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% A linear on-shell repair is proved, but no nonlinear or interacting repair
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\title{\textbf{Landmark VII-B}\\[0.5em]
Cauchy harmonic insertions and second-order retarded products:\\
a sharp-boundary obstruction in an off-shell polynomial domain}
\author{}
\date{}

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

\begin{abstract}
We investigate the first geometric coefficient of a relational scalar
defined using harmonic coordinates with fixed zero Cauchy data.
The quantum fields are a scalar and a gauge-fixed spin-two field,
together with the ghost and auxiliary sectors of perturbative gravity.
The proposed insertion space is the algebra of compactly supported
microcausal polynomial kernels, with a fixed graded Hadamard contraction.

For the coordinate-response operator \(L=\mathsf E_0C\), we compute its
distributional adjoint exactly.  It contains a hypersurface delta
distribution whose nonzero conormal covectors are timelike.  We prove an
exact criterion for a linearly smeared coordinate response to belong to
the declared domain.  More importantly, for every nonzero real test
function supported strictly to the future of the initial surface, the
first geometric insertion fails the microcausal condition.  The proof
exhibits a smooth scalar configuration at which the metric derivative
has a nonvanishing hypersurface delta term.  This excludes a
coefficientwise, classically normalized quantization of that insertion
in the declared off-shell algebra, and hence excludes the proposed
termwise second-order construction in that algebra.

The obstruction is not a claim that the relevant distributions do not
exist.  Smooth-edge regularization is constructed and its sharp limit
is analyzed.  A smooth spacetime representative is also constructed for
each linear coordinate observable on the free mass shell.  Neither
result supplies the missing nonlinear interacting products.  The paper
therefore closes the boundary-domain alternative, not the
second-order existence alternative.
\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) {Classical relational scalar};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Reference coordinates};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Quantum product domain};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.35) {A classical construction does not by itself name the product domain.};
\draw[->] (a) -- (b);
\draw[->] (b) -- (c);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A relational scalar is not a product domain. 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-B}
\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) {Classical relational scalar};
\node[draw,rounded corners,align=center,text width=4.3cm,font=\small,inner sep=3pt] (b) at (6.4,0) {Quantum product domain};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (3.2,-2.35) {A classical construction does not by itself name the product domain.};
\draw[->] (a) -- (s);
\draw[->] (b) -- (s);
\end{tikzpicture}
\caption{A relational scalar is not a product domain. 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-B}
\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) {Classical relational scalar};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (b) at (0,-1.7) {Reference coordinates};
\node[draw,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (c) at (0,-3.4) {Quantum product domain};
\node[draw,dashed,rounded corners,align=center,text width=7.2cm,font=\small,inner sep=3pt] (s) at (0,-5.2) {A classical construction does not by itself name the product domain.};
\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-B}
\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) {Classical relational scalar};
\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) {Quantum product domain};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5.2,-2.35) {Reference coordinates. A classical construction does not by itself name the product domain.};
\draw (a) -- (m);
\draw (m) -- (b);
\draw[->] (m) -- (s);
\end{tikzpicture}
\caption{Classical relational scalar is not Quantum product domain. 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-B}
\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: Classical relational scalar};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (b) at (5,0) {Used here: Reference coordinates};
\node[draw,rounded corners,align=center,text width=3.15cm,font=\small,inner sep=3pt] (c) at (10,0) {Not claimed: Quantum product domain};
\node[draw,dashed,rounded corners,align=center,text width=9.6cm,font=\small,inner sep=3pt] (s) at (5,-2.5) {A classical construction does not by itself name the product domain.};
\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-B}
\end{figure}

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


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



\tableofcontents

\section{Question, result, and scope}

A classical relational construction does not, by itself, specify a
quantum product domain.  This distinction is especially important when
the reference coordinates are obtained by solving a hyperbolic
equation with data prescribed on a sharp initial hypersurface.

We consider the geometric functional
\begin{equation}
 \Bcal_f[\mathbf g,\phi]
 =
 \int_M f(y)\,
 \phi\bigl(\mathsf X_{\mathbf g}^{-1}(y)\bigr)\,\dd^4y,
 \label{eq:geometric-functional}
\end{equation}
where the four functions \(\mathsf X_{\mathbf g}^{\mu}\) are harmonic
for \(\mathbf g\), with their values and first time derivatives fixed
on an initial surface.  All classical ingredients used below are
specified explicitly in Section~\ref{sec:classical}; no theorem about
their diffeomorphism intertwining is assumed or reproved.

The intended second-order interacting coefficient is formally
\begin{equation}
 \Bcal_f^{(2)}
 +R_1(V_1;\Bcal_f^{(1)})
 +R_1(V_2;\Bcal_f^{(0)})
 +\frac12 R_2(V_1,V_1;\Bcal_f^{(0)}).
 \label{eq:four-pieces-intro}
\end{equation}
Before this expression can be constructed term by term, its geometric
insertions must be admitted by the chosen domain.  That prerequisite
fails for the domain studied here.

The main result is Theorem~\ref{thm:main}.  For every
\[
 0\ne f\in C_c^\infty(\{t>0\};\R),
\]
the first geometric coefficient
\[
 \Bcal_f^{(1)}[h,\phi]
 =
 -\int f(x)\,(\Cau C h)^\mu(x)\,\partial_\mu\phi(x)\,\dd^4x
\]
is not a microcausal functional on the full smooth configuration
space.  The obstruction is a \emph{proved wavefront component},
rather than a failure to verify a sufficient criterion for multiplying
distributions.

We use a finite polynomial-kernel domain.  This avoids relying on
closure assertions for arbitrary smooth microcausal functionals:
Hawkins, Rejzner, and Visser identify a closure problem for that larger
class, introduce equicausal functionals, and establish equicausality
for Wick polynomials with admissible fixed kernels
\cite{HRV}.  The present obstruction is more elementary: it violates
the microcausal cone condition itself, and therefore also excludes an
equicausal representative that is equal to the original functional
off shell.

The gravitational interpretation is deliberately limited.  The
gauge-fixed field content belongs to perturbative gravity, rather
than to a scalar theory with a prescribed external metric.  However,
admission to a gauge-fixed algebra is not the same as physicality.
The latter requires a separate gauge-cohomological analysis, as in
the perturbative framework of Brunetti, Fredenhagen, and Rejzner
\cite{BFR}.

\paragraph{What is and is not proved.}
The following statements are established:
\begin{enumerate}[label=(\roman*),leftmargin=2.2em]
 \item an exact sharp-Cauchy adjoint formula;
 \item an exact domain criterion for linearly smeared coordinate
       responses;
 \item exclusion of every nontrivially smeared first geometric
       insertion from the declared off-shell domain;
 \item impossibility of an off-shell, classically normalized
       quantization in that domain;
 \item a smooth-edge approximation and a linear on-shell repair.
\end{enumerate}
No Ward identity, radiative response, classification of interaction
counterterms, or second-order product-existence theorem is asserted.
In particular, the result does not exclude a construction in an
appropriately enlarged boundary algebra or an interacting on-shell
formulation.

\section{Classical input and geometric bookkeeping}
\label{sec:classical}

\subsection{Spacetime, fields, and zero-Cauchy response}

Let
\[
 M=\R^{1,3},\qquad
 \eta=\diag(1,-1,-1,-1),\qquad
 \Sigma_0=\{t=0\},\qquad
 M_+=\{t\geq0\}.
\]
Indices are raised and lowered with \(\eta\), and
\[
 \square=\partial_t^2-\Delta_{\bx}.
\]
The metric perturbation \(h_{\mu\nu}\) is a smooth symmetric
covariant tensor.  Write
\[
 h=\eta^{\mu\nu}h_{\mu\nu},\qquad
 C^\mu[h]=\partial_\nu h^{\nu\mu}
              -\frac12\partial^\mu h.
 \label{eq:C-definition}
\]
The symbol \(h\) without indices denotes the trace only where no
confusion can arise.

