% XX-D.tex
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%   pdflatex -interaction=nonstopmode -halt-on-error XX-D.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XX-D.tex
%
% No compilation, machine verification, or independent execution
% is asserted.
%
% RELEASE QUALIFICATION:
% The supplied XX-C does not select a covariance representative with
% verified global infrared estimates. The native classical insertion
% is constructed below, as are compact-source-cutoff Wick symbols and
% a precise conditional native one-contraction construction. The missing
% infrared and noncompact product hypotheses are not silently assumed
% verified. Consequently the requested native quantum Q3 gate is not
% announced closed.
%
% Main debugging corrections:
% 1. The lapse is calibrated to the matter scalar, not to its exponential
%    clock relabeling.
% 2. Second jets are retained; they are not assigned new free covariances.
% 3. Y_2 is absent for the exact reason partial_i N_0 = 0.
% 4. The Newton kernel is spatial, with the original divergence source.
% 5. An equal-time off-diagonal conormal does not by itself exclude
%    a compact quadratic kernel from the microcausal domain.
% 6. Microcausal admission of a compact quadratic kernel does not license
%    its unrenormalized internal contraction with W.
% 7. Compact output smearing does not make the native Hessian compact.
% 8. A Hadamard germ does not control the smooth infrared remainder.
% 9. Fixed classical pullback is not quantum field-dependent substitution.
% 10. A formal classical symbol is not an expectation-value limit.
% 11. One-contraction order is not a complete interacting one-loop order.
% 12. Native two-rod commutators have an inverse-distance component.

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\hypersetup{
  pdftitle={XX-D: The Original-Rod Relational Lapse:
  Composite Formula and Product Domains},
  pdfauthor={},
  pdfsubject={Gate Q3; nonlocal composite insertion;
  Hadamard subtraction; Newton composition}
}

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\title{\textbf{XX-D}\\[4pt]
The Original-Rod Relational Lapse:\\
Composite Formula and Product Domains\\[4pt]
\large A Native Classical Construction, Compact Wick Realizations,\\
and a Conditional Noncompact One-Contraction Extension}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We derive the complete kinematic second jet of the
original-rod relational lapse on the power-law
Einstein--scalar background reproduced in XX-C.
The calculation retains every metric component,
both second-jet fields, the clock embedding,
the induced spatial metric, and the original
slice-elliptic rod equation. The second spatial
rod does not enter the quadratic lapse.

The unsmoothed classical coefficient is a continuous
polynomial on a spatial Schwartz configuration domain.
Its Hessian has noncompact dipole tails. After a
compact source cutoff, that Hessian is microcausal:
its equal-time off-diagonal covectors occur with
opposite signs. Nevertheless its raw internal
Hadamard contraction fails the sufficient product
criterion at coincidence. We give both a
Hadamard-subtracted one-contraction prescription
and a transverse Taylor extension of the
corresponding punctured numerical kernel.

A conditional theorem constructs the native,
cutoff-free Wick symbol through quadratic
perturbation order and first order in Planck's
constant, together with products against compact
regular probes. The theorem states separately
the infrared hypotheses for internal and external
contractions. Those hypotheses have not been
verified for a selected XX-B/XX-C representative.
Products requiring two noncompact rod legs need
additional control and cannot be assigned a
universal inverse-square separation bound.

The formal classical symbol recovers the printed
classical coefficients exactly. This is not an
expectation-value limit. The requested native
quantum Q3 gate remains open wherever its
noncompact contraction hypotheses remain
unverified; no universal extension no-go is claimed.
\end{abstract}

\tableofcontents

%=====================================================================
\section{Scope, fixed target, and the meaning of completion}
\label{sec:scope}
%=====================================================================

The requested deliverable concerns the native
unsmoothed insertion, not merely a numerical
finite part, a compact approximation, or a
classical functional. These objects are separated
throughout this manuscript.

The supplied registered records distinguish the
original-rod lapse from other relational prescriptions
and explicitly leave its quantum product audit
incomplete \cite{CheungEm02E,CheungXIXA,CheungXIXD}.
Their statements are documentary scope controls;
the calculations below use the equations reproduced
here.

XX-C, supplied in the present exchange \cite{XXC},
does not select a covariance with verified global
infrared estimates. Therefore a complete
unconditional quantum Q3 certificate cannot be
deduced from its Hadamard class alone.

\begin{definition}[Three different outputs]
\label{def:outputs}
The outputs considered here are:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a native classical polynomial, with the exact
      original Newton inverse;
\item compact-source-cutoff Wick symbols in the
      free algebra, without spatial ultraviolet
      smoothing;
\item a native quantum Wick symbol and its stated
      products, conditional on explicit noncompact
      contraction estimates.
\end{enumerate}
Only an unconditional realization of the third
output for an actually specified covariance and
product domain would close the quantum Q3 gate.
\end{definition}

\paragraph{Perturbative orders.}
All formulas are taken modulo \(s^3\) and
\(\hbar^2\). The parameter \(s\) grades the
kinematic Taylor jet; it is not silently identified
with a gravitational coupling. First order in
\(\hbar\) means one free contraction here, not a
complete interacting one-loop contribution inventory.

