field equation};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {New insertion domain. Changing the domain does not replace the field equation.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Inherited sharp dressing is not A new field equation. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:pair-VII-C}
\end{figure}

\begin{figure}[ht]
\centering
\begin{tikzpicture}[x=1cm,y=1cm]
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (a) at (0,0) {Named: Inherited sharp dressing};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: New insertion domain};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: A new field equation};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {Changing the domain does not replace the field equation.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{Three statuses stay separate. The boxes name objects this manuscript keeps separate. The dashed box is the stated stop. Not a proof, and not an admission of a quantum coefficient.}
\label{fig:gate-VII-C}
\end{figure}

Figure~\ref{fig:split-VII-C} records one view of the same separation. Figure~\ref{fig:chain-VII-C} records one view of the same separation. Figure~\ref{fig:pair-VII-C} records one view of the same separation. Figure~\ref{fig:gate-VII-C} records one view of the same separation. The drawings add no hypothesis.


Figure~\ref{fig:scope-VII-C} records that separation.
The drawing adds no hypothesis.



\section{Problem, fixed model, and precise scope}

\[
\boxed{\text{VII-C changes the insertion domain only.}}
\]

The sharp dressing considered here is
\begin{equation}
    L=\Ezero C,\qquad w[h]=Lh,
    \qquad
    B_f[h,\phi]=-\int_M f(x)w^\mu[h](x)\partial_\mu\phi(x)\dd^4x.
    \label{eq:B}
\end{equation}
The minus sign agrees with the first geometric coefficient
\(\mathcal B_f^{(1)}\).
All classical configurations are smooth and unrestricted by their
equations of motion.

The preceding obstruction is not interpreted as closing the
interacting product problem. Here we ask a narrower question:
does admitting the actual compact distributional kernel of
\eqref{eq:B}, with a specified normal topology, permit a continuous
extension of the fixed free Wick product?

The answer proved below is negative for a named mandatory graph.
The continuity assumptions are part of the theorem.
An abstract algebra with a different topology, a smaller collection of
bulk observables, or additional renormalized boundary operations is not
excluded.

\subsection{Conventions}

Let
\[
    M=\R^{1+3},\qquad
    \eta=\diag(-1,1,1,1),\qquad
    \square=-\partial_t^2+\Delta_{\mathbf x},
    \qquad
    \Sigma_0=\{t=0\},\qquad M_+=\{t>0\}.
\]
Indices are raised with \(\eta\), and
\begin{equation}
    (Ch)^\mu
    =\partial_\nu h^{\mu\nu}
      -\frac12\partial^\mu(\eta^{\alpha\beta}h_{\alpha\beta}).
    \label{eq:C}
\end{equation}
The zero-Cauchy inverse satisfies
\[
    \square\Ezero j=j,\qquad
    (\Ezero j)|_{\Sigma_0}=0,\qquad
    \partial_t(\Ezero j)|_{\Sigma_0}=0.
\]
We take \(f\in C_c^\infty(M_+)\). Thus its support has a positive
distance in the time coordinate from \(\Sigma_0\).

\paragraph{Signature dictionary.}
Relative to the convention
\(\eta_{\mathrm B}=\diag(1,-1,-1,-1)\),
\[
    \eta=-\eta_{\mathrm B},\qquad
    \square=-\square_{\mathrm B},\qquad
    \Ezero=-\Ezero^{\mathrm B}.
\]
All raised-index expressions are recomputed using \(\eta\).
If one also changes \(h=-h_{\mathrm B}\) so that the full metric
changes sign, then \(C[h]=-C_{\mathrm B}[h_{\mathrm B}]\) and
\(\Ezero C[h]=\Ezero^{\mathrm B}C_{\mathrm B}[h_{\mathrm B}]\).

\subsection{The fixed free matrix}

The active even blocks are
\begin{align}
 H_{hh\,\alpha\beta\rho\sigma}(x,y)
   &=P_{\alpha\beta\rho\sigma}\Delta_0^+(x-y),
 \label{eq:Hhh}\\
 P_{\alpha\beta\rho\sigma}
   &=\eta_{\alpha(\rho}\eta_{\sigma)\beta}
       -\frac12\eta_{\alpha\beta}\eta_{\rho\sigma},
 \label{eq:P}\\
 H_{\phi\phi}(x,y)&=\Delta_m^+(x-y),\qquad
 H_{h\phi}=H_{\phi h}=0,
 \label{eq:Hscalar}
\end{align}
where \(m\geq0\) is fixed and
\begin{equation}
 \Delta_m^+(x-y)
 =\int_{\R^3}\frac{\dd^3p}{(2\pi)^3\,2\omega_m(\mathbf p)}
   e^{-i\omega_m(\mathbf p)(t-s)
      +i\mathbf p\cdot(\mathbf x-\mathbf y)},
 \qquad
 \omega_m(\mathbf p)=\sqrt{|\mathbf p|^2+m^2}.
 \label{eq:positive}
\end{equation}
No positivity is assumed for the graviton block.

The remaining gauge-fixed multiplet components are spectators in every
calculation below. An enlargement of the full multiplet algebra must,
in particular, admit the displayed even-sector products. Therefore an
obstruction in this sector is sufficient for the stated conclusion.
No inference about physical observables in gauge cohomology is made;
that is a separate problem in the perturbative gravitational
framework \cite{BFR}.
% Framework reference checked against the primary source: ([arxiv.org](https://arxiv.org/pdf/1306.1058))

We use the Wick-symbol convention
\begin{equation}
 F\star_H G
 =\sum_{r\geq0}\frac{\hbar^r}{r!}
   \la F^{(r)},H^{\otimes r}G^{(r)}\ra.
 \label{eq:star}
\end{equation}
For finite polynomials this sum is finite. A local expression such as
\(\int g h_{00}^2\) denotes its Wick symbol; no self-contraction is
included inside that expression.

\subsection{The old polynomial domain}

Let \(\Pol\) denote finite polynomial functionals with compact
distributional kernels satisfying the usual microcausal exclusion:
for a kernel of degree \(n\),
\[
 \WF(u_n)\cap
 \left(
   \widetilde{\cV}_+^{\,n}
   \cup
   \widetilde{\cV}_-^{\,n}
 \right)=\varnothing,
 \qquad
 \widetilde{\cV}_\pm=\overline{\cV}_\pm\cup\{0\},
\]
with the total zero covector omitted.
Let \(\Preg\subset\Pol\) denote the finite polynomials with smooth
compact kernels.