Let \(\Ret\) and \(\Adv\) denote the retarded and advanced Green
operators for \(\square\), respectively:
\[
 \square\Ret u=u,\qquad
 \square\Adv u=u,\qquad
 \Ret^*=\Adv.
\]
The zero-Cauchy solution operator \(\Cau\) is defined by
\begin{equation}
 \square \Cau s=s\quad(t>0),\qquad
 (\Cau s)|_{\Sigma_0}=0,\qquad
 \partial_t(\Cau s)|_{\Sigma_0}=0.
 \label{eq:cauchy-operator}
\end{equation}
For smooth \(s\), it can be realized for \(t>0\) as
\begin{equation}
 \Cau s=\Ret(\theta s),
 \qquad
 \theta(t)=
 \begin{cases}
  0,&t<0,\\
  1,&t>0.
 \end{cases}
 \label{eq:cauchy-retarded}
\end{equation}
The right-hand side is understood as the retarded solution of the
past-compact distributional source \(\theta s\).  Its restriction to
\(M_+\) has the prescribed zero Cauchy data.

The first coordinate response is
\begin{equation}
 w^\mu[h]=L^\mu h,\qquad L=\Cau C.
 \label{eq:L-definition}
\end{equation}
This operator is not switched.

For a compact set \(K\Subset\{t>0\}\), set
\begin{equation}
 \D_K=J^-(K)\cap M_+.
 \label{eq:DK}
\end{equation}
In Minkowski spacetime this is compact.  Finite propagation speed
implies that \(w[h]|_K\) depends only on \(h\) in \(\D_K\), including
its trace on the initial surface.

\subsection{Harmonic expansion}

Introduce a single metric field by
\[
 \mathbf g=\eta+\kappa h.
\]
Define
\[
 d(\kappa)=\sqrt{-\det(\eta+\kappa h)},\qquad
 A^{\mu\nu}(\kappa)=d(\kappa)\,\mathbf g^{\mu\nu}.
\]
Their first coefficients are
\begin{align}
 d_1&=\frac12 h,
 &
 d_2&=\frac18h^2-\frac14h_{\rho\sigma}h^{\rho\sigma},
 \label{eq:dcoeff}\\
 A_1^{\mu\nu}
 &=\frac12h\,\eta^{\mu\nu}-h^{\mu\nu},
 \label{eq:A1}\\
 A_2^{\mu\nu}
 &=h^\mu{}_\rho h^{\rho\nu}
   -\frac12h\,h^{\mu\nu}
   +d_2\eta^{\mu\nu}.
 \label{eq:A2}
\end{align}

The harmonic-coordinate equations are written in density form:
\begin{equation}
 \partial_\nu\!
 \left(A^{\nu\rho}(\kappa)
       \partial_\rho\mathsf X^\mu_{\mathbf g}\right)=0.
 \label{eq:harmonic-density}
\end{equation}
Fix
\[
 \mathsf X^\mu_{\mathbf g}|_{\Sigma_0}=x^\mu|_{\Sigma_0},
 \qquad
 \partial_t\mathsf X^\mu_{\mathbf g}|_{\Sigma_0}
 =\partial_t x^\mu|_{\Sigma_0}.
\]
Writing
\[
 \mathsf X^\mu_{\mathbf g}
 =x^\mu+\kappa w^\mu+\kappa^2v^\mu+O(\kappa^3)
\]
gives \eqref{eq:L-definition} and
\begin{equation}
 v^\mu[h,h]
 =
 -\Cau\left\{
       \partial_\nu A_2^{\nu\mu}
       +\partial_\nu
        \left(A_1^{\nu\rho}\partial_\rho w^\mu\right)
      \right\}.
 \label{eq:v-definition}
\end{equation}
Here \(v\) denotes the coefficient of \(\kappa^2\), without a
factorial.

Formal inversion yields
\begin{equation}
 (\mathsf X_{\mathbf g}^{-1})^\mu
 =
 x^\mu-\kappa w^\mu
 +\kappa^2\left(w^\nu\partial_\nu w^\mu-v^\mu\right)
 +O(\kappa^3).
 \label{eq:inverse}
\end{equation}
Consequently,
\begin{align}
 \Bcal_f^{(0)}[h,\phi]
 &=\int f\phi\,\dd^4x,
 \label{eq:B0}\\
 \Bcal_f^{(1)}[h,\phi]
 &=-\int f w^\mu\partial_\mu\phi\,\dd^4x,
 \label{eq:B1}\\
 \Bcal_f^{(2)}[h,\phi]
 &=\int f\left[
    \left(w^\nu\partial_\nu w^\mu-v^\mu\right)
       \partial_\mu\phi
    +\frac12 w^\mu w^\nu\partial_\mu\partial_\nu\phi
   \right]\dd^4x.
 \label{eq:B2}
\end{align}
These are geometric coefficients, expanded before any interacting
field equations are solved.

\subsection{No double counting}

For clarity, consider arbitrary smooth formal curves
\[
 H_{\mathrm{int}}=h+\kappa k+O(\kappa^2),\qquad
 \phi_{\mathrm{int}}
 =\phi_0+\kappa\phi_1+\kappa^2\phi_2+O(\kappa^3).
\]
Then direct substitution gives
\begin{align}
 \Bcal_f[\eta+\kappa H_{\mathrm{int}},\phi_{\mathrm{int}}]
 ={}&
 \int f\phi_0\,\dd^4x
 +\kappa\int f\left(\phi_1-w[h]^\mu\partial_\mu\phi_0\right)
       \dd^4x
 \nonumber\\
 &+\kappa^2\int f\Big[
 \phi_2-w[h]^\mu\partial_\mu\phi_1
 -w[k]^\mu\partial_\mu\phi_0
 \nonumber\\
 &\hspace{5.8em}
 +\left(w[h]^\nu\partial_\nu w[h]^\mu
        -v[h,h]^\mu\right)\partial_\mu\phi_0
 \nonumber\\
 &\hspace{5.8em}
 +\frac12w[h]^\mu w[h]^\nu
       \partial_\mu\partial_\nu\phi_0
 \Big]\dd^4x
 +O(\kappa^3).
 \label{eq:classical-bookkeeping}
\end{align}
Thus the second metric coefficient \(k\) enters through the
interacting expansion of one graviton field.  It is not a second
independent quantum field.

Equation~\eqref{eq:classical-bookkeeping} is an algebraic identity on
smooth configurations.  It is not a theorem asserting the existence
of quantum retarded products.  In particular, \(\phi_1\) and
\(\phi_2\) are not included in the bare coefficients
\eqref{eq:B1}--\eqref{eq:B2}.

\section{Quantum model and declared insertion domain}
\label{sec:domain}

\subsection{Gauge-fixed free multiplet}

The quantum field multiplet is
\[
 \Psi=(h_{\mu\nu},\phi,c^\mu,\bar c_\mu,\beta_\mu).
\]
The fields \(c,\bar c\) are odd; the remaining fields are even.
The scalar has mass \(m>0\).

Let the trace-reversal operator be
\begin{equation}
 (\mathsf T h)_{\mu\nu}
 =h_{\mu\nu}-\frac12\eta_{\mu\nu}h,
 \qquad
 \mathsf T^2=\id.
 \label{eq:trace-reversal}
\end{equation}
The free action, modulo an unswitched total divergence, is
\begin{equation}
 S_0
 =
 \int\left[
 \frac12\partial_\lambda h_{\mu\nu}
             \partial^\lambda h^{\mu\nu}
 -\frac14\partial_\lambda h\,\partial^\lambda h
 +\frac12\partial_\lambda\phi\,\partial^\lambda\phi
 -\frac12m^2\phi^2
 -\bar c_\mu\square c^\mu
 -\frac14\beta_\mu\beta^\mu
 \right]\dd^4x.
 \label{eq:free-action}
\end{equation}
This is the linear de Donder gauge with a decoupled, shifted
Nakanishi--Lautrup field.  An explicit nonlinear action representative
and its switched vertices are specified in
Appendix~\ref{app:vertices}.