\paragraph{Second jets.}
The second-jet fields \(k,\psi\) are retained as
independent smooth prescribed jets. They are
central parameters in the free Wick construction.
If they are instead defined as interacting
operator-valued responses, their own construction
and cross-contractions must be added. No such
identification is made.

\paragraph{Analytic references.}
The compact Wick-kernel framework and local
subtraction templates are those of
Brunetti--Fredenhagen and Hollands--Wald
\cite{BF,HWlocal,HWexistence}.
The noncompact Newton estimates and native
coefficient calculations are proved below.
Those references do not automatically admit
a nonlocal lapse.

%=====================================================================
\section{The frozen Einstein--scalar specification}
\label{sec:freeze}
%=====================================================================

\subsection{Background, action, and clock calibration}

Use signature \((-,+,+,+)\) and the action
\begin{equation}
 S[g,\phi]
 =\int\sqrt{-g}
   \left(\frac{M^2}{2}R[g]
          -\frac12g^{\mu\nu}\partial_\mu\phi
                                      \partial_\nu\phi
          -V(\phi)\right)\dd^4x.
 \label{eq:action}
\end{equation}
Fix \(M>0\), \(p>1\), \(t_*>0\), and
\begin{align}
 A&=\sqrt{2p}\,M,
 &
 a(t)&=(t/t_*)^p,
 &
 H(t)&=p/t,
 \notag\\
 \barphi(t)&=\phi_*+A\log(t/t_*),
 &
 v(t)&=\dot{\barphi}(t)=A/t,
 \notag\\
 \barg&=-\dd t^2+a(t)^2\delta_{ij}\dd x^i\dd x^j,
 &
 V(\phi)&=\frac{M^2p(3p-1)}{t_*^2}
                         e^{-2(\phi-\phi_*)/A}.
 \label{eq:background}
\end{align}
The spacetime is
\[
 \mathcal M=I\times\R^3,\qquad
 I=(t_-,t_+),\qquad 0<t_-<t_+<\infty.
\]
The background solves the Einstein--scalar
Euler equations: substitution uses
\(A^2=2pM^2\), \(H=p/t\), and
\(V(\barphi)=M^2p(3p-1)/t^2\).

Retain the factorial jet
\begin{equation}
 g_s=\barg+s h+\frac{s^2}{2}k,\qquad
 \phi_s=\barphi+s\varphi+\frac{s^2}{2}\psi.
 \label{eq:jet}
\end{equation}
The coordinate clock is the monotone relabeling
\begin{equation}
 T_s=t_*e^{(\phi_s-\phi_*)/A}.
 \label{eq:T}
\end{equation}
The lapse itself remains
\begin{equation}
 N_s=\left(-g_s^{-1}(\dd\phi_s,\dd\phi_s)\right)^{-1/2}.
 \label{eq:rawN}
\end{equation}
It is not replaced by
\(\left(-g_s^{-1}(\dd T_s,\dd T_s)\right)^{-1/2}\).
In particular,
\[
 N_0=t/A.
\]

All output tests use the fixed background measure
\[
 \dd V_{\barg}=a(t)^3\dd t\,\dd^3x.
\]
No field-dependent volume factor is inserted.

\subsection{Native free normalization}

Write \(z=(h,\varphi)\) and
\(\zeta=(k,\psi)\).
The field bundle is
\[
 \mathcal V=S^2T^*\mathcal M\oplus\underline{\R}.
\]
The integrated pairing is
\[
 (z,w)_0=\int
       (h_{\mu\nu}j^{\mu\nu}+\varphi\omega)\dd V_{\barg}.
\]
Define the action Hessian by
\[
 \calH z=
     \left(-\frac12\delta E_{\mu\nu}[z],
                      \delta E_\phi[z]\right),
\]
where
\[
 E_{\mu\nu}=M^2G_{\mu\nu}-T_{\mu\nu},
 \qquad E_\phi=\Box_g\phi-V'(\phi).
\]

The gauge generator, gauge condition, and native
gauge-fixed operator are
\begin{align}
 K\xi&=(2\nabla_{(\mu}\xi_{\nu)},
                   \partial_\mu\barphi\,\xi^\mu),
 \notag\\
 \calF_\nu z
 &=\nabla^\mu h_{\mu\nu}
      -\frac12\nabla_\nu\tr_{\barg}h
      -M^{-2}\partial_\nu\barphi\,\varphi,
 \notag\\
 r&=M^2/2,
 &
 \calB&=\calH-r\calF^\dagger\calF,
 &
 Q&=\calF K.
 \label{eq:operator}
\end{align}
With four-dimensional trace reversal
\[
 \mathsf T h=h-\frac12\barg\,\tr_{\barg}h,
\]
retain
\begin{equation}
 \calA=\operatorname{diag}
        \left(\frac{M^2}{4}\mathsf T,\id\right),
 \qquad \calL=\calA^{-1}\calB.
 \label{eq:A}
\end{equation}
The principal differential part of \(\calL\) is
\(\Box_{\barg}\id\). Thus its Cauchy theory and
causal inverses are those of a normally hyperbolic
bundle operator \cite{BGP}.