Only finite polynomials are used. Results about unrestricted smooth
microcausal functionals are not presumed. In particular, the
distinction between polynomial and general functional domains
identified by Hawkins, Rejzner, and Visser is respected
\cite{HRV}. Their generalized-cone framework motivates a cone test,
but supplies no premise asserting that \(B_f\) passes it.
% Polynomial-domain and generalized-cone references: ([arxiv.org](https://arxiv.org/html/2312.15203v2))

\subsection{What will be proved}

The principal result is the following statement, whose topology and
coefficient are defined precisely in Sections~\ref{sec:topology}
and~\ref{sec:triple}.

\begin{theorem}[Main result: a specified C-obstruction]
\label{thm:main}
For the fixed matrix \eqref{eq:Hhh}--\eqref{eq:Hscalar}, there exist
real nonnegative \(f,f',g\in C_c^\infty(M_+)\), with pairwise disjoint
supports, and regular polynomials \(B_{f,\eps},B_{f',\eps}\) such that:
\begin{enumerate}[label=\textup{(\roman*)}]
 \item their mixed kernels converge to the exact kernels of
 \(B_f,B_{f'}\) in the normal-topology stages specified in
 Section~\ref{sec:topology};

 \item with \(W_g=\int g(y)h_{00}(y)^2\dd^4y\),
 \begin{equation}
 \ev[\hbar^3]\!
 \left(B_{f,\eps}\star_H W_g\star_H B_{f',\eps}\right)
 =
 \mathcal C(f,g,f')\log(\eps^{-1})+O(1),
 \label{eq:mainasymptotic}
 \end{equation}
 where \(\mathcal C(f,g,f')>0\);

 \item the corresponding fully contracted coefficient has no
 sequentially continuous extension at the pair of sharp kernels
 that agrees with the old Wick product on regular kernels.
\end{enumerate}
In particular, no algebra extension satisfying
Assumption~\ref{ass:extension} exists.
\end{theorem}

The evaluation \(\ev\) sets all classical fields to zero and hence
extracts the field-independent coefficient. It is not a replacement
of the algebra-valued target by a matrix element: it is a necessary
continuous coefficient map of the proposed algebra extension.

\section{Exact off-shell reconstruction and causal support}

Write
\[
    \gamma_0h=h|_{\Sigma_0},\qquad
    \gamma_1h=\partial_t h|_{\Sigma_0},\qquad
    j=\square h.
\]
All formulas act componentwise on \(h\).

\begin{proposition}[Off-shell reconstruction]
\label{prop:reconstruction}
For every smooth \(h\), on \(M_+\),
\begin{align}
 h&=\U(\gamma_0h,\gamma_1h)+\Ezero j,
 \label{eq:reconstruction}\\
 w[h]&=\Ezero C\U(\gamma_0h,\gamma_1h)
        +\Ezero C\Ezero j.
 \label{eq:wsplit}
\end{align}
Consequently,
\[
 B_f=B_{f,\mathrm{data}}+B_{f,\mathrm{source}}
\]
is an exact off-shell identity.

For \(K=\supp f\), set
\[
 \mathcal D_K
 =
 J^-(K)\cap\{0\leq t\leq \max_{x\in K}t(x)\}.
\]
Then \(\mathcal D_K\) is compact, and \(B_f\) depends on \(h\) only
in \(\mathcal D_K\) and on \(\phi\) only in \(K\).
\end{proposition}

\begin{proof}
Let \(A=\sqrt{-\Delta_{\mathbf x}}\). In spatial Fourier notation,
\[
 \U(a,b)(t)=\cos(tA)a+\frac{\sin(tA)}{A}b,
\]
and
\[
 (\Ezero j)(t)
 =-\int_0^t\frac{\sin((t-u)A)}{A}j(u)\dd u.
\]
The quotients at zero spatial frequency are interpreted by their
smooth limits.

The difference between the two sides of
\eqref{eq:reconstruction} solves the homogeneous wave equation with
zero Cauchy data, and therefore vanishes. Applying \(\Ezero C\)
gives \eqref{eq:wsplit}, and substituting into \eqref{eq:B} gives the
claimed split.

Finite propagation speed implies that \(w[h](x)\), for \(x\in K\),
depends only on the restriction of \(h\) to the causal segment between
\(\Sigma_0\) and \(x\). The same holds for its derivatives.
The union of these segments is contained in \(\mathcal D_K\).
Compactness follows directly from compactness of \(K\) and the
Minkowski causal bound.
\end{proof}

\begin{remark}[The source is retained]
The triple \((\gamma_0h,\gamma_1h,j)\) is a presentation of the same
off-shell field, not a new collection of propagating fields.
In particular, \(j\) is not set to zero.

For linear source smearings, the bisolution identities imply
\[
 H_{jh}=0,\qquad H_{jj}=0.
\]
Thus such source variables are central for the free Wick product.
This does not make them zero classical functionals, and it does not
by itself define products involving arbitrary singular source-sector
kernels.
\end{remark}

\section{The full insertion kernel and the characteristic-cone test}

\subsection{A joint wavefront upper bound}

We order the mixed kernel variables as \((x,z)\), with \(x\) the scalar
slot and \(z\) the graviton slot. There is a compact tensor-valued
distribution \(b_f(x,z)\) such that
\[
 B_f[h,\phi]=\la b_f,\phi\otimes h\ra.
\]
The compact-kernel assertion follows either from the explicit
construction below or from continuity of the bilinear functional
\eqref{eq:B} and Proposition~\ref{prop:reconstruction}.

Let \(E_{\mathrm{ret}}\) denote the retarded fundamental solution of
\(\square\). For \(x^0>0\),
\begin{equation}
 (Lh)(x)
 =
 \int E_{\mathrm{ret}}(x-z)\theta(z^0)(Ch)(z)\dd^4z.
 \label{eq:Lret}
\end{equation}
The identity is local in \(x\), so harmless compact cutoffs can be
inserted in the integration variables.

For covectors in the fixed Cartesian coordinates, define
\begin{align}
 \Lambda_{\mathrm{diag}}
   &=\{(\xi,\zeta)\neq0:\zeta=-\xi\},\\
 \Lambda_{\partial}
   &=\{(\xi,\zeta)\neq0:
          \eta^{-1}(\xi,\xi)=0,\ 
          \xi+\zeta\in\R\,\dd t\},\\
 \Lambda_B&=\Lambda_{\mathrm{diag}}\cup\Lambda_{\partial},
 \qquad
 \Gamma_B=(M\times M)\times\Lambda_B.
 \label{eq:GammaB}
\end{align}
The case \(\xi=0\) is included in \(\Lambda_{\partial}\).

\begin{lemma}[Joint-kernel control]
\label{lem:WFbound}
The kernel \(b_f\) satisfies
\[
 \WF(b_f)\subset\Gamma_B.
\]
More precisely, its possible singular branches arise from:
\begin{enumerate}[label=\textup{(\alph*)}]
 \item the bulk diagonal \(x=z\);
 \item direct null propagation from \(z\) to \(x\);
 \item \(z^0=0\) together with null propagation, with covectors
 \[
      (\xi,-\xi+\tau\,\dd t);
 \]
 \item pure boundary conormals
 \[
      (0,\tau\,\dd t).
 \]
\end{enumerate}
This is an upper bound, not an assertion that every listed covector
occurs.
\end{lemma}

\begin{proof}
The retarded wave kernel has diagonal conormals and, away from the
diagonal, the null propagation relation
\((x,\xi;z,-\xi)\).
Multiplication by \(\theta(z^0)\) is permitted: a wavefront covector
of that factor has zero \(x\)-component, whereas a wavefront covector
of the retarded kernel has nonzero \(x\)-component. Hence opposite
covectors cannot sum to zero.

The product theorem yields the direct branches, the boundary sums
\((\xi,-\xi+\tau\,\dd t)\), and the pure boundary conormals.
At \(z^0=0\), the bulk diagonal does not meet the support of \(f\),
because \(f\) is supported strictly in \(M_+\).