With the metric used to identify dual tensor bundles, the free
Hessian has diagonal even blocks
\[
 P_{hh}=-\mathsf T\square,\qquad
 P_{\phi\phi}=-(\square+m^2),\qquad
 P_{\beta\beta}=-\frac12\eta,
\]
and odd block, in the order \((c,\bar c)\),
\[
 P_{\mathrm{odd}}
 =
 \begin{pmatrix}
  0&\square\\
  -\square&0
 \end{pmatrix}.
\]
The displayed odd signs correspond to writing the quadratic action
as \(-\bar c_\mu\square c^\mu\).

For \(m\geq0\), write
\begin{equation}
 \Delta_m^+(x-y)
 =
 \int_{\R^3}
 \frac{\dd^3\bk}{(2\pi)^3\,2\omega_m(\bk)}
 e^{-i\omega_m(\bk)(x^0-y^0)+i\bk\cdot(\bx-\by)},
 \qquad
 \omega_m(\bk)=\sqrt{|\bk|^2+m^2}.
 \label{eq:wightman}
\end{equation}
Fix the graded two-point matrix by
\begin{align}
 \Htwo_{hh\,\mu\nu\rho\sigma}
 &=\mathsf T_{\mu\nu\rho\sigma}\Delta_0^+,
 &
 \Htwo_{\phi\phi}&=\Delta_m^+,
 \label{eq:H-even}\\
 \Htwo^{c^\mu\bar c_\nu}
 &=\delta^\mu_\nu\Delta_0^+,
 &
 \Htwo^{\bar c_\nu c^\mu}
 &=-\delta^\mu_\nu\Delta_0^+.
 \label{eq:H-odd}
\end{align}
All remaining blocks vanish.  In particular, the algebraic auxiliary
field has zero two-point function.

No positivity assertion is made for this gauge-fixed matrix.
Moreover, gauge fixing does not impose \(C[h]=0\) as an identity on
the off-shell configuration space.

\subsection{The polynomial kernel domain}

Let \(\overline V_\pm\subset T^*M\) be the future/past nonspacelike
covector cones, including the zero covector in each fibre.  For
\(n\geq1\), define
\begin{equation}
 \Xi_n
 =
 T^*M^n\setminus
 \left(\overline V_+^{\,n}\cup\overline V_-^{\,n}\right).
 \label{eq:Xi}
\end{equation}
Wavefront sets never contain the total zero covector.

\begin{definition}[Admissible polynomial kernels]
\label{def:poly}
The space \(\Pol\) consists of finite polynomials in the field
multiplet whose coefficient kernels are compactly supported
distributions \(u_n\) satisfying
\[
 \WF(u_n)\subset\Xi_n.
\]
The kernels have the graded symmetry appropriate to their field
slots.  In a purely even sector,
\[
 F(\Psi)=\sum_{n=0}^N\frac1{n!}
          \langle u_n,\Psi^{\otimes n}\rangle.
\]
The quantum polynomial algebra is
\[
 \Alg=\Pol[[\hbar]]
\]
with the graded Wick product determined by \(\Htwo\).
\end{definition}

For fixed polynomial degree the Wick expansion has only finitely
many contractions.  The usual wavefront calculus gives the Wick
product on this domain; the polynomial setting also falls within
the fixed-kernel results of \cite{HRV}.  Background on these
contractions and their microlocal formulation can be found in
\cite{FR}.

We shall use only a necessary property of this domain.

\begin{lemma}[Necessary derivative condition]
\label{lem:derivative-condition}
If \(F\in\Pol\), then every functional derivative satisfies
\[
 \WF\bigl(F^{(r)}(\Psi)\bigr)\subset\Xi_r
\]
at every smooth configuration \(\Psi\).  In particular,
\(F^{(1)}(\Psi)\) has no nonzero timelike covector in its wavefront
set.
\end{lemma}

\begin{proof}
Differentiation produces finite sums of partial contractions of
the compact kernels \(u_n\) with smooth configurations.
In the wavefront bound for a partial contraction, the covectors in
the contracted slots must be zero.  If the remaining covectors
were all future nonspacelike, or all past nonspacelike, adjoining
the zero covectors would give a forbidden element of
\(\WF(u_n)\).  This proves the claim.  The same argument applies
componentwise in the graded theory.
\end{proof}

The geometric coefficients are initially considered as functionals
on the full space of smooth configurations.  Quotienting by the
free field equations is not part of Definition~\ref{def:poly}.

\begin{definition}[Off-shell classical normalization]
\label{def:normalization}
An admissible quantization of a classical insertion \(F\) is an
element
\[
 Q(F)=\sum_{r\geq0}\hbar^r Q_r\in\Alg
\]
whose coefficient \(Q_0\) equals \(F\) on every smooth off-shell
configuration.
\end{definition}

This requirement distinguishes equality of the original functional
from equality only after imposing field equations.

\section{Exact sharp-Cauchy adjoint}
\label{sec:adjoint}

Let
\[
 a\in C_c^\infty(\{t>0\};T^*M)
\]
be a test covector, and define
\begin{equation}
 \ell_a[h]
 =
 \int a_\mu(x)(Lh)^\mu(x)\,\dd^4x,
 \qquad
 b_\mu=\Adv a_\mu.
 \label{eq:ell-a}
\end{equation}
The field \(b\) is smooth.  It is supported in
\(J^-(\supp a)\), and it solves
\[
 \square b_\mu=a_\mu.
\]
In a full neighbourhood of \(\Sigma_0\), it solves the homogeneous
wave equation.

The formal adjoint of \(C\), using the pairing
\(\int b_\mu C^\mu[h]\), is
\begin{equation}
 (C^*b)^{\alpha\beta}
 =
 -\partial^{(\alpha}b^{\beta)}
 +\frac12\eta^{\alpha\beta}\partial^\mu b_\mu.
 \label{eq:C-adjoint}
\end{equation}

\begin{proposition}[Exact adjoint and boundary coefficient]
\label{prop:adjoint}
Let \(n_\mu=\partial_\mu t\).  The compactly supported distributional
adjoint \(q_a=L^*a\) is
\begin{align}
 q_a^{\alpha\beta}
 &=C^*(\theta b)^{\alpha\beta}
 \nonumber\\
 &=\theta(C^*b)^{\alpha\beta}
   +\delta(t)\,\mathsf B^{\alpha\beta}(b|_{\Sigma_0}),
 \label{eq:q-adjoint}\\
 \mathsf B^{\alpha\beta}(v)
 &=
 -n^{(\alpha}v^{\beta)}
 +\frac12\eta^{\alpha\beta}n^\mu v_\mu.
 \label{eq:boundary-map}
\end{align}
Furthermore,
\begin{equation}
 \supp q_a\subset J^-(\supp a)\cap M_+.
 \label{eq:q-support}
\end{equation}
There is no \(\delta'(t)\) term in this adjoint.
\end{proposition}

\begin{proof}
For smooth \(h\), the retarded representation and formal
adjointness give
\[
 \ell_a[h]
 =
 \langle a,\Ret(\theta Ch)\rangle
 =
 \langle \Adv a,\theta Ch\rangle
 =
 \langle C^*(\theta b),h\rangle.
\]
The pairings involve only the compact causal set
\(J^-(\supp a)\cap M_+\).  Hence the calculation is valid without
a global compact-support assumption on \(h\).