The contraction representative, when specified, obeys
\begin{equation}
 \calB W=W\calB=0,\qquad
 W^*=W,\qquad W-W^{\mathrm T}=iE_{\calB},
 \qquad \WF'(W)\subset\mathscr N_+.
 \label{eq:W}
\end{equation}
Here
\[
 E_{\calB}
 =G_{\calL}^{\ret}\calA^{-1}
                   -G_{\calL}^{\adv}\calA^{-1}.
\]
The factor of \(\hbar\) is not included in \(W\);
a free contraction is \(\hbar W\).

Write
\[
 \mu=\frac12(W+W^{\mathrm T}).
\]
It is real and symmetric in a real field
trivialization. No positivity is assumed.

The complete auxiliary and ghost multiplet is retained.
The kinematic lapse has zero functional derivatives
with respect to those components. Hence they contribute
no direct free internal contraction to the insertion.
This observation does not eliminate ghost or auxiliary
contributions to an interacting calculation.

%=====================================================================
\section{Complete classical coefficient through second order}
\label{sec:classical}
%=====================================================================

\subsection{Inverse metric and raw lapse}

Set
\[
 g_s^{-1}=R_0+sR_1+\frac{s^2}{2}R_2.
\]
Matrix inversion gives
\begin{align}
 R_0&=\barg^{-1},
 &
 R_1&=-\barg^{-1}h\barg^{-1},
 \notag\\
 R_2&=2\barg^{-1}h\barg^{-1}h\barg^{-1}
                         -\barg^{-1}k\barg^{-1}.
 \label{eq:inverse}
\end{align}
For
\[
 -g_s^{-1}(\dd\phi_s,\dd\phi_s)
        =v^2+sS_1+\frac{s^2}{2}S_2,
\]
the exact coefficients are
\begin{align}
 S_1
 &=-R_1^{\mu\nu}\partial_\mu\barphi\,
                               \partial_\nu\barphi
       -2R_0^{\mu\nu}\partial_\mu\barphi\,
                               \partial_\nu\varphi,
 \notag\\
 S_2
 &=-R_2^{\mu\nu}\partial_\mu\barphi\,
                               \partial_\nu\barphi
       -4R_1^{\mu\nu}\partial_\mu\barphi\,
                               \partial_\nu\varphi
 \notag\\
 &\quad
       -2R_0^{\mu\nu}\partial_\mu\varphi\,
                               \partial_\nu\varphi
       -2R_0^{\mu\nu}\partial_\mu\barphi\,
                               \partial_\nu\psi.
 \label{eq:S}
\end{align}
Consequently
\begin{equation}
 N_1=-\frac{S_1}{2v^3},\qquad
 N_2=\frac{3S_1^2}{4v^5}-\frac{S_2}{2v^3}.
 \label{eq:N12}
\end{equation}

On the displayed background,
\begin{align}
 S_1&=v^2h_{00}+2v\dot\varphi,
 \notag\\
 S_2
 &=2v^2h_{00}^2-2v^2a^{-2}h_{0i}h_{0i}
          +v^2k_{00}
          +4vh_{00}\dot\varphi
          -4va^{-2}h_{0i}\partial_i\varphi
 \notag\\
 &\quad
          +2\dot\varphi^2
          -2a^{-2}\partial_i\varphi\partial_i\varphi
          +2v\dot\psi.
 \label{eq:Sexplicit}
\end{align}
Thus no shift component has been dropped.

\subsection{Clock embedding and induced metric}

Let the embedding of the \(T_s=t\) slice be
\[
 \iota_s(t,\mathbf x)
   =\left(t+s u_1+\frac{s^2}{2}u_2,\mathbf x\right).
\]
The clock equation is equivalently
\(\phi_s\circ\iota_s=\barphi(t)\).
It gives
\begin{align}
 u_1&=-\frac{\varphi}{v}=-\frac tA\varphi,
 \notag\\
 u_2
 &=-\frac{\psi+2u_1\dot\varphi+\dot v\,u_1^2}{v}
 \notag\\
 &=-\frac tA\psi
       +\frac{2t^2}{A^2}\varphi\dot\varphi
       +\frac t{A^2}\varphi^2.
 \label{eq:u}
\end{align}
Expanding the exponential clock directly gives
\[
 T_s=t+s\frac tA\varphi
       +\frac{s^2}{2}
          \left(\frac tA\psi+\frac t{A^2}\varphi^2\right),
\]
and its inverse yields the same \eqref{eq:u}.
This verifies the retained calibration.

Write
\[
 \gamma_s=\iota_s^*g_s
     =a^2I+s\gamma_1+\frac{s^2}{2}\gamma_2.
\]
The complete induced coefficients are
\begin{align}
 (\gamma_1)_{ij}
 &=h_{ij}+2a^2Hu_1\delta_{ij},
 \label{eq:gamma1}\\
 (\gamma_2)_{ij}
 &=k_{ij}+2u_1\dot h_{ij}
       +2a^2\left(Hu_2+(\dot H+2H^2)u_1^2\right)\delta_{ij}
 \notag\\
 &\quad
       +2h_{0i}\partial_j u_1
       +2h_{0j}\partial_i u_1
       -2\partial_i u_1\partial_j u_1.
 \label{eq:gamma2}
\end{align}
The last line is the embedding contribution. Its
last term uses \(\bar g_{00}=-1\).