Integrating the first-order operator \(C\) by parts in \(z\) gives
the kernel of \(L\). This can produce a \(\delta(z^0)\), but not a
\(\delta'(z^0)\). Finally, moving \(\partial_\mu\) from \(\phi\)
onto the \(x\)-kernel and multiplying by \(f(x)\) do not enlarge the
wavefront set. This proves the assertion.
\end{proof}

The distinctions between multiplication, pullback, and pushforward
used here are the standard distributional ones; membership in a
wavefront-restricted vector space is not itself an operation theorem
\cite{BDH}.
% Distributional-operation reference: ([arxiv.org](https://arxiv.org/pdf/1409.7662))

\subsection{The characteristic-cone enlargement does not admit the insertion}

Let \(\cN_\pm\) be the two null cones in the cotangent bundle and put
\(\widetilde{\cN}_\pm=\cN_\pm\cup\{0\}\).
The proposed characteristic-cone condition is
\[
 \WF(u_n)\cap
 \left(
   \widetilde{\cN}_+^{\,n}
   \cup
   \widetilde{\cN}_-^{\,n}
 \right)=\varnothing.
\]

\begin{proposition}[Failure of the characteristic-cone test]
\label{prop:char}
For every nonzero \(f\in C_c^\infty(M_+)\),
\[
             B_f\notin\Pchar.
\]
In particular, its full mixed kernel cannot be admitted merely by
replacing the causal exclusion cones by the null exclusion cones.
\end{proposition}

\begin{proof}
Fix \(\mathbf n\in S^2\), let \(r>0\), and consider the smooth
complexified configuration
\[
 h^{(r)}_{\alpha\beta}(z)
 =
 P_{\alpha\beta00}
 e^{-irz^0+ir\mathbf n\cdot\mathbf z}.
\]
A direct use of \eqref{eq:C} gives
\[
 (Ch^{(r)})^\mu
 =
 -ir\,\delta^\mu_0
 e^{-irz^0+ir\mathbf n\cdot\mathbf z}.
\]
Solving the zero-Cauchy equation yields
\begin{equation}
 w^0[h^{(r)}](t,\mathbf x)
 =
 \left(
   -\frac t2e^{-irt}
   +\frac{\sin(rt)}{2r}
 \right)e^{ir\mathbf n\cdot\mathbf x},
 \qquad
 w^i[h^{(r)}]=0.
 \label{eq:plane-w}
\end{equation}

Choose \(a\in C_c^\infty(M_+)\) with
\(\int fa\dd^4x\neq0\), and set
\[
 \phi_r(x)=a(x)e^{-irt-ir\mathbf n\cdot\mathbf x}.
\]
Substitution into \eqref{eq:B} gives
\[
 B_f[h^{(r)},\phi_r]
 =
 \frac14\int fa\dd^4x+O(r^{-1}).
\]
Indeed, the nonoscillating term is obtained from
\[
 \sin(rt)e^{-irt}=\frac{1-e^{-2irt}}{2i};
\]
all terms containing \(e^{-2irt}\) decrease rapidly after the
\(t\)-integration, including the term with the prefactor \(r t\).
The term containing \(\partial_t a\) is \(O(r^{-1})\).

This is a Fourier test of the compact kernel \(a(x)b_f(x,z)\),
contracted with the constant tensor \(P_{\alpha\beta00}\), along
a ray for which both slot covectors are null and have the same
temporal orientation. If the characteristic-cone condition held,
compactness and the definition of the wavefront set would imply
rapid decrease along that ray. The nonzero limit contradicts it.

If desired, the plane wave in the graviton slot can be multiplied
by a compact cutoff equal to one on a neighborhood of
\(\mathcal D_{\supp f}\), without changing the calculation.
\end{proof}

\section{Cauchy covariances and the full tensor audit}

\subsection{Cauchy pullbacks}

Define
\[
 H_\Sigma^{ij}
 =
 (\gamma_i\otimes\gamma_j)H_{hh},
 \qquad i,j\in\{0,1\}.
\]
These pullbacks exist because the conormals of the spacelike Cauchy
surface are timelike, whereas the nonzero Hadamard covectors are
null. The same argument permits the bulk--boundary blocks.
This is an application of the pullback criterion, not a multiplication
of boundary delta distributions \cite{BDH}.

For the scalar massless factor, spatial Fourier transformation gives
\begin{equation}
 \widehat H_\Sigma(\mathbf p)
 =
 \frac1{2r}
 \begin{pmatrix}
    1&ir\\
    -ir&r^2
 \end{pmatrix},
 \qquad r=|\mathbf p|,
 \label{eq:Cauchymatrix}
\end{equation}
multiplied by \(P_{\alpha\beta\rho\sigma}\) in the graviton sector.
The entries are locally integrable at \(r=0\) in three spatial
dimensions. The full matrix, including the off-diagonal phases, is
retained.

\subsection{The vanishing \(ww\) contraction}

\begin{proposition}[Tensor cancellation]
\label{prop:tensor}
For the matrix \eqref{eq:Hhh},
\begin{align}
 C_x^\mu H_{hh\,\rho\sigma}(x,y)
 &=
 \delta^\mu_{(\rho}\partial^x_{\sigma)}
       \Delta_0^+(x-y),
 \label{eq:CH}\\
 C_x^\mu C_y^\nu H_{hh}(x,y)
 &=
 -\frac12\eta^{\mu\nu}\square_x\Delta_0^+(x-y)=0.
 \label{eq:CHC}
\end{align}
The induced free Cauchy contractions satisfy
\[
                         H_{ww}=0.
\]
\end{proposition}

\begin{proof}
In four dimensions,
\[
 \eta^{\alpha\beta}P_{\alpha\beta\rho\sigma}
 =-\eta_{\rho\sigma}.
\]
The trace contribution in \(C_x^\mu\) cancels the trace term in the
first divergence of \(P\), leaving \eqref{eq:CH}.
Applying \(C_y^\nu\), using
\(\partial^y=-\partial^x\) on \(\Delta_0^+(x-y)\), cancels the two
mixed second-derivative terms and gives \eqref{eq:CHC}.

To justify the Cauchy operations without multiplying singular
unsmeared kernels, use \eqref{eq:positive}.
On each massless momentum mode, the twice-contracted tensor
numerator in \eqref{eq:CHC} vanishes before either zero-Cauchy
solution operator is applied. The remaining mode factors have
polynomial growth and locally integrable infrared behavior.
Thus the smeared contraction is zero. Equivalently, the Cauchy
covariance construction based on \eqref{eq:Cauchymatrix} gives the
zero distribution.
\end{proof}

\begin{remark}[Three limitations]
The preceding result does not say that \(w[h]\) is the zero
off-shell functional. It does not say that \(H_{wh}=0\).
It depends on the fixed matrix \eqref{eq:Hhh}; no claim is made for
an arbitrary gauge choice or an arbitrary smooth change of covariance.
\end{remark}

\subsection{The mixed covariance and its reflected branch}

The component needed for the obstruction is
\[
             K(x,y):=H_{wh}^{0}{}_{00}(x,y).
\]
By \eqref{eq:CH},
\[
 K(x,y)=\Ezero^x\partial_t\Delta_0^+(x-y).
\]
For \(r>0\),
\[
 \Ezero(e^{-irt})
 =
 -\frac{it}{2r}e^{-irt}
 +\frac{i\sin(rt)}{2r^2}.
\]
Consequently,
\begin{align}
 K(x,y)
 &=
 \int_{\R^3}\frac{\dd^3p}{(2\pi)^3}
 e^{i\mathbf p\cdot(\mathbf x-\mathbf y)}
 \Bigg[
   \left(-\frac{t}{4r}+\frac{i}{8r^2}\right)
       e^{-ir(t-s)}
   -\frac{i}{8r^2}e^{ir(t+s)}
 \Bigg].
 \label{eq:K}
\end{align}
The terms \(r^{-1}\) and \(r^{-2}\) are locally integrable at
\(\mathbf p=0\). At large momentum this is an oscillatory
distribution with symbol amplitudes.