Applying the product rule in distributions and using
\[
 \partial_\mu\theta=n_\mu\delta(t)
\]
gives \eqref{eq:q-adjoint}.  Since \(C^*\) has differential order
one, it differentiates the step function at most once.
This proves the absence of a delta derivative.  Differentiation
does not enlarge support, giving \eqref{eq:q-support}.
\end{proof}

\begin{lemma}[Nondegeneracy of the boundary map]
\label{lem:B-injective}
The map \(v\mapsto\mathsf B(v)\) is injective.
In the chosen coordinates,
\begin{equation}
 \mathsf B^{00}(v)=-\frac12v_0,\qquad
 \mathsf B^{0i}(v)=\frac12v_i,\qquad
 \mathsf B^{ij}(v)=-\frac12\delta^{ij}v_0.
 \label{eq:B-components}
\end{equation}
\end{lemma}

\begin{proof}
The displayed components follow directly from
\eqref{eq:boundary-map}.  The \(00\) and \(0i\) components recover
all components of \(v\).
\end{proof}

\subsection{Wavefront set}

Let
\[
 N^*\Sigma_0
 =
 \{(0,\bx;\tau\,\dd t):\tau\ne0\}.
\]
Every element of this conormal bundle is timelike.

\begin{lemma}[Actual conormal singularities]
\label{lem:wavefront}
For every \(a\) as above,
\begin{equation}
 \WF(q_a)\subset N^*\Sigma_0.
 \label{eq:q-wf-upper}
\end{equation}
If \(b(0,\bx_0)\ne0\), then
\begin{equation}
 (0,\bx_0;\tau\,\dd t)\in\WF(q_a)
 \qquad\text{for every }\tau\ne0.
 \label{eq:q-wf-lower}
\end{equation}
In particular, the hypersurface delta term cannot be cancelled by
the step-function term in \eqref{eq:q-adjoint}.
\end{lemma}

\begin{proof}
Both coefficients in \eqref{eq:q-adjoint} are smooth.  The standard
wavefront bounds for a smooth multiple of a step or delta
distribution therefore give \eqref{eq:q-wf-upper}; see
\cite{BDHwavefront}.

For the lower bound, choose a tensor component for which
\(\mathsf B(b(0,\bx_0))\ne0\), possible by
Lemma~\ref{lem:B-injective}.  Locally that component is
\[
 u(t,\bx)=\delta(t)v(\bx)+\theta(t)r(t,\bx),
 \qquad v(\bx_0)\ne0,
\]
with \(r,v\) smooth.

After compact localization, Fourier transformation in the normal
variable has the form
\[
 \widehat u(\tau,\boldsymbol\zeta)
 =
 \widehat v_{\mathrm{loc}}(\boldsymbol\zeta)
 +O(|\tau|^{-1})
\]
at fixed tangential frequency \(\boldsymbol\zeta\), in either
normal direction.  For any localization nonzero at
\((0,\bx_0)\), the localized \(v\) is not identically zero, so
its tangential Fourier transform is nonzero at some fixed
\(\boldsymbol\zeta\).  Thus rapid decay fails in every conic
neighbourhood of each nonzero normal covector.  The step term
has one lower normal Fourier order and cannot cancel the
nonzero leading term.
\end{proof}

\subsection{An exact criterion for linear observables}

\begin{theorem}[Linear domain criterion]
\label{thm:linear-criterion}
For \(a\in C_c^\infty(\{t>0\};T^*M)\), the following are equivalent:
\begin{enumerate}[label=(\roman*),leftmargin=2.2em]
 \item \(\ell_a\in\Pol\);
 \item \(q_a=L^*a\) is smooth;
 \item the advanced solution \(b=\Adv a\) has vanishing full Cauchy
       data on \(\Sigma_0\):
       \[
        b|_{\Sigma_0}=0,\qquad
        \partial_t b|_{\Sigma_0}=0.
       \]
\end{enumerate}
When these conditions hold, \(q_a\) is a smooth compactly supported
test tensor.
\end{theorem}

\begin{proof}
For a linear functional, domain membership is exactly the
condition \(\WF(q_a)\subset\Xi_1\).
By \eqref{eq:q-wf-upper}, every possible singular covector of
\(q_a\) is timelike.  Hence domain membership is equivalent to
smoothness.

Suppose \(q_a\) is smooth.  Its delta coefficient must vanish:
a distribution supported on the surface cannot cancel a locally
bounded step term.  Lemma~\ref{lem:B-injective} gives
\(b|_{\Sigma_0}=0\).
Smoothness of the remaining distribution
\(\theta C^*b\) implies
\[
 (C^*b)|_{\Sigma_0}=0.
\]
Since the trace of \(b\) vanishes identically on the surface, all
of its tangential derivatives vanish there.  Formula
\eqref{eq:C-adjoint} then gives
\[
 (C^*b)|_{\Sigma_0}
 =
 \mathsf B(\partial_t b|_{\Sigma_0}).
\]
Injectivity again yields \(\partial_t b|_{\Sigma_0}=0\).

Conversely, choose \(\epsilon>0\) such that \(a=0\) for
\(t<\epsilon\).  On that region \(b\) solves the homogeneous wave
equation.  Vanishing Cauchy data imply, by uniqueness, that
\(b=0\) for \(t<\epsilon\).  Thus \(\theta b=b\) is smooth, and
\(q_a=C^*b\) is smooth.  Its compact support follows from
\eqref{eq:q-support}.
\end{proof}

\begin{remark}
The theorem does not say that a distribution with timelike
wavefront set is intrinsically meaningless as a quantum smearing.
It says that this smearing is excluded by the specific off-shell
domain in Definition~\ref{def:poly}.  Section~\ref{sec:repairs}
constructs a linear on-shell representative.
\end{remark}

\section{Exclusion of the first geometric insertion}
\label{sec:main}

The obstruction for the bilinear insertion is stronger than the
existence of exceptional bad test covectors.  Every nonzero real
smearing \(f\) produces a failure somewhere in the smooth
configuration space.

\begin{lemma}[A nonzero boundary trace for every nonzero smearing]
\label{lem:witness}
Let \(0\ne f\in C_c^\infty(\{t>0\};\R)\).  There exists
\(\phi_f\in C_c^\infty(M;\R)\) such that, for
\[
 a_\mu=f\,\partial_\mu\phi_f,\qquad b_\mu=\Adv a_\mu,
\]
the trace \(b_0|_{\Sigma_0}\) is not identically zero.
One explicit choice is
\begin{equation}
 \phi_f=-\partial_t(tf).
 \label{eq:phi-witness}
\end{equation}
\end{lemma}

\begin{proof}
Let
\[
 F(t)=\int_{\R^3}a_0(t,\bx)\,\dd^3\bx,\qquad
 B(t)=\int_{\R^3}b_0(t,\bx)\,\dd^3\bx.
\]
On every bounded time interval, the spatial supports of \(a_0\)
and \(b_0\) are contained in a common compact set.  Integrating
\(\square b_0=a_0\) over space therefore yields
\[
 B''(t)=F(t).
\]
Since \(b_0\) is advanced, \(B=B'=0\) for sufficiently large
positive \(t\).  Consequently,
\begin{equation}
 B(0)=\int_0^\infty tF(t)\,\dd t
     =\int_M tf\,\partial_t\phi_f\,\dd^4x.
 \label{eq:moment}
\end{equation}
Set \(u=\partial_t(tf)\), so that \(\phi_f=-u\).
Integration by parts gives
\begin{equation}
 B(0)=\int_M u^2\,\dd^4x.
 \label{eq:positive-moment}
\end{equation}
This number is strictly positive.  Indeed, \(u=0\) would imply
that \(tf\) is independent of \(t\); compact time support would
then force \(f=0\).  Hence \(b_0|_{\Sigma_0}\) cannot vanish
identically.
\end{proof}

\begin{theorem}[Sharp-boundary obstruction]
\label{thm:main}
Fix the field multiplet and the off-shell domain of
Section~\ref{sec:domain}.  For every
\(0\ne f\in C_c^\infty(\{t>0\};\R)\):
\begin{enumerate}[label=(\roman*),leftmargin=2.2em]
 \item \(\Bcal_f^{(1)}\) is a continuous bilinear functional of
       smooth \(h,\phi\), with compact functional support
       contained in \(\D_{\supp f}\).
 \item It is not microcausal and, in particular,
       \[
        \Bcal_f^{(1)}\notin\Pol.
       \]
 \item There is no admissible quantization of
       \(\Bcal_f^{(1)}\) satisfying
       Definition~\ref{def:normalization}.
 \item There is no termwise retarded-product construction of
       \eqref{eq:four-pieces-intro} in \(\Alg\) that admits the
       original geometric insertion coefficientwise and
       preserves its off-shell classical normalization.
\end{enumerate}
\end{theorem}

\begin{proof}
Continuity of \(L\) on smooth fields, locally in the future
half-space, follows from the continuous dependence of the
inhomogeneous Cauchy problem on its source.  Multiplication by
\(f\), differentiation of \(\phi\), and integration are continuous.
This proves the first assertion, including polynomial smoothness.
The support statement follows from finite propagation and
\eqref{eq:DK}.