\subsection{Original spatial rods}

Let
\[
 C=a^{-2}\gamma_1,\qquad D=a^{-2}\gamma_2.
\]
Expanding the original elliptic density gives
\[
 \sqrt{\det\gamma_s}\,\gamma_s^{-1}
     =a\left(I+sB_1+\frac{s^2}{2}B_2\right),
\]
where
\begin{align}
 B_1&=\frac12(\tr C)I-C,
 \notag\\
 B_2
 &=\left(\frac12\tr D+\frac14(\tr C)^2
                         -\frac12\tr(C^2)\right)I
        -(\tr C)C+2C^2-D.
 \label{eq:B12}
\end{align}
There is no lapse multiplier in this density.

The rods satisfy
\begin{equation}
 \partial_i\left(
       \sqrt{\det\gamma_s}\,
           \gamma_s^{ij}\partial_j R_s^a\right)=0,
 \qquad R_s^a-x^a\longrightarrow0
         \quad(\abs{\mathbf x}\to\infty).
 \label{eq:rod}
\end{equation}
For
\[
 R_s^a=x^a+sY_1^a+\frac{s^2}{2}Y_2^a,
\]
retain the spatial inverse
\begin{equation}
 (\Gsp b)(\mathbf x)
   =-\frac1{4\pi}\int_{\R^3}
          \frac{b(\mathbf y)}{\abs{\mathbf x-\mathbf y}}
                                        \dd^3y,
 \qquad \Delta\Gsp b=b.
 \label{eq:Newton}
\end{equation}
It follows that
\begin{align}
 Y_1^a&=-\Gsp\partial_iB_1^{ia},
 \notag\\
 Y_2^a&=-\Gsp\partial_i
               \left(B_2^{ia}+2B_1^{ij}\partial_jY_1^a\right).
 \label{eq:Y}
\end{align}

Define the local rod-source map
\begin{equation}
 (\calCz)^{ia}
 =\frac1{2a^2}\delta^{ia}\delta^{jk}h_{jk}
       -a^{-2}h_{ia}-\frac pA\delta^{ia}\varphi.
 \label{eq:C}
\end{equation}
Then \(B_1=\calCz\), and
\[
 Y_1=\calRz,\qquad
 (\calRz)^a=-\Gsp\partial_i(\calCz)^{ia}.
\]
After integration by parts,
\begin{align}
 (\calRz)^a(t,\mathbf x)
 &=\int_{\R^3}
        P_i(\mathbf x-\mathbf y)
                     (\calCz)^{ia}(t,\mathbf y)\dd^3y,
 \notag\\
 P_i(\mathbf r)&=-\frac{r_i}{4\pi\abs{\mathbf r}^3}.
 \label{eq:P}
\end{align}
The kernel \(P_i\) is locally integrable and has
spatial scaling degree two.

\subsection{Relational composition and absence of \(Y_2\)}

The spacetime rod label is obtained by transporting
\(R_s\) from its clock slice. Let \(\alpha_s\) be
the inverse clock-and-rod map and write
\[
 \alpha_s=x+s\alpha_1+\frac{s^2}{2}\alpha_2.
\]
At first order,
\[
 \alpha_1^0=u_1,\qquad
 \alpha_1^i=-Y_1^i.
\]
At second order the clock equation gives
\[
 \alpha_2^0=u_2-2Y_1^i\partial_i u_1.
\]

\begin{theorem}[Complete relational second jet]
\label{thm:jet}
The relational lapse
\[
 \mathcal N_s=N_s\circ\alpha_s
      =N_0+s n_1+\frac{s^2}{2}n_2
\]
has coefficients
\begin{align}
 n_1&=N_1+\dot N_0u_1,
 \notag\\
 n_2&=N_2+2u_1\dot N_1+\dot N_0u_2
       +\ddot N_0u_1^2
       -2Y_1^i\partial_i n_1.
 \label{eq:composition}
\end{align}
In particular \(Y_2\) does not occur.
\end{theorem}

\begin{proof}
Taylor expansion gives
\[
 n_2=N_2+2\alpha_1^\mu\partial_\mu N_1
       +\alpha_2^\mu\partial_\mu N_0
       +\alpha_1^\mu\alpha_1^\nu
                        \partial_\mu\partial_\nu N_0.
\]
Because \(N_0\) is spatially homogeneous,
\(\alpha_2^i\partial_iN_0=0\).
Substitute the first spatial displacement and the
displayed temporal second displacement. The spatial
terms combine to
\(-2Y_1^i\partial_i(N_1+\dot N_0u_1)\).
This proves \eqref{eq:composition} and the stated
absence. No quantum substitution was used.
\end{proof}

This derivation reproduces the composition structure
in the supplied XIX-A passage
\cite[locators \sid{prop:lapse-composition},
\sid{eq:n1}, \sid{eq:n2}]{CheungXIXA}.