The last term in \eqref{eq:K} is the
\emph{Cauchy-reflected branch}. Its first time frequency is opposite
to that of the direct Hadamard branch, while its second time
frequency is unchanged. This is the branch responsible for the
obstruction.

Reality of \(\Ezero C\) and Hermiticity of the fixed covariance imply
\[
 H_{hw\,00}{}^0(y,x)=\overline{K(x,y)}.
\]

\section{The finite tests that do pass}
\label{sec:smalltests}

The results in this section are existence statements for specified
compact polynomial kernels. They are not a closure theorem.

\subsection{Linear generators and insertion--linear products}

Independently smeared Cauchy fields are admitted using
\eqref{eq:Cauchymatrix}. Their pairings with ordinary spacetime
smearings are well defined. This does not define coincident products
of boundary fields.

For a compact smooth graviton test tensor \(a\), write
\[
 h(a)=\int a^{\alpha\beta}h_{\alpha\beta}.
\]
Since \(H_{hh}a\) is smooth,
\[
 B_f\star_H h(a)
 =
 B_fh(a)
 -\hbar\int f\,
       \bigl(L(H_{hh}a)\bigr)^\mu
       \partial_\mu\phi.
\]
Likewise, for \(u\in C_c^\infty(M)\),
\[
 B_f\star_H\phi(u)
 =
 B_f\phi(u)
 -\hbar\int f\,w^\mu[h]\,
       \partial_\mu(H_{\phi\phi}u).
\]
The reverse products use the transposed covariance.
All displayed coefficients are compact distributional polynomial
functionals with the causal support inherited from
Proposition~\ref{prop:reconstruction}.

\subsection{The insertion self-product}

\begin{proposition}[A well-defined self-product kernel]
\label{prop:self}
For \(f,f'\in C_c^\infty(M_+)\), the following compact
distributional polynomial is well defined:
\begin{align}
 B_fB_{f'}
 +\hbar\,T_{f,f'}[h],\qquad
 T_{f,f'}[h]
 &=
 \int f(x)f'(x')\,
 w^\mu[h](x)w^\nu[h](x')
 \partial_\mu^x\partial_\nu^{x'}
 \Delta_m^+(x-x')\dd^4x\dd^4x'.
 \label{eq:self}
\end{align}
It is the Wick contraction formula for the proposed
\(B_f\star_H B_{f'}\). The potential \(\hbar^2\) coefficient vanishes.
\end{proposition}

\begin{proof}
For fixed smooth \(h\), the factor multiplying the differentiated
Hadamard distribution in \eqref{eq:self} is a smooth compact test
function. Hence the pairing exists.

The map from \(h\) to \(w[h]\), restricted to any fixed compact
observation region, is continuous in the smooth topology and depends
only on a fixed compact causal region.
Polarization of \eqref{eq:self} therefore gives a continuous bilinear
form on the corresponding smooth graviton configurations.
The Schwartz kernel theorem gives a compact distributional
quadratic kernel.

There are no \(h\)--\(\phi\) contractions. Every term involving a
\(w\)--\(w\) contraction vanishes by
Proposition~\ref{prop:tensor}. This leaves exactly \eqref{eq:self}.
\end{proof}

\begin{remark}
The conclusion is stronger than a merely formal inventory of
self-contractions, but weaker than an algebra theorem.
The mixed triple below shows why the distinction is necessary.
\end{remark}

\subsection{The mixed quadratic test}

Let
\[
 Q_q=\int q^{\rho\sigma\nu}(y)
        h_{\rho\sigma}(y)\partial_\nu\phi(y)\dd^4y,
 \qquad q\in C_c^\infty.
\]

\begin{proposition}[The mixed quadratic test]
\label{prop:quadratic}
The finite contraction formulas for
\(B_f\star_H Q_q\) and \(Q_q\star_H B_f\) define compact
distributional polynomial kernels. In particular, the double
contraction
\[
 H_{wh}^{\mu}{}_{\rho\sigma}(x,y)
 \,\partial_\mu^x\partial_\nu^y\Delta_m^+(x-y)
\]
is a well-defined distribution.
\end{proposition}

\begin{proof}
Each component of \(H_{wh}\) has a direct and a reflected Fourier
representation of the form illustrated in \eqref{eq:K}, with
polynomial tensor numerators. In either branch, every nonzero
wavefront covector in the \(y\)-slot has the same temporal orientation
as the second slot of \(\Delta_m^+\); it is nonzero and null.
This follows directly from the phase
\(ir s-i\mathbf p\cdot\mathbf y\).
The direct branch has the usual null relation and the reflected
branch has the relation obtained by reflecting the first time
coordinate.

The two factors in the displayed product therefore cannot have
opposite total covectors: their \(y\)-components already have the
same orientation in a pointed cone. The multiplication criterion
applies.

A single \(h\)-contraction leaves a distribution multiplied by the
smooth classical scalar factors. A single scalar contraction leaves
a differentiated Hadamard distribution multiplied by the smooth
factors \(w[h](x)h(y)\). As in
Proposition~\ref{prop:self}, continuity in the smooth configurations
and the kernel theorem give the remaining compact polynomial
kernels. The reverse product has the reversed, but again common,
orientation.
\end{proof}

The product with
\[
                 W_g=\int g\,h_{00}^2
\]
also exists at the level of \(B_f\star_H W_g\): at most one graviton
leg of \(B_f\) can contract with \(W_g\), and the resulting mixed
covariance is a distribution paired with smooth remaining
configuration factors. What fails is the mandatory subsequent
multiplication by a second insertion in the topology specified next.

\section{The named topology and regular approximations}
\label{sec:topology}

\subsection{Normal-topology stages}

For a compact \(K\subset M\times M\), write
\[
 \E'_{\Gamma_B}(K)
 =
 \{u\in\E'(M\times M):
       \supp u\subset K,\ \WF(u)\subset\Gamma_B\},
\]
componentwise in the finite-dimensional tensor indices.
It carries the normal topology.

Explicitly, this topology is generated by strong distribution
seminorms
\[
 p_{\mathcal B}(u)
   =\sup_{\psi\in\mathcal B}|\la u,\psi\ra|,
\]
where \(\mathcal B\) is a bounded set of test functions, and by
\[
 p_{\chi,V,N}(u)
 =
 \sup_{\zeta\in V}
 (1+|\zeta|)^N|\widehat{\chi u}(\zeta)|,
\]
where \(V\) is a closed cone and
\[
 (\supp\chi\times V)\cap\Gamma_B=\varnothing.
\]
The normal topology and the distinction between continuity and
hypocontinuity are important here \cite{DB,BDH}.
% Normal-topology references: ([arxiv.org](https://arxiv.org/pdf/1308.1061))

The translation-invariant angular cone \(\Lambda_B\) in
\eqref{eq:GammaB} is deliberately an upper-bound cone. The theorem
does not presume that it is the smallest possible cone.

\subsection{A compact regularization of the kernels}

Choose a real, nonnegative, even, Euclidean-radial
\(\rho\in C_c^\infty(\R^4)\) with \(\int\rho=1\), and put
\[
 \rho_\eps(x)=\eps^{-4}\rho(x/\eps),
 \qquad
 S_\eps u=\rho_\eps*u.
\]
Define
\begin{equation}
 B_{f,\eps}[h,\phi]
 =
 B_f[S_\eps h,S_\eps\phi].
 \label{eq:Beps}
\end{equation}
Its mixed kernel is
\[
 b_{f,\eps}
 =
 (\rho_\eps\otimes\rho_\eps)*b_f.
\]
In particular, \(B_{f,\eps}\in\Preg\).