For the second assertion, differentiate with respect to the metric
field:
\begin{equation}
 \frac{\delta\Bcal_f^{(1)}}{\delta h}(h,\phi)
 =
 -L^*(f\,\dd\phi).
 \label{eq:metric-derivative}
\end{equation}
This derivative is independent of \(h\).

Choose \(\phi=\phi_f\) from Lemma~\ref{lem:witness}.  Then
\(b=\Adv(f\,\dd\phi_f)\) has a nonzero trace at some point of
\(\Sigma_0\).  Lemma~\ref{lem:wavefront} shows that
\(-L^*(f\,\dd\phi_f)\) has nonzero timelike covectors in its
wavefront set.  This contradicts the necessary microcausal
condition of Lemma~\ref{lem:derivative-condition}.

If an admissible quantization existed, its coefficient of
\(\hbar^0\) would belong to \(\Pol\) and would equal
\(\Bcal_f^{(1)}\) on every smooth configuration.  Differentiating
this equality gives the same forbidden derivative, proving the
third assertion.

Finally, a coefficientwise construction that admits the geometric
insertion in \(\Alg[[\kappa]]\) must admit its coefficient
\(\Bcal_f^{(1)}\), before applying
\(R_1(V_1;\,\cdot\,)\).  The third assertion excludes this
prerequisite.  This proves the fourth assertion under its stated
termwise and off-shell requirements.
\end{proof}

\begin{corollary}[The problem is already in the quadratic kernel]
\label{cor:quadratic}
The compact bilinear coefficient kernel of
\(\Bcal_f^{(1)}\) does not satisfy the cone condition
\(\Xi_2\).
\end{corollary}

\begin{proof}
If that kernel satisfied \(\Xi_2\), contraction of its scalar slot
with the smooth field \(\phi_f\) would produce a metric kernel
satisfying \(\Xi_1\), by the proof of
Lemma~\ref{lem:derivative-condition}.  This contradicts
\eqref{eq:metric-derivative} and Lemma~\ref{lem:witness}.
\end{proof}

Thus the obstruction is not an artefact of evaluating a nonlinear
functional far from the zero configuration.  The fixed degree-two
kernel itself fails the required wavefront condition.  The witness
configuration can also be multiplied by any nonzero real constant,
including arbitrarily small ones.

\subsection{What local changes cannot accomplish}

\begin{proposition}[Changes supported away from the initial surface]
\label{prop:no-local-repair}
Let \(Q_f\) be a smooth functional such that, for every smooth
configuration, its metric derivative is supported in a fixed closed
set disjoint from a neighbourhood of \(\Sigma_0\).  Then
\[
 \Bcal_f^{(1)}+Q_f
\]
is not microcausal.

In particular, adding a bilinear kernel supported on the ordinary
diagonal, with its scalar projection contained in \(\supp f\),
cannot remove the obstruction.
\end{proposition}

\begin{proof}
At the witness configuration, the metric derivative of \(Q_f\)
vanishes in a neighbourhood of \(\Sigma_0\).  Therefore the
derivative of the sum has exactly the same nonzero boundary
wavefront component as \eqref{eq:metric-derivative}.

For the final statement, a diagonal-supported kernel with scalar
projection in \(\supp f\) also has its metric projection in
\(\supp f\), which is a positive distance in time from
\(\Sigma_0\).
\end{proof}

\begin{remark}[Normal ordering and quantum corrections]
Changing a prescription only in positive powers of \(\hbar\)
cannot change the contradiction in the classical coefficient.
Allowing a new order-\(\hbar^0\) boundary term changes the
off-shell insertion unless that term vanishes as a functional.
Theorem~\ref{thm:main} does not prohibit changing the observable;
it prohibits calling such a change an off-shell quantization of
the unchanged observable under
Definition~\ref{def:normalization}.
\end{remark}

\section{Two precise repairs and their limitations}
\label{sec:repairs}

\subsection{Smooth-edge approximation}

Choose a smooth function \(\chi:\R\to[0,1]\) such that
\[
 \chi(s)=0\quad(s\leq0),\qquad
 \chi(s)=1\quad(s\geq1),
\]
and set
\[
 \theta_\epsilon(t)=\chi(t/\epsilon),\qquad \epsilon>0.
\]
Define
\begin{equation}
 L_\epsilon h=\Ret(\theta_\epsilon Ch),\qquad
 q_{a,\epsilon}=C^*(\theta_\epsilon\Adv a).
 \label{eq:smooth-edge}
\end{equation}

\begin{proposition}[Linear smoothing and the sharp limit]
\label{prop:smoothing}
For each fixed \(a\):
\begin{enumerate}[label=(\roman*),leftmargin=2.2em]
 \item \(q_{a,\epsilon}\) is a smooth compactly supported test
       tensor, and \(\ell_{a,\epsilon}[h]
       =\langle q_{a,\epsilon},h\rangle\) is a regular linear
       element of \(\Pol\);
 \item the supports of \(q_{a,\epsilon}\) are contained in the
       fixed compact set \(J^-(\supp a)\cap M_+\);
 \item \(q_{a,\epsilon}\to q_a\) in the strong topology of
       compactly supported distributions;
 \item if \(b|_{\Sigma_0}\ne0\), this sequence cannot converge
       to an element of any fixed prescribed-wavefront space
       \(\E'_\Gamma\) with \(\Gamma\subset\Xi_1\).
\end{enumerate}
\end{proposition}

\begin{proof}
Smoothness and support follow directly from
\eqref{eq:smooth-edge}.  For a smooth test tensor \(h\),
\[
 \langle q_{a,\epsilon}-q_a,h\rangle
 =
 \langle(\theta_\epsilon-\theta)b,Ch\rangle.
\]
The integrand is supported in a slab of time thickness
\(\epsilon\), inside a common spatial compact set.  It follows
that
\[
 |\langle q_{a,\epsilon}-q_a,h\rangle|
 \leq C_a\epsilon\,
       \sup_{\D_{\supp a}}|Ch|.
\]
The estimate is uniform on bounded sets of smooth test tensors,
which proves strong distributional convergence.

If convergence held in \(\E'_\Gamma\) to an element of that
space, continuity of its inclusion into distributions would
identify the limit with \(q_a\).  But
Lemma~\ref{lem:wavefront} gives a timelike wavefront component of
\(q_a\), incompatible with \(\Gamma\subset\Xi_1\).
\end{proof}

The prescribed-wavefront topology in the last assertion can be
taken to be the normal topology used for distributional operations
in \cite{BDHtopology}.  The assertion only requires that its
inclusion into distributions be continuous.

At fixed \(\epsilon\), the source equation is
\[
 \square L_\epsilon h=\theta_\epsilon Ch,
\]
rather than the unswitched equation on the whole future
half-space.  Thus a fixed nonzero edge width changes the
coordinate prescription.  Proposition~\ref{prop:smoothing}
establishes a distributional sharp limit for linear smearings,
not a sharp limit inside the declared insertion domain and not
a theorem for the bilinear composite or its retarded products.