\subsection{Fully expanded native coefficient}

Define the local linear map
\begin{equation}
 \ell z=-\frac t{2A}h_{00}
         -\frac{t^2}{A^2}\dot\varphi
         -\frac t{A^2}\varphi.
 \label{eq:ell}
\end{equation}
Then the complete result is
\begin{equation}
 n_1=\ell z,\qquad
 n_2=\ell\zeta+q_{\mathrm{loc}}(z)
                     -2(\calRz)^i\partial_i(\ell z),
 \label{eq:ncomplete}
\end{equation}
where
\begin{align}
 q_{\mathrm{loc}}(z)
 &=-\frac t{4A}h_{00}^2
       +\frac t{Aa^2}h_{0i}h_{0i}
       +\frac{t^2}{A^2}h_{00}\dot\varphi
       +\frac{2t^3}{A^3}\dot\varphi^2
 \notag\\
 &\quad
       +\frac{2t^2}{A^2a^2}h_{0i}\partial_i\varphi
       +\frac{t^3}{A^3a^2}
                        \partial_i\varphi\partial_i\varphi
 \notag\\
 &\quad
       +\frac{t^2}{A^2}\varphi\dot h_{00}
       +\frac t{A^2}\varphi h_{00}
       +\frac{2t^3}{A^3}\varphi\ddot\varphi
 \notag\\
 &\quad
       +\frac{6t^2}{A^3}\varphi\dot\varphi
       +\frac t{A^3}\varphi^2.
 \label{eq:qloc}
\end{align}
The spatial metric enters through the rod term,
not through an artificially restricted scalar
metric subclass.

%=====================================================================
\section{Native classical domain and distributional operation rules}
\label{sec:domains}
%=====================================================================

\subsection{A concrete unsmoothed classical domain}

Let
\begin{equation}
 \mathscr X=C^\infty\bigl(I;\Sch(\R^3;\mathcal V)\bigr)
 \label{eq:X}
\end{equation}
with seminorms
\[
 p_{J,N,m,\beta}(z)
 =\sup_{t\in J,\mathbf x}
      (1+\abs{\mathbf x})^N
           \abs{\partial_t^m\partial_{\mathbf x}^\beta z},
 \qquad J\Subset I.
\]
Use the same domain for the prescribed second jet
\(\zeta\). Spatially compact smooth configurations
are included.

\begin{lemma}[Native Newton domain and tails]
\label{lem:Newton}
For \(z\in\mathscr X\), \(\calRz\) is smooth and
\[
 \partial_t^m\partial_{\mathbf x}^\beta\calRz
       =O\bigl((1+\abs{\mathbf x})^{-2-\abs{\beta}}\bigr)
\]
uniformly on \(J\Subset I\).
The map is continuous in the stated seminorms.
It is the unique decaying solution of its rod equation.
The same conclusions hold for uniformly compact
smooth sources.
\end{lemma}

\begin{proof}
The source \(\calCz\) is spatial Schwartz, with
smooth bounded time coefficients on \(J\).
For compact sources the derivative bounds follow
directly from \eqref{eq:P} outside twice the source
radius.

For Schwartz sources split the integral into
\(\abs{\mathbf y}\le\abs{\mathbf x}/2\) and its
complement. In the first region the differentiated
kernel has the claimed decay and the source has
uniform finite moments. In the second region
the source decreases to arbitrary order.
Near \(\mathbf y=\mathbf x\), use local
integrability of \(P\), transferring derivatives
onto the smooth source. Away from that region
differentiate the kernel directly.
These estimates use finitely many Schwartz
seminorms for each desired bound, proving
continuity.

The equation follows from \(\Delta\Gsp=\id\).
A difference of two smooth decaying solutions is
an entire harmonic function tending to zero;
the maximum principle makes it zero.
\end{proof}

The products in \eqref{eq:gamma2} and
\eqref{eq:B12} are Schwartz. The product
\(B_1\partial Y_1\) is also Schwartz because
\(B_1\) is Schwartz and \(Y_1\) has bounded
derivatives. Thus \(Y_2\) is defined on this
domain as well.

For \(f\in\Dc(\mathcal M)\), set
\begin{equation}
 F_f[z,\zeta;s]
 =\int f\left[
       \frac tA+s\ell z
       +\frac{s^2}{2}
          \left(\ell\zeta+q_{\mathrm{loc}}(z)
                        -2\calRz\cdot\nabla\ell z\right)
                      \right]\dd V_{\barg}.
 \label{eq:Fclass}
\end{equation}

\begin{theorem}[Native classical insertion]
\label{thm:classicaldomain}
Equation \eqref{eq:Fclass} is a continuous
polynomial on \(\mathscr X\times\mathscr X\).
As \(f\) varies it defines a distribution-valued
polynomial section, without spatial smoothing
or a replacement of the original Newton inverse.
\end{theorem}

\begin{proof}
Every local term uses finitely many derivatives
and products of Schwartz functions.
Lemma~\ref{lem:Newton} controls the sole nonlocal
factor. On the compact support of \(f\), all
these operations are continuous in finitely
many seminorms. The dependence on \(f\) is
continuous in the ordinary test topology.
\end{proof}

\subsection{Rules for generalized coefficients}

For distributions, the following are sufficient
operation rules.

\begin{enumerate}[label=\textup{D\arabic*.}]
\item \textbf{Multiplication.}
      The product \(uv\) is defined by the usual
      wavefront criterion if no
      \((x,\xi)\in\WF(u)\) and
      \((x,-\xi)\in\WF(v)\) occur simultaneously.