These are approximation devices for the exact sharp kernel.
They are not proposed as replacement observables.

\begin{lemma}[Normal convergence]
\label{lem:normalconv}
For every fixed \(f\), choose a compact \(K\) containing a neighborhood
of \(\supp b_f\). Then, for all sufficiently small \(\eps>0\),
\[
 \supp b_{f,\eps}\subset K,\qquad
 b_{f,\eps}\longrightarrow b_f
 \quad\hbox{in }\E'_{\Gamma_B}(K)
\]
with its normal topology.
\end{lemma}

\begin{proof}
Smoothness and the fixed compact support follow from convolution
with a compact approximate identity.

Strong distribution convergence follows from the finite order of
\(b_f\) and the convergence
\[
 (\rho_\eps\otimes\rho_\eps)*\psi\longrightarrow\psi
\]
in all smooth seminorms, uniformly for \(\psi\) in a bounded set of
test functions.

For completeness, the microlocal estimates can be checked directly
because \(\Gamma_B\) has a translation-invariant angular cone.
Compactness and
\(\WF(b_f)\subset(M\times M)\times\Lambda_B\)
imply rapid decrease of \(\widehat b_f\) in every closed cone
separated from \(\Lambda_B\).
Moreover,
\[
 \widehat b_{f,\eps}(\xi,\zeta)
 =
 \widehat\rho(\eps\xi)\widehat\rho(\eps\zeta)
 \widehat b_f(\xi,\zeta).
\]
The multipliers are uniformly bounded and tend to one on compact
frequency sets.

On a closed cone separated from \(\Lambda_B\), split frequency space
into a fixed ball and its complement. On the ball, the multiplier
converges uniformly to one. On the complement, arbitrarily high
rapid-decay bounds for \(\widehat b_f\) give a uniformly small tail.
Thus the weighted Fourier seminorms converge to zero.

For a localized seminorm, use
\[
 \widehat{\chi(b_{f,\eps}-b_f)}
 =
 \widehat\chi*
 \bigl[
   (\widehat\rho(\eps\xi)\widehat\rho(\eps\zeta)-1)
   \widehat b_f
 \bigr].
\]
A slightly larger cone still disjoint from \(\Lambda_B\) controls
the nearby-frequency part. On the complementary part, the angular
separation forces the argument of \(\widehat\chi\) to be large;
its arbitrary rapid decrease dominates the uniform polynomial
growth of \(\widehat b_f\).
This proves convergence of every normal-topology seminorm.
\end{proof}

\begin{remark}[Support and the role of the regulator]
The approximants have support in arbitrarily small neighborhoods of
the exact compact support; the limit retains the exact causal support
of Proposition~\ref{prop:reconstruction}.
The theorem uses an ordinary fixed-support distribution stage
containing those neighborhoods.
It does not impose an equation of motion and does not replace
\(\theta\) by a new smooth-edge observable.
\end{remark}

\section{The mandatory mixed triple}
\label{sec:triple}

\subsection{The exact regularized coefficient}

Let \(f,f',g\) be real compact smooth functions, with \(f,f'\)
supported in \(M_+\), and set
\[
 W_g[h]=\int g(y)h_{00}(y)^2\dd^4y.
\]
Define
\begin{equation}
 \cG_\eps(f,g,f')
 =
 \ev[\hbar^3]
 \left(
    B_{f,\eps}\star_H W_g\star_H B_{f',\eps}
 \right).
 \label{eq:Gdef}
\end{equation}
This coefficient is unambiguously defined in the old polynomial
domain.

There is exactly one fully contracted graph type:
the scalar legs of the two insertions are joined, and the two
graviton legs are joined to the two legs of \(W_g\).
The two assignments of the \(W_g\) legs give a factor of \(2\).
No \(ww\) contraction is present in this graph.

Let
\[
 \mu(u)=\widehat\rho(-u,u\mathbf n),
 \qquad \mathbf n\in S^2.
\]
Euclidean radiality makes this independent of \(\mathbf n\).
It is real, smooth, rapidly decreasing, and \(\mu(0)=1\).
For the scalar field define
\[
 M_\eps(r)
 =
 \widehat\rho\bigl(\eps(-\omega_m(r),r\mathbf n)\bigr)
 =
 \mu\!\left(\eps\sqrt{r^2+\frac{m^2}{2}}\right).
 \label{eq:M}
\]
Here and below \(\omega_m(r)=\sqrt{r^2+m^2}\).

Let \(K_\eps\) be the right-hand side of \eqref{eq:K} with the
additional multiplier \(\mu(\eps|\mathbf p|)\) in the momentum
integral. This is exactly the covariance between
\(w[S_\eps h](x)\) and \(h_{00}(y)\).

For \(q\in\R^3\), put
\begin{equation}
 F_{f,\eps}(y,q)
 =
 \int f(x)K_\eps(x,y)
       e^{-i\omega_m(q)t+iq\cdot\mathbf x}\dd^4x.
 \label{eq:F}
\end{equation}

\begin{lemma}[Spectral form of the triple]
\label{lem:spectral}
For every \(\eps>0\),
\begin{equation}
 \cG_\eps(f,g,f')
 =
 \frac{1}{(2\pi)^3}
 \int g(y)\dd^4y
 \int_{\R^3}
   \omega_m(q)M_\eps(|q|)^2
   F_{f,\eps}(y,q)
   \overline{F_{f',\eps}(y,q)}
   \dd^3q.
 \label{eq:spectral}
\end{equation}
For \(f'=f\) and \(g\geq0\), this expression is nonnegative.
\end{lemma}

\begin{proof}
By \eqref{eq:CH}, a contraction with \(h_{00}\) forces the
insertion vector index to be \(0\). Thus the scalar contraction is
\[
 \partial_t\partial_{t'}(S_\eps\otimes S_\eps)
             \Delta_m^+(x-x')
 =
 \int\frac{\dd^3q}{(2\pi)^3}
 \frac{\omega_m(q)}2 M_\eps(|q|)^2
 e^{-i\omega_m(q)(t-t')+iq\cdot(\mathbf x-\mathbf x')}.
\]
The factor \(2\) from the two \(W_g\)-leg assignments cancels the
factor \(1/2\) in this spectral density. The two minus signs in the
definitions of the insertions cancel as well.
Hermiticity gives the complex conjugate factor in
\eqref{eq:spectral}.