\subsection{A linear on-shell spacetime representative}

The failure of off-shell membership does not preclude an on-shell
representation by smooth test tensors.

Let
\[
 E=\Ret-\Adv
\]
be the causal propagator of \(\square\), acting componentwise.

\begin{lemma}[Smoothing by the causal propagator]
\label{lem:causal-smoothing}
If \(q\) is a compactly supported tensor distribution with
\[
 \WF(q)\subset N^*\Sigma_0,
\]
then \(u=Eq\) is a smooth, spacelike-compact solution of
\(\square u=0\).
\end{lemma}

\begin{proof}
The causal propagator extends to compactly supported
distributions.  Its distribution kernel has only paired
characteristic covectors: both nonzero covectors in each
wavefront pair are null.  In the kernel-composition wavefront
bound, a singularity in \(Eq\) would require a null input
covector from \(\WF(q)\), or a zero input covector in the
wavefront set of the propagator kernel.  Neither occurs:
\(\WF(q)\) is timelike, and the causal propagator has no such
zero-input wavefront component.  Hence \(Eq\) is smooth.
This is an application of the standard kernel wavefront calculus
reviewed in \cite{BDHwavefront,FR}.

The bisolution identity gives \(\square Eq=0\), and the support
bound
\[
 \supp Eq\subset J(\supp q)
\]
gives spacelike compactness.
\end{proof}

\begin{theorem}[Linear on-shell repair]
\label{thm:onshell}
Let \(q=q_a=L^*a\).  There exist a smooth compactly supported
tensor \(q_{\mathrm{sm}}\) and a compactly supported tensor
distribution \(r\) such that
\begin{equation}
 q-q_{\mathrm{sm}}=\square r.
 \label{eq:on-shell-relation}
\end{equation}
Consequently,
\begin{equation}
 \langle q,h\rangle=\langle q_{\mathrm{sm}},h\rangle
 \label{eq:on-shell-equality}
\end{equation}
for every smooth solution of \(\square h=0\).

A construction is obtained by setting \(u=Eq\), choosing a smooth
time function \(\rho(t)\) equal to zero sufficiently far in the
past and one sufficiently far in the future, and defining
\begin{equation}
 q_{\mathrm{sm}}=[\square,\rho]u
 =\rho''u+2\rho'\partial_tu.
 \label{eq:time-slice-representative}
\end{equation}
\end{theorem}

\begin{proof}
Lemma~\ref{lem:causal-smoothing} gives a smooth,
spacelike-compact homogeneous solution \(u\).
The derivatives of \(\rho\) have compact time support.
Therefore \(q_{\mathrm{sm}}\) is smooth and compactly supported.

The identities
\[
 \Ret q_{\mathrm{sm}}=\rho u,\qquad
 \Adv q_{\mathrm{sm}}=(\rho-1)u
\]
follow from the wave equation and uniqueness with zero data in
the appropriate time direction.  Thus
\[
 E q_{\mathrm{sm}}=u=Eq.
\]
Set \(v=q-q_{\mathrm{sm}}\).  Then \(v\) is compactly supported
and \(Ev=0\), so
\[
 r:=\Ret v=\Adv v.
\]
Its support is contained in
\[
 J^+(\supp v)\cap J^-(\supp v),
\]
a compact set.  Moreover, \(\square r=v\), proving
\eqref{eq:on-shell-relation}.  Pairing that identity with a
smooth solution \(h\) and using the formal self-adjointness of
\(\square\) proves \eqref{eq:on-shell-equality}.
\end{proof}

\begin{remark}[Support and scope of the repair]
The construction gives compact support, but it does not assert
that \(q_{\mathrm{sm}}\) has support in the original
\(\D_{\supp a}\).  More importantly,
\eqref{eq:on-shell-equality} is an equality on the free mass shell,
not on the full configuration space.

For the gauge-fixed graviton in
\eqref{eq:free-action}, the free equation is
\(\mathsf T\square h=0\), equivalent to \(\square h=0\).
Thus Theorem~\ref{thm:onshell} applies to the linear graviton
observable.  It does not construct an off-shell polynomial
representative of the bilinear insertion, and it does not
control contact terms in interacting retarded products.
\end{remark}

\subsection{Linear coordinate two-point pairings}

For each fixed pair of coordinate tests \(a,a'\), the preceding
construction also gives well-defined free on-shell pairings
\begin{align}
 \Htwo_{wh}(a,t)
 &:=\Htwo_{hh}(q_{\mathrm{sm},a},t),
 \label{eq:linear-wh}\\
 \Htwo_{ww}(a,a')
 &:=\Htwo_{hh}
       (q_{\mathrm{sm},a},q_{\mathrm{sm},a'}),
 \label{eq:linear-ww}
\end{align}
where \(t\) is a smooth compactly supported graviton test tensor.
These expressions are independent of the time cutoff used to
construct the smooth representatives.

Indeed, two such representatives have a smooth compactly
supported difference \(d\) with \(Ed=0\).  Then
\[
 d=\square(\Ret d),
\]
and \(\Ret d=\Adv d\) is smooth and compactly supported.
The bisolution property of \(\Htwo_{hh}\) annihilates this
difference in either argument.

Equations~\eqref{eq:linear-wh}--\eqref{eq:linear-ww} are
statements about smooth-test pairings of free linear observables.
They are not a licence to multiply the corresponding unsmeared
kernels with scalar propagators, or to pull them back to the
coincidences required by a composite insertion.

\subsection{Refusal of an exact smooth off-shell replacement}

\begin{proposition}
\label{prop:no-smooth-offshell}
If \(q_a\) is singular, there is no smooth compactly supported
tensor \(s\) such that
\[
 \langle q_a,h\rangle=\langle s,h\rangle
\]
for every smooth \(h\).
\end{proposition}

\begin{proof}
The equality would in particular hold for all compactly
supported smooth \(h\), and hence would imply \(q_a=s\) as
distributions, contradicting singularity.
\end{proof}

Thus the on-shell repair and the off-shell obstruction are
compatible statements, with different equality requirements.

\section{Consequences for second-order retarded products}
\label{sec:consequences}

Suppose
\[
 V=\kappa V_1+\kappa^2V_2+O(\kappa^3)
\]
and a retarded map is normalized formally by
\[
 R_V(F)=F+R_1(V;F)+\frac12R_2(V,V;F)+\cdots.
\]
Substituting the geometric expansion produces
\begin{equation}
 [\kappa^2]R_V(\Bcal_f)
 =
 \Bcal_f^{(2)}
 +R_1(V_1;\Bcal_f^{(1)})
 +R_1(V_2;\Bcal_f^{(0)})
 +\frac12R_2(V_1,V_1;\Bcal_f^{(0)}).
 \label{eq:four-pieces}
\end{equation}
This is a formal coefficient identity conditional on the existence
of its terms.