\item \textbf{Fixed pullback.}
      For a smooth map \(j\), \(j^*u\) is defined
      if
      \[
       \WF(u)\cap N_j=\varnothing,\qquad
       N_j=\{(j(x),\eta):(\dd j_x)^{\mathrm T}\eta=0\}.
      \]
      A fixed diffeomorphism has \(N_j\) empty
      away from the zero section.

\item \textbf{Kernel composition.}
      Products of kernels on the intermediate
      space must satisfy D1, and the subsequent
      pushforward must have proper support or
      an independently proved convergent
      noncompact integration.

\item \textbf{Newton inversion.}
      Local elliptic distributional inversion
      does not by itself impose the global
      decaying boundary condition. We use
      Lemma~\ref{lem:Newton}, or a separately
      stated weighted domain, for that condition.

\item \textbf{Extension.}
      Failure of D1 or D2 does not prove that
      every extension is impossible. A punctured
      distribution with finite transverse scaling
      degree can instead be extended, subject to
      its normalization and symmetry conditions.
\end{enumerate}

These are the microlocal operation principles used
in the analytic frameworks cited above
\cite{BF,HWlocal}; their hypotheses are checked
for the particular kernels below.

\subsection{Classical pullback versus quantum substitution}

The coefficients in Section~\ref{sec:classical}
come from a classical Taylor expansion of
\(N_s\circ\alpha_s\).
If a classical configuration is small enough that
the clock-and-rod map is invertible, its fixed
inverse \(\alpha_s\) is an ordinary diffeomorphism.
Distributional pullback by that fixed map is
allowed by D2.

For a fixed spacelike slice embedding, the
conormal covectors are timelike. A free
normally hyperbolic solution with null wavefront
set can therefore be restricted to that slice.
The corresponding simultaneous equal-time
restriction of \(W\) also exists.

Neither assertion defines
\[
 \widehat N_s\circ\widehat\alpha_s.
\]
A field-dependent quantum substitution requires
the products appearing in its expansion. In
this manuscript its retained jet is represented
only by \eqref{eq:ncomplete} with the specified
Wick and extension rules.

%=====================================================================
\section{The Hessian: microcausality, noncompactness, and internal
contraction}
\label{sec:Hessian}
%=====================================================================

Put
\[
 q(z)=q_{\mathrm{loc}}(z)
                     -2\calRz\cdot\nabla\ell z.
\]
Let \(b(z,w)\) be its symmetric polarization:
\[
 b(z,w)=\frac12\bigl(q(z+w)-q(z)-q(w)\bigr).
\]
Thus
\begin{equation}
 b(z,w)=b_{\mathrm{loc}}(z,w)
       -\calRz\cdot\nabla\ell w
       -\calRw\cdot\nabla\ell z.
 \label{eq:b}
\end{equation}
Define \(K_f\) by
\[
 \pair{K_f}{z\otimes w}
           =\int f\,b(z,w)\dd V_{\barg}.
\]
Then
\[
 F_f^{(2)}=s^2K_f
\]
with respect to the free first jet \(z\).

All component kernels may be written relative to
coordinate measures by absorbing the smooth
\(a^3\) factors. This does not change wavefront
sets or scaling degrees.

\subsection{A compact source cutoff, not a native replacement}

Choose \(\chi\in\Dc(\R^3)\), equal to one near
the origin, and set
\[
 \chi_R(\mathbf y)=\chi(\mathbf y/R).
\]
Define
\begin{equation}
 \calR_Rz=-\Gsp\partial_i
                    \bigl(\chi_R(\calCz)^{ia}\bigr).
 \label{eq:RRcut}
\end{equation}
The derivative includes the cutoff derivative.
Equivalently, \eqref{eq:P} has source
\(\chi_R\calCz\).
Let \(F_{f,R}\) and \(K_{f,R}\) replace
\(\calR\) by \(\calR_R\).

This is a compact source approximation.
It is not the native rod prescription at finite
\(R\), and it is not ultraviolet smoothing:
the equal-time delta and Newton singularity
are unchanged.