For fixed \(\eps>0\), the mollifier multipliers and the compact
smooth smearings make the resulting integrals convergent.
The positivity assertion uses only the positive scalar spectral
measure, not positivity of the graviton covariance.
\end{proof}

\subsection{The reflected light-ray transform}

For \(y=(s,\mathbf y)\) and \(\mathbf n\in S^2\), define
\begin{equation}
 A_f(y,\mathbf n)
 =
 \int_\R f\bigl(t,\mathbf y+\mathbf n(t+s)\bigr)\dd t.
 \label{eq:ray}
\end{equation}
Only a compact interval contributes.
Geometrically, this samples a future observation ray issuing from
the spatial point \(\mathbf y+s\mathbf n\) on \(\Sigma_0\).

\begin{lemma}[Uniform high-frequency asymptotics]
\label{lem:asymptotic}
Let \(Y\subset M\) be compact. Uniformly for
\(y\in Y\), \(\mathbf n\in S^2\), \(r\geq1\), and
\(0<\eps\leq1\),
\begin{equation}
 F_{f,\eps}(y,r\mathbf n)
 =
 -\frac{i}{8r^2}
 e^{ir(s+\mathbf n\cdot\mathbf y)}
 \left[
    \mu(\eps r)A_f(y,\mathbf n)
    +E_{f,\eps}(y,r,\mathbf n)
 \right],
 \label{eq:Fasymptotic}
\end{equation}
where
\begin{equation}
 |E_{f,\eps}(y,r,\mathbf n)|
 \leq C_{f,Y,m}(r^{-1}+\eps).
 \label{eq:Ebound}
\end{equation}
For \(|q|\leq1\), \(F_{f,\eps}(y,q)\) is uniformly bounded on
\(Y\), independently of \(0<\eps\leq1\).
\end{lemma}

\begin{proof}
Separate \eqref{eq:K} into its direct and reflected terms.

\paragraph{Direct term.}
After integration against \(f\), the direct term contains the
Fourier transform of \(f\), or of \(tf\), at temporal frequency
\(|p|+\omega_m(r)\) and spatial frequency \(p+r\mathbf n\).
Since \(f\) is smooth and compact,
\[
 \left|\widehat f\bigl(|p|+\omega_m(r),p+r\mathbf n\bigr)\right|
 \leq C_N(1+|p|+r)^{-N},
\]
and the same holds with \(tf\).
Integrating against \(|p|^{-1}+|p|^{-2}\) proves arbitrary rapid
decrease in \(r\), uniformly in \(\eps\) and in \(y\in Y\).

\paragraph{Reflected term.}
Write
\[
 p=-r\mathbf n+a.
\]
On \(|a|\leq r/2\),
\begin{align*}
 |p|&=r-\mathbf n\cdot a+O(|a|^2/r),\\
 r^2|p|^{-2}&=1+O(|a|/r),\\
 \mu(\eps|p|)
   &=\mu(\eps r)+O(\eps|a|),\\
 \omega_m(r)&=r+O(r^{-1}).
\end{align*}
The constants in the last estimate may depend on the fixed mass.
The phase is
\begin{align*}
 &|p|(t+s)-\omega_m(r)t
   +p\cdot(\mathbf x-\mathbf y)
   +r\mathbf n\cdot\mathbf x\\
 &\qquad
 =r(s+\mathbf n\cdot\mathbf y)
   +a\cdot\bigl(\mathbf x-\mathbf y-\mathbf n(t+s)\bigr)
   +O\!\left((|a|^2+1)/r\right),
\end{align*}
uniformly for \(t\) in the support of \(f\) and \(y\in Y\).

The spatial Fourier transform of \(f(t,\cdot)\) decreases rapidly
in \(a\), uniformly in \(t\), with all required derivatives.
The errors above therefore contribute
\(O(r^{-1}+\eps)\) after extracting \(r^{-2}\).

The leading \(a\)-integral is Fourier inversion:
\[
 \int_{\R^3}\frac{\dd^3a}{(2\pi)^3}
 \int_{\R^3}f(t,\mathbf x)
 e^{ia\cdot(\mathbf x-\mathbf y-\mathbf n(t+s))}
 \dd^3x
 =
 f(t,\mathbf y+\mathbf n(t+s)).
\]
Its \(t\)-integral is \(A_f\).

On \(|a|>r/2\), arbitrary rapid decrease in \(a\) dominates the
locally integrable factor \(|p|^{-2}\).
After extracting \(r^{-2}\), that tail is \(O(r^{-N})\) for any
prescribed \(N\), by taking sufficiently many Fourier-decay powers.
The direct term can be absorbed into the same error.
This proves \eqref{eq:Fasymptotic}--\eqref{eq:Ebound}.

For bounded \(q\), integrate the spatial Fourier-decay bounds
against \(|p|^{-1}+|p|^{-2}\). Both are locally integrable, and the
large-\(|p|\) tails are rapidly decreasing. This proves the last
assertion.
\end{proof}

\subsection{The logarithmic coefficient}

\begin{theorem}[Explicit mixed-graph divergence]
\label{thm:log}
For real \(f,f',g\) as above,
\begin{equation}
 \cG_\eps(f,g,f')
 =
 \mathcal C(f,g,f')\log(\eps^{-1})+O(1),
 \label{eq:log}
\end{equation}
where
\begin{equation}
 \boxed{
 \mathcal C(f,g,f')
 =
 \frac{1}{64(2\pi)^3}
 \int_M g(y)\dd^4y
 \int_{S^2}
 A_f(y,\mathbf n)A_{f'}(y,\mathbf n)
 \dd\Omega(\mathbf n).
 }
 \label{eq:residue}
\end{equation}
The \(O(1)\) bound is for the fixed three test functions and fixed
mass. The coefficient of the logarithm is independent of the chosen
normalized radial mollifier.
\end{theorem}

\begin{proof}
Insert Lemma~\ref{lem:asymptotic} into \eqref{eq:spectral} and use
polar coordinates \(q=r\mathbf n\).
For \(r\geq1\),
\[
 r^2\omega_m(r)=r^3(1+O(r^{-2})).
\]
The leading contribution is therefore
\[
 \mathcal C(f,g,f')\,I_\eps,
 \qquad
 I_\eps
 =
 \int_1^\infty
 \frac{M_\eps(r)^2\mu(\eps r)^2}{r}\dd r.
\]

We first check that the remainder is uniformly bounded.
By \eqref{eq:Ebound}, the error in the product of the two square
brackets in \eqref{eq:Fasymptotic} is bounded by a constant times
\[
 (r^{-1}+\eps)+(r^{-1}+\eps)^2.
\]
Also, for every \(N\),
\[
 |M_\eps(r)|\leq C_N(1+\eps r)^{-N}.
\]
Thus the error integral is controlled by
\[
 C\int_1^\infty
 \frac{(1+\eps r)^{-N}}{r}
 \left[
   r^{-1}+\eps+(r^{-1}+\eps)^2
 \right]\dd r.
\]
This is uniformly bounded: terms containing \(r^{-1}\) are
integrable without the cutoff, and
\[
 \eps\int_1^\infty\frac{(1+\eps r)^{-N}}r\dd r
 =
 O\bigl(\eps\log(\eps^{-1})+\eps\bigr).
\]
The correction from
\(\omega_m(r)/r-1=O(r^{-2})\) is integrable as well.
The region \(0\leq r\leq1\) is uniformly bounded by the final
assertion of Lemma~\ref{lem:asymptotic}.