The present result stops before the first mixed product:
\[
 R_1(V_1^{\mathrm{sc}};\Bcal_f^{(1)}).
\]
Its proposed geometric argument is not an element of the declared
off-shell domain.  No graph inventory or extension prescription
for this product is therefore claimed here.

\begin{center}
\begin{tabular}{p{0.31\textwidth}p{0.60\textwidth}}
\toprule
Item & Established status\\
\midrule
Free multiplet and vertices
&
Specified, including ghosts and auxiliaries; see
Section~\ref{sec:domain} and Appendix~\ref{app:vertices}.
\\[0.4em]
Geometric bookkeeping
&
Equations~\eqref{eq:B0}--\eqref{eq:B2} and
\eqref{eq:classical-bookkeeping}; no double counting.
\\[0.4em]
Linear Cauchy adjoint
&
Computed exactly; its domain criterion is
Theorem~\ref{thm:linear-criterion}.
\\[0.4em]
First geometric composite
&
Excluded from the declared off-shell domain by
Theorem~\ref{thm:main}.
\\[0.4em]
Mixed and two-vertex products
&
Not constructed; the stated termwise off-shell route is blocked
at its insertion-domain prerequisite.
\\[0.4em]
Repairs
&
Linear smoothing and a free on-shell representative are proved.
No nonlinear interacting repair is asserted.
\\
\bottomrule
\end{tabular}
\end{center}

The obstruction is not a failure of a diagonal extension theorem.
The distribution \(L^*a\) already exists, with compact support and
finite transverse singular order.  It fails the selected
wavefront condition.  Distribution-extension methods such as
those of Brunetti and Fredenhagen \cite{BF}, and local covariant
time-ordered products of local Wick polynomials as constructed
by Hollands and Wald \cite{HW}, do not by themselves place this
nonlocal sharp-Cauchy insertion in the declared domain.

Nor is the result an off-diagonal multiplication obstruction:
no claim is made that a characteristic product of a coordinate
kernel and a quantum two-point function is intrinsically
undefined.  Such a claim would require an additional analysis
of the actual tensor-valued graph distributions.

The precise conclusion is therefore
\begin{equation}
 \boxed{
 \begin{gathered}
 \text{sharp zero-Cauchy dressing}
 +\text{exact off-shell classical normalization}\\
 +\text{the polynomial microcausal domain}
 \quad\Longrightarrow\quad
 \text{failure at }\Bcal_f^{(1)}
 \quad(f\ne0).
 \end{gathered}}
 \label{eq:conclusion}
\end{equation}
A continuation must change at least one of these three
requirements and prove that the new formulation preserves the
intended observable in its stated sense.

No connected two-point response is declared in this paper.
Accordingly,
\(\Bcal_f^{(1)}\star\Bcal_{f'}^{(1)}\) is not constructed.
The boundary result neither computes a quantum response nor
settles a later Ward or renormalization problem.

\appendix

\section{An explicit action and vertex ledger}
\label{app:vertices}

This appendix fixes a perturbative Einstein--scalar model with
linear de Donder gauge fixing.  Its purpose is to identify the
quantum continuation unambiguously.  None of the vertices is
needed for the boundary obstruction, which is already present
in the geometric insertion.

\subsection{Fixed gravitational density representative}

For \(\mathbf g=\eta+\kappa h\), let
\[
 \Gamma^\rho_{\mu\nu}(\mathbf g)
 =
 \frac12\mathbf g^{\rho\lambda}
 \left(
 \partial_\mu\mathbf g_{\lambda\nu}
 +\partial_\nu\mathbf g_{\lambda\mu}
 -\partial_\lambda\mathbf g_{\mu\nu}
 \right).
\]
Choose the first-derivative Einstein density representative
\begin{equation}
 \mathcal L_{\mathrm{grav}}(\kappa)
 =
 -\frac{2}{\kappa^2}
 A^{\mu\nu}(\kappa)
 \left(
  \Gamma^\rho_{\mu\sigma}\Gamma^\sigma_{\nu\rho}
  -\Gamma^\rho_{\mu\nu}\Gamma^\sigma_{\rho\sigma}
 \right).
 \label{eq:grav-density}
\end{equation}
The curvature-sign convention is the one for which this is the
Einstein density modulo a total divergence.  The representative
\eqref{eq:grav-density}, rather than its equivalence class modulo
divergences, is used before switching.

Because \(\Gamma=O(\kappa)\), the density has a regular formal
expansion
\[
 \mathcal L_{\mathrm{grav}}(\kappa)
 =
 \mathcal L_{\mathrm{grav},0}
 +\kappa\mathcal L_{\mathrm{grav},1}
 +\kappa^2\mathcal L_{\mathrm{grav},2}
 +O(\kappa^3).
\]
Its quadratic part, modulo an unswitched divergence, is
\begin{align}
 \mathcal L_{\mathrm{grav},0}
 ={}&
 \frac12\partial_\lambda h_{\mu\nu}
             \partial^\lambda h^{\mu\nu}
 -(\partial_\mu h^{\mu\nu})(\partial^\rho h_{\rho\nu})
 \nonumber\\
 &+(\partial_\mu h^{\mu\nu})\partial_\nu h
 -\frac12\partial_\lambda h\,\partial^\lambda h.
 \label{eq:FP}
\end{align}

\subsection{Gauge fixing, ghosts, and auxiliary field}

For an odd vector field \(c^\mu\), define
\begin{align}
 (\mathsf D_\kappa(h)c)_{\mu\nu}
 ={}&
 \partial_\mu c_\nu+\partial_\nu c_\mu
 \nonumber\\
 &+\kappa\left(
 c^\rho\partial_\rho h_{\mu\nu}
 +h_{\rho\nu}\partial_\mu c^\rho
 +h_{\mu\rho}\partial_\nu c^\rho
 \right).
 \label{eq:gauge-generator}
\end{align}
The notation \(\mathsf D_\kappa(h)\) here denotes the
infinitesimal metric gauge generator, not an Euler operator.

With an unshifted auxiliary field \(B_\mu\), choose
\begin{equation}
 \mathcal L_{\mathrm{gf+gh}}
 =
 B_\mu C^\mu[h]-\frac14B_\mu B^\mu
 -\bar c_\mu C^\mu[\mathsf D_\kappa(h)c].
 \label{eq:gf-ghost}
\end{equation}
The identity
\[
 C^\mu[\partial c+\partial c]=\square c^\mu
\]
gives
\begin{equation}
 \mathcal L_{\mathrm{gf+gh}}
 =
 C_\mu[h]C^\mu[h]-\frac14\beta_\mu\beta^\mu
 -\bar c_\mu\square c^\mu
 -\kappa\bar c_\mu C^\mu[\Lie_c h],
 \qquad
 \beta_\mu=B_\mu-2C_\mu[h].
 \label{eq:shifted-gf}
\end{equation}
Adding \(C_\mu C^\mu\) to \eqref{eq:FP} gives the graviton
quadratic density in \eqref{eq:free-action}, modulo an
unswitched divergence.  The shifted auxiliary field has no
interaction vertices in this parametrization.

The chosen action is sufficient to define the gauge-fixed
perturbative field content.  No switched BRST or Ward identity
is inferred from these formulas.

\subsection{Scalar density and switching}

Let
\begin{equation}
 \mathcal L_{\mathrm{sc}}(\mathbf g,\phi)
 =
 \frac12 A^{\mu\nu}(\kappa)
               \partial_\mu\phi\,\partial_\nu\phi
 -\frac12m^2d(\kappa)\phi^2.
 \label{eq:scalar-density}
\end{equation}
Choose real compactly supported switches
\(g,g_{\mathrm{grav}},g_{\mathrm{gh}}\).
The scalar interaction is
\begin{equation}
 \int g\,
 \left(
 \mathcal L_{\mathrm{sc}}(\eta+\kappa h,\phi)
 -\mathcal L_{\mathrm{sc}}(\eta,\phi)
 \right)\dd^4x.
 \label{eq:scalar-switch}
\end{equation}
Thus one switch multiplies the complete curved-minus-flat
density.  The quadratic metric vertex is multiplied by \(g\),
not by \(g^2\).

The full switched interaction is
\begin{align}
 V(\kappa)
 ={}&
 \int g\,
 \left(
 \mathcal L_{\mathrm{sc}}(\eta+\kappa h,\phi)
 -\mathcal L_{\mathrm{sc}}(\eta,\phi)
 \right)\dd^4x
 \nonumber\\
 &+\int g_{\mathrm{grav}}\,
 \left(
 \mathcal L_{\mathrm{grav}}(\kappa)
 -\mathcal L_{\mathrm{grav},0}
 \right)\dd^4x
 \nonumber\\
 &-\kappa\int g_{\mathrm{gh}}\,
       \bar c_\mu C^\mu[\Lie_c h]\,\dd^4x.
 \label{eq:full-interaction}
\end{align}
All representatives in this formula are fixed before multiplying
by their switches.