\subsection{The relevant wavefront calculation}

In the two field slots \(u=(t_u,\mathbf y)\),
\(v=(t_v,\mathbf x)\), the nonlocal kernel,
before transferring differential operators to
the slots, has the form
\begin{equation}
 f(v)\,\delta(t_u-t_v)\,
       P_i(\mathbf x-\mathbf y)\,
       \calC_u^{ia}\,\partial_{x^a}\ell_v,
 \label{eq:Kform}
\end{equation}
together with its transposed term.
For finite \(R\), multiply it by \(\chi_R(\mathbf y)\).

\begin{proposition}[Compact Hessian admission]
\label{prop:microcausal}
For finite \(R\), \(K_{f,R}\) is a compactly
supported distributional kernel satisfying
\[
 \WF(K_{f,R})\cap
       \left(\widetilde V_+^{\,2}
                    \cup\widetilde V_-^{\,2}\right)
       =\varnothing.
 \label{eq:microcausal}
\]
The same exclusion is locally valid for the
native kernel \(K_f\), but \(K_f\) is generally
not compactly supported.
\end{proposition}

\begin{proof}
Away from \(\mathbf x=\mathbf y\), the Newton
kernel is smooth. The only wavefront covectors
from \eqref{eq:Kform} are
\[
 ((t,\mathbf y;\tau,\mathbf0),
                    (t,\mathbf x;-\tau,\mathbf0)),
 \qquad \tau\ne0.
\]
They have opposite time orientations.

At coincidence, use difference coordinates
\((t_u-t_v,\mathbf y-\mathbf x)\).
The singular factors are a time delta and a
spatial distribution in those independent
variables. Their wavefront covectors, transferred
to the two slots, have the form
\[
 ((x;\tau,\boldsymbol\xi),
                      (x;-\tau,-\boldsymbol\xi)).
\]
They cannot be simultaneously future or
simultaneously past causal.
The local terms are derivatives of a full
diagonal delta and have the same opposite-slot
property.

Differentiation and smooth multiplication do not
enlarge these wavefront upper bounds.
Both slots are compact for finite \(R\).
For the native kernel only the local slot is
confined by \(f\); the other slot has the
dipole tail of \(P\).
\end{proof}

\begin{remark}[An important correction]
An instantaneous off-diagonal timelike conormal
is not by itself a microcausal exclusion:
its sign pattern matters. Here the signs in the
two field slots are opposite.
The obstruction to directly importing the
compact Wick theorem for \(K_f\) is global
noncompactness, not an asserted same-sign
wavefront pair.
\end{remark}

\subsection{Why raw internal contraction is different}

The kernel contraction
\[
 \pair{K_{f,R}}{W}
\]
is not justified merely by
\eqref{eq:microcausal}.
At coincidence \(K_{f,R}\) has opposite-slot
covectors with arbitrary spatial component,
including null ones. The Hadamard kernel has
opposite-slot null covectors. These can cancel
in the product criterion D1.

Similarly, the diagonal pullback of \(W\),
or of its differentiated entries, meets the
conormal of the diagonal. Local Wick
subtraction is required.

Thus a compact normal-ordered quadratic element
can exist even though its raw internal
contraction is undefined by the ordinary
product criterion. This distinction is retained
in the compact Wick framework
\cite{HWlocal}.

%=====================================================================
\section{Explicit one-contraction normalization}
\label{sec:Wick}
%=====================================================================

\subsection{A declared subtraction germ}

Fix the native bundle-valued Hadamard subtraction
germ of XX-C, with the action normalization
\(\calA^{-1}\) in the source slot.
Only its following property is used:
\[
 W-H^+\quad\text{is smooth near the diagonal}.
\]
This is a transport-parametrix normalization
statement, not a conclusion from wavefront
upper bounds alone.

Let \(H_{\mathrm{sub}}\) be a real symmetric
extension of the symmetric part of that germ,
using a smooth symmetric cutoff equal to one
in a neighborhood of the diagonal relevant
to the compact support of \(f\).
Its extension away from that neighborhood is
part of the prescription.

Define
\begin{equation}
 D=\mu-H_{\mathrm{sub}}.
 \label{eq:D}
\end{equation}
Then \(D\) is smooth near the full diagonal.

For the equal-time map
\[
 j(t,\mathbf y,\mathbf x)
          =((t,\mathbf y),(t,\mathbf x)),
\]
all differentiated entries of \(D\) used below
have smooth pullbacks:
near \(\mathbf x=\mathbf y\) this follows from
full-diagonal smoothness; away from coincidence,
the two points on this homogeneous slice are
spacelike separated, so neither \(W\) nor the
chosen near-diagonal subtraction has a null
singularity there.

Derivatives in the two spacetime slots are
taken before equal-time restriction.

\subsection{Complete scalar correction}

For components, let \(D_{00,00}\) denote the
\(h_{00},h_{00}\) entry, and similarly for the
other displayed field components. A prime
indicates the second slot.