It remains to evaluate \(I_\eps\).
For \(r\geq1\),
\[
 \left|
 \sqrt{r^2+\frac{m^2}{2}}-r
 \right|\leq C_m r^{-1},
\]
so boundedness of \(\mu'\) gives
\[
 |M_\eps(r)-\mu(\eps r)|\leq C_m\eps r^{-1}.
\]
Consequently,
\[
 I_\eps
 =
 \int_1^\infty\frac{\mu(\eps r)^4}{r}\dd r+O(\eps).
\]
Changing variables \(u=\eps r\),
\[
 \int_1^\infty\frac{\mu(\eps r)^4}{r}\dd r
 =
 \int_\eps^\infty\frac{\mu(u)^4}{u}\dd u
 =
 \log(\eps^{-1})+O(1),
\]
because \(\mu(0)=1\), \(\mu(u)^4-1=O(u)\) near zero, and \(\mu\)
decreases rapidly at infinity.
This proves \eqref{eq:log} with the coefficient
\eqref{eq:residue}.
\end{proof}

\subsection{Nonzero coefficient away from all observation coincidences}

\begin{lemma}[Separated-support witness]
\label{lem:separated}
There exist nonnegative \(f,f',g\in C_c^\infty(M_+)\) with
pairwise disjoint supports and
\[
                  \mathcal C(f,g,f')>0.
\]
Their supports may also be chosen disjoint from \(\Sigma_0\).
\end{lemma}

\begin{proof}
Choose \(S,T,T'>0\) pairwise distinct and \(\mathbf n_0\in S^2\).
Consider
\[
 y_0=(S,\mathbf0),\qquad
 x_0=(T,(T+S)\mathbf n_0),\qquad
 x_0'=(T',(T'+S)\mathbf n_0).
\]
These three spacetime points are distinct.
Choose nonnegative smooth bumps \(g,f,f'\), respectively positive
near these points, with sufficiently small pairwise disjoint
supports contained in \(M_+\).

For \(y\) near \(y_0\) and \(\mathbf n\) near \(\mathbf n_0\),
the rays in \eqref{eq:ray} meet the regions where \(f\) and \(f'\)
are positive. Hence both ray transforms are positive on an open
set on which \(g\) is positive. Formula \eqref{eq:residue} then
gives the result.
\end{proof}

The coefficient \(\mathcal C\) is supported on the reflected
collinearity relation
\begin{equation}
 \begin{split}
 \cR_\Sigma=\bigl\{(x,y,x'):\;&
 y=(s,\mathbf y),\\
 &x=(t,\mathbf y+\mathbf n(t+s)),\\
 &x'=(t',\mathbf y+\mathbf n(t'+s)),\
 \mathbf n\in S^2
 \bigr\}.
 \end{split}
 \label{eq:reflectionrelation}
\end{equation}
The witness in Lemma~\ref{lem:separated} lies on its noncoincident
part.

\section{The continuity obstruction}
\label{sec:obstruction}

\subsection{The coefficient-level statement}

For regular mixed kernels \(u,v\), let \(F_u,F_v\) denote the
associated degree-\((1,1)\) polynomials, and define
\[
 T_g(u,v)
 =
 \ev[\hbar^3]\bigl(F_u\star_H W_g\star_H F_v\bigr).
\]
This is a bilinear map on regular kernels.

\begin{theorem}[No normal-topology extension of the mandatory coefficient]
\label{thm:coefficient}
Choose \(f,f',g\) as in Lemma~\ref{lem:separated}, and let \(K\)
be a compact set containing neighborhoods of both mixed-kernel
supports.
The map \(T_g\) has no extension to a domain containing
\[
 b_f,\ b_{f'},\
 \{b_{f,\eps},b_{f',\eps}:0<\eps\leq\eps_0\}
\]
which:
\begin{enumerate}[label=\textup{(\roman*)}]
 \item agrees with \(T_g\) on the regular kernels;
 \item is sequentially continuous at \((b_f,b_{f'})\) for the
 product of the subspace normal topologies inherited from
 \(\E'_{\Gamma_B}(K)\).
\end{enumerate}
\end{theorem}

\begin{proof}
By Lemma~\ref{lem:normalconv},
\[
 (b_{f,\eps},b_{f',\eps})\longrightarrow(b_f,b_{f'})
\]
in the stated product topology.
Sequential continuity would force
\(T_g(b_{f,\eps},b_{f',\eps})\) to converge to a finite complex
number.

Agreement on regular kernels and Theorem~\ref{thm:log} instead give
\[
 \operatorname{Re}T_g(b_{f,\eps},b_{f',\eps})
 =
 \mathcal C(f,g,f')\log(\eps^{-1})+O(1)
 \longrightarrow+\infty.
\]
This is a contradiction.
\end{proof}

\begin{remark}[Why this is more than a divergent regulator]
Three independently established facts enter the contradiction:
the approximants belong to the old domain; they converge to the
sharp kernels in the named topology; and the proposed extension is
required to agree with the old products and be continuous.
The divergence alone would not constitute an obstruction.
\end{remark}

\subsection{Consequence for algebra extensions}

\begin{assumption}[Normal-topology Wick extension]
\label{ass:extension}
A proposed extension consists of a Hausdorff locally convex
\(\C[[\hbar]]\)-algebra \(\cA\), with the following properties.
\begin{enumerate}[label=\textup{(E\arabic*)}]
 \item \(\Pol[[\hbar]]\) embeds into \(\cA\), and its product is the
 fixed Wick product \eqref{eq:star}.

 \item The algebra admits the sharp insertions \(Q_\Sigma(B_f)\),
 with
 \[
 [Q_\Sigma(B_f)]_{\hbar^0}=B_f
 \]
 as functionals on all smooth configurations.

 \item On the degree-\((1,1)\) kernel stages used above, admission
 is sequentially continuous for the normal topology and agrees
 with the old inclusion on regular kernels. In particular,
 \[
 B_{f,\eps}\longrightarrow Q_\Sigma(B_f)
 \quad\hbox{in }\cA.
 \]

 \item Multiplication is hypocontinuous with respect to bounded
 sets. The field-independent \(\hbar^3\) coefficient map
 \[
       a\longmapsto \ev[\hbar^3]a
 \]
 is continuous and agrees with the usual coefficient extraction
 on the old polynomial domain.
\end{enumerate}
\end{assumption}

\begin{corollary}[No algebra extension under the stated requirements]
\label{cor:noalgebra}
There is no extension satisfying
Assumption~\ref{ass:extension}.
\end{corollary}

\begin{proof}
Let
\[
 X_\eps=B_{f,\eps},\quad
 Y_\eps=B_{f',\eps},\quad
 X=Q_\Sigma(B_f),\quad
 Y=Q_\Sigma(B_{f'}).
\]
By assumption \(X_\eps\to X\) and \(Y_\eps\to Y\), so both
convergent families are bounded.

Hypocontinuity implies separate continuity, hence
\(X_\eps\star_H W_g\to X\star_H W_g\).
It also gives joint sequential continuity along convergent
sequences: subtract the limit product, split the difference into
two terms, and use continuity uniformly over the bounded family
in the other factor. Therefore
\[
 (X_\eps\star_H W_g)\star_H Y_\eps
 \longrightarrow
 (X\star_H W_g)\star_H Y.
\]
Applying the continuous coefficient map gives convergence of
\(\cG_\eps(f,g,f')\), contradicting Theorem~\ref{thm:log} and
Lemma~\ref{lem:separated}.
\end{proof}

Theorem~\ref{thm:main} follows from
Lemma~\ref{lem:normalconv},
Theorem~\ref{thm:log},
Lemma~\ref{lem:separated}, and
Theorem~\ref{thm:coefficient}.

\subsection{Counterterms and the scope of the conclusion}

The extension rule considered here is:
\begin{quote}
Keep the fixed Wick product on the old domain, admit sharp kernels
continuously in the specified normal topology, and introduce no
new counterterms on noncoincident characteristic relations.
\end{quote}

The separated-support witness shows that a counterterm supported
only on the ordinary observation diagonals
\[
 x=y,\qquad y=x',\qquad x=x'
\]
or on observation boundary faces cannot subtract the coefficient
\eqref{eq:residue}. The divergence persists when the three
observation supports avoid all those sets.