In particular,
\begin{align}
 V_1^{\mathrm{sc}}
 &=
 \frac12\int g\left(
 A_1^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi
 -m^2d_1\phi^2
 \right)\dd^4x,
 \label{eq:V1sc}\\
 V_2^{\mathrm{sc}}
 &=
 \frac12\int g\left(
 A_2^{\mu\nu}\partial_\mu\phi\,\partial_\nu\phi
 -m^2d_2\phi^2
 \right)\dd^4x,
 \label{eq:V2sc}\\
 V_1^{\mathrm{gh}}
 &=
 -\int g_{\mathrm{gh}}\,
       \bar c_\mu C^\mu[\Lie_c h]\,\dd^4x,
 &
 V_2^{\mathrm{gh}}&=0,
 \label{eq:Vghost}\\
 V_1^{\mathrm{aux}}&=0,
 &
 V_2^{\mathrm{aux}}&=0.
 \label{eq:Vaux}
\end{align}

\subsection{Finite formulas for the gravitational vertices}

The following formulas specify the gravitational vertices without
suppressing integrations by parts.

Define
\[
 T_{\lambda\mu\nu}(h)
 =
 \partial_\mu h_{\lambda\nu}
 +\partial_\nu h_{\lambda\mu}
 -\partial_\lambda h_{\mu\nu}.
\]
Write
\[
 \Gamma^\rho_{\mu\nu}
 =
 \kappa\gamma_1^\rho{}_{\mu\nu}
 +\kappa^2\gamma_2^\rho{}_{\mu\nu}
 +\kappa^3\gamma_3^\rho{}_{\mu\nu}
 +O(\kappa^4),
\]
where
\begin{equation}
 \gamma_r^\rho{}_{\mu\nu}
 =
 \frac12(-1)^{r-1}
 (h^{r-1})^\rho{}_\sigma\eta^{\sigma\lambda}
 T_{\lambda\mu\nu}(h),
 \qquad r=1,2,3,
 \label{eq:gamma-coeff}
\end{equation}
and \(h^0=\id\).

For two Christoffel-type arrays \(P,Q\), set
\[
 \mathcal Q_{\mu\nu}(P,Q)
 =
 P^\rho{}_{\mu\sigma}Q^\sigma{}_{\nu\rho}
 -P^\rho{}_{\mu\nu}Q^\sigma{}_{\rho\sigma}.
\]
Then
\begin{align}
 \mathcal L_{\mathrm{grav},1}
 =-2\Big[
 &A_1^{\mu\nu}\mathcal Q_{\mu\nu}(\gamma_1,\gamma_1)
 \nonumber\\
 &+\eta^{\mu\nu}
 \bigl(
 \mathcal Q_{\mu\nu}(\gamma_1,\gamma_2)
 +\mathcal Q_{\mu\nu}(\gamma_2,\gamma_1)
 \bigr)
 \Big],
 \label{eq:grav1}\\
 \mathcal L_{\mathrm{grav},2}
 =-2\Big[
 &A_2^{\mu\nu}\mathcal Q_{\mu\nu}(\gamma_1,\gamma_1)
 \nonumber\\
 &+A_1^{\mu\nu}
 \bigl(
 \mathcal Q_{\mu\nu}(\gamma_1,\gamma_2)
 +\mathcal Q_{\mu\nu}(\gamma_2,\gamma_1)
 \bigr)
 \nonumber\\
 &+\eta^{\mu\nu}
 \bigl(
 \mathcal Q_{\mu\nu}(\gamma_1,\gamma_3)
 +\mathcal Q_{\mu\nu}(\gamma_2,\gamma_2)
 +\mathcal Q_{\mu\nu}(\gamma_3,\gamma_1)
 \bigr)
 \Big].
 \label{eq:grav2}
\end{align}
Finally,
\[
 V_j^{\mathrm{grav}}
 =
 \int g_{\mathrm{grav}}\,
       \mathcal L_{\mathrm{grav},j}\,\dd^4x,
 \qquad j=1,2.
\]
These expressions are finite local polynomials with smooth compact
switches.  Their local coefficient kernels satisfy the
microcausal cone condition: the covectors in a diagonal
conormal sum to zero, and hence cannot all lie in the same
nonspacelike cone unless they all vanish.  Derivatives do not
enlarge that wavefront bound.

\section{Kernel support and the location of the obstruction}
\label{app:kernel}

For \(x\in\{t>0\}\), let
\(\Lambda^{\mu,\alpha\beta}(x,z)\) be the distribution kernel
of \(L\):
\[
 (Lh)^\mu(x)
 =
 \int
 \Lambda^{\mu,\alpha\beta}(x,z)h_{\alpha\beta}(z)\,\dd^4z.
\]
Equivalently, it is characterized by
\[
 \int a_\mu(x)\Lambda^{\mu,\alpha\beta}(x,z)\,\dd^4x
 =
 \bigl[C^*(\theta\Adv a)\bigr]^{\alpha\beta}(z).
\]
The first geometric insertion has mixed kernel
\begin{equation}
 K_f^{\alpha\beta}(z,x)
 =
 \partial_{x^\mu}
 \left(
 f(x)\Lambda^{\mu,\alpha\beta}(x,z)
 \right),
 \label{eq:mixed-kernel}
\end{equation}
so that
\[
 \Bcal_f^{(1)}[h,\phi]
 =
 \langle K_f,h\otimes\phi\rangle.
\]
Its support satisfies
\begin{equation}
 \supp K_f
 \subset
 \left\{
 (z,x):
 x\in\supp f,\quad
 z\in J^-(x),\quad z^0\geq0
 \right\}.
 \label{eq:mixed-support}
\end{equation}
This is a compact subset of \(M\times M\).

The boundary part is located at \(z^0=0\), whereas
\(x^0>0\) on \(\supp f\).  It is therefore away from the
ordinary diagonal \(z=x\).  Corollary~\ref{cor:quadratic}
proves the failure of its required wavefront cone without
needing to assert a complete wavefront equality for
\eqref{eq:mixed-kernel}.

For the contracted adjoint kernel
\[
 q_a=\theta C^*b+\delta(t)\mathsf B(b|_{\Sigma_0}),
\]
the transverse scaling degree of the delta term is one wherever
its coefficient is nonzero.  The step term has transverse
scaling degree at most zero.  These are finite-order
distributions.  Thus the obstruction is not excessive
transverse scaling and is not the nonexistence of a
distributional extension: it is incompatibility with the
off-shell cone condition.

\section{Logical boundary of the result}
\label{app:logic}

The proof separates four assertions that should not be identified:
\begin{enumerate}[label=(\arabic*),leftmargin=2.2em]
 \item The sharp-Cauchy insertion is a well-defined smooth
       polynomial functional on smooth classical fields.
 \item Its fixed kernels are compactly supported distributions.
 \item Those kernels belong to a prescribed quantum insertion
       domain.
 \item Retarded products with those kernels exist in a common
       normalization prescription.
\end{enumerate}
The first two assertions hold for
\(\Bcal_f^{(1)}\).  The third fails in the domain of
Definition~\ref{def:poly}.  The fourth is therefore not reached
by the stated termwise construction.

Theorem~\ref{thm:onshell} establishes that the analogous failure
for a \emph{linear} coordinate observable can be bypassed after
restricting to the free equations of motion.  It does not
supply the nonlinear extension of that bypass.  In particular,
replacing a distributional metric smearing by an on-shell
equivalent smooth smearing inside an interacting composite
requires an additional proof controlling field dependence,
support, and equation-of-motion contact terms.

The resulting manuscript is a closed boundary-domain result.
It leaves the existence of a suitably repaired second-order
relational insertion open.

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\end{document}