The local one-contraction correction is
\begin{align}
 \gamma_{\mathrm{loc}}(x)
 &=-\frac t{4A}\diag{D_{00,00}}
       +\frac t{Aa^2}\delta^{ij}\diag{D_{0i,0j}}
       +\frac{t^2}{A^2}\diag{\partial_{t'}D_{00,\varphi}}
 \notag\\
 &\quad
       +\frac{2t^3}{A^3}
                    \diag{\partial_t\partial_{t'}D_{\varphi,\varphi}}
       +\frac{2t^2}{A^2a^2}
                    \diag{\partial_{i'}D_{0i,\varphi}}
 \notag\\
 &\quad
       +\frac{t^3}{A^3a^2}\delta^{ij}
                    \diag{\partial_i\partial_{j'}D_{\varphi,\varphi}}
       +\frac{t^2}{A^2}\diag{\partial_{t'}D_{\varphi,00}}
 \notag\\
 &\quad
       +\frac t{A^2}\diag{D_{\varphi,00}}
       +\frac{2t^3}{A^3}
                    \diag{\partial_{t'}^2D_{\varphi,\varphi}}
 \notag\\
 &\quad
       +\frac{6t^2}{A^3}\diag{\partial_{t'}D_{\varphi,\varphi}}
       +\frac t{A^3}\diag{D_{\varphi,\varphi}}.
 \label{eq:gammaloc}
\end{align}
These are all the monomials of
\eqref{eq:qloc}; no state positivity is needed
to form these smooth diagonal values.

Define the differentiated equal-time matrix
\begin{equation}
 d^{ia}(t,\mathbf y,\mathbf x)
   =\left[
       \calC_y^{ia}\,
           \partial_{x^a}\ell_x\,D(y,x)
     \right]_{t_y=t_x=t},
 \label{eq:drod}
\end{equation}
with the index \(a\) summed when paired with
\(P_i\). The rod correction is
\begin{align}
 \gamma_{\mathrm{rod},R}(t,\mathbf x)
   &=-2\int_{\R^3}
         P_i(\mathbf x-\mathbf y)\chi_R(\mathbf y)
                     d^{ia}(t,\mathbf y,\mathbf x)\dd^3y,
 \notag\\
 \Gamma_{f,R}
   &=\int f\left(
           \gamma_{\mathrm{loc}}
                      +\gamma_{\mathrm{rod},R}\right)\dd V_{\barg}.
 \label{eq:Gamma}
\end{align}
The integrand is locally integrable at
\(\mathbf y=\mathbf x\), because \(d^{ia}\)
is smooth there and \(P_i\) is locally
integrable.

\begin{definition}[Minimal Gaussian normalization]
\label{def:Gaussian}
The unit and linear fields are unchanged.
In the \(W\)-normal-ordered symbol coordinates,
define the finite-\(R\) insertion by
\begin{equation}
 \mathcal I_{f,R}
   =F_{f,R}+\frac{\hbar s^2}{2}\Gamma_{f,R}
       \pmod{(s^3,\hbar^2)}.
 \label{eq:IR}
\end{equation}
This fixes a subtraction choice. Additional
finite changes, if authorized, must be stated
separately; none are silently used.
\end{definition}

\begin{theorem}[Compact one-contraction realization]
\label{thm:compactWick}
For every finite \(R\), \eqref{eq:IR} is a
well-defined free Wick symbol through the
retained orders. Its quadratic field kernel
is \(K_{f,R}\), and its scalar correction is
exactly \eqref{eq:Gamma}.
All terms in the clock-and-rod jet
\eqref{eq:ncomplete} are represented.
\end{theorem}

\begin{proof}
Proposition~\ref{prop:microcausal} gives compact
microcausal kernels. The finite-component
algebraic Wick construction is therefore
available, using the contractions of
\eqref{eq:W}; its distributional multiplication
uses the positive-frequency relation and does
not require a positive state representation.

The change from the declared symmetric
subtraction to the \(W\)-symbol coordinates
contracts the symmetric Hessian once with
\(D\).
For the local terms this is
\eqref{eq:gammaloc}; for the rod term it is
\eqref{eq:Gamma}.
The latter is finite by compact source support
and local integrability.

Since \(F_{f,R}^{(2)}=s^2K_{f,R}\), the
normal-ordering change is
\((\hbar/2)\pair{D}{F_{f,R}^{(2)}}\), giving
the coefficient in \eqref{eq:IR}.
The central linear second-jet term generates
no internal free contraction.
\end{proof}

This theorem constructs algebraic Wick symbols.
It does not assert that the indefinite
gauge-fixed contraction is a physical state.

\subsection{An alternative extension of the punctured numerical kernel}

There is also an explicit extension route
when one starts with the raw punctured
equal-time contraction.

Let \(\mathbf r=\mathbf x-\mathbf y\).
Off \(\mathbf r=0\), the differentiated
Hadamard contraction in the rod term is smooth.
Its spatial singularity has an upper bound
of the form
\[
 \abs{\beta(t,\mathbf x,\mathbf r)}
 \le C\abs{\mathbf r}^{-6}
                    (1+\abs{\log\abs{\mathbf r}})^N
\]
on compact tangential sets. Indeed, the
four-dimensional Hadamard germ has leading
degree two, the differentiated field factor
has at most two derivatives, and the
dipole kernel has degree two.

The bound is conservative: cancellations or
equal-time parity can lower the actual degree.

\begin{proposition}[Transverse Taylor extension]
\label{prop:Taylor}
Suppose the punctured coefficient \(\beta\)
has the stated Hadamard bound with smooth