There is an additional distinction worth making explicit.
A distribution local in auxiliary Cauchy variables can acquire
nonlocal support after Cauchy propagation. Such a modification is
not automatically an ordinary observation-diagonal counterterm.
Its pushforward, topology, and compatibility with the old Wick
product would have to be specified separately.

More fundamentally, any subtraction performed on the regular
approximants in \eqref{eq:Gdef} changes products that are already
fixed in \(\Pol\). Assigning a subtracted value only at the sharp
kernel does not restore the continuity contradicted by
Theorem~\ref{thm:coefficient}.

Thus the theorem excludes the stated normal-topology extension.
It does not exclude a different boundary theory with additional
renormalized operations, a stronger topology in which the
approximants do not converge, or a smaller bulk algebra that omits
\(W_g\).

\section{Outcome and the remaining interacting problem}

\subsection{Audit of the finite tests}

\begin{center}
\begin{tabular}{@{}p{0.13\textwidth}p{0.43\textwidth}p{0.35\textwidth}@{}}
\toprule
Test & Object & Result in this manuscript\\
\midrule
A &
Smoothly smeared Cauchy and bulk linear fields &
Cauchy covariance construction exists.\\[0.3em]
B &
\(B_f\) against bulk linear fields &
Compact polynomial representatives exist.\\[0.3em]
C &
\(B_f\star_H B_{f'}\) &
The kernel \eqref{eq:self} exists;
the \(ww\) terms vanish.\\[0.3em]
D &
\(B_f\star_H Q_q\) and reverse order &
The finite contraction kernels exist.\\[0.3em]
E &
\(B_f\star_H W_g\star_H B_{f'}\) &
No extension of the fully contracted coefficient with the stated
normal-topology continuity and old-product agreement.\\
\bottomrule
\end{tabular}
\end{center}

Neither the vanishing of \(H_{ww}\) nor success of Tests A--D is a
closure theorem. Test E supplies the named obstruction.

The characteristic-cone alternative also fails for the actual
insertion kernel by Proposition~\ref{prop:char}. Merely allowing
timelike linear boundary conormals misses the same-orientation
null pairs in the bilinear kernel.

\subsection{Off-shell normalization and presentation independence}

Throughout the argument:
\begin{itemize}
 \item \(B_f\) is exactly the functional \eqref{eq:B};
 \item \(j=\square h\) is retained in \eqref{eq:wsplit};
 \item no free-equation quotient is taken;
 \item no independent coordinate field is introduced;
 \item all regularized coefficients are computed in the original
 Wick-symbol presentation.
\end{itemize}

The bulk representation \eqref{eq:Lret}, the Cauchy-data/source
representation \eqref{eq:wsplit}, and the Fourier expression
\eqref{eq:K} describe the same fixed operator.
The obstruction is therefore not obtained by assigning different
products to two presentations of the same classical kernel.

\subsection{Separation from time-ordered and retarded products}

The source-sector observation in Section~2 uses the bisolution
identity
\[
                    P H=0.
\]
A Feynman Green function instead satisfies
\[
                    P H_F=c\,\delta
\]
with a convention-dependent nonzero constant \(c\).
Consequently, source variables with zero free Wick contractions can
produce contact terms in time-ordered constructions.
The free Wick and renormalized time-ordered constructions are
distinct \cite{FR}.
% Free versus time-ordered construction reference: ([arxiv.org](https://arxiv.org/pdf/1208.1428))

No statement here constructs any of
\[
 R_1(V_1^{\mathrm{sc}};B_f),\qquad
 \mathcal B_f^{(2)},\qquad
 R_1(V_2;\mathcal B_f^{(0)}),\qquad
 \frac12R_2(V_1,V_1;\mathcal B_f^{(0)}).
\]
No interaction vertex is needed for the obstruction, and none is
introduced as an unused premise.

\subsection{Closed statement}

The result of this manuscript is the following restricted but
complete conclusion:
\[
\boxed{
\begin{gathered}
 \text{sharp zero-Cauchy dressing}
 +\text{ exact off-shell insertion}\\
 +\text{ the fixed de Donder Wick matrix}
 +\text{ normal-topology admission}\\
 +\text{ continuity and agreement with the old Wick product}\\
 \Longrightarrow
 \text{ failure of the mandatory mixed triple coefficient.}
\end{gathered}
}
\]

This is a C-obstruction for a named boundary-extension scheme.
It is not a proof that every possible enlargement fails, not a
bimodule construction, and not a closure of the interacting product
problem.

\appendix

\section{A compact derivation of the boundary-frequency reversal}

The mechanism can be isolated at one massless momentum.
For \(r>0\), the scalar zero-Cauchy inverse is
\[
 \Ezero(e^{-irt})
 =
 -\frac{it}{2r}e^{-irt}
 +\frac{1}{4r^2}\bigl(e^{irt}-e^{-irt}\bigr).
\]
The second term enforces the initial conditions. It introduces an
\(e^{irt}\) component although the source has only \(e^{-irt}\)
frequency.

For the component selected by \(h_{00}\), the derivative in
\eqref{eq:CH} multiplies the source by \(-ir\), and the Hadamard
spectral measure contributes \(1/(2r)\). The reflected mixed
covariance therefore has amplitude
\[
                       -\frac{i}{8r^2}.
\]
In the mixed triple, two such amplitudes contribute \(r^{-4}\).
The differentiated scalar covariance contributes \(r\), and the
remaining radial measure contributes \(r^2\dd r\).
Thus the leading radial behavior is
\[
                       \frac{\dd r}{r}.
\]
This power count alone is not the proof of divergence:
the tensor selection, exact graph identity, nonzero light-ray
coefficient, uniform remainder estimate, and topology argument
are supplied in Sections~4--7.

\section{Why no additional boundary field is hidden in the proof}

The variables
\[
             (\gamma_0h,\gamma_1h,\square h)
\]
are used only through the identity
\[
 h=\U(\gamma_0h,\gamma_1h)+\Ezero\square h.
\]
Their compatibility is inherited from the original smooth
configuration \(h\). No independent integration over boundary
data or source variables is introduced.

Similarly, \(S_\eps h\) and \(S_\eps\phi\) in \eqref{eq:Beps} are
ordinary smoothings of the original fields. The resulting
\(B_{f,\eps}\) are regular elements of the old domain and serve
only to test continuity at the exact sharp kernel.
The proof does not declare any fixed \(\eps>0\) observable to be
the sharp insertion.

\begin{thebibliography}{99}

\bibitem{DB}
Y.~Dabrowski and C.~Brouder,
\emph{Functional properties of H\"ormander's space of distributions
having a specified wavefront set},
arXiv:1308.1061v2, 2014.

\bibitem{BDH}
C.~Brouder, N.~V.~Dang, and F.~H\'elein,
\emph{Continuity of the fundamental operations on distributions
having a specified wave front set
(with a counter example by Semyon Alesker)},
Studia Mathematica \textbf{232} (2016), 201--226;
arXiv:1409.7662v2.

\bibitem{HRV}
E.~Hawkins, K.~Rejzner, and B.~Visser,
\emph{A novel class of functionals for perturbative algebraic
quantum field theory},
arXiv:2312.15203v2, 2024.
The generalized-cone discussion and the polynomial results of this
specified version are the background used here.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory},
arXiv:1208.1428v2, 2013.

\bibitem{BFR}
R.~Brunetti, K.~Fredenhagen, and K.~Rejzner,
\emph{Quantum gravity from the point of view of locally covariant
quantum field theory},
arXiv:1306.1058.

\end{thebibliography}

\end{document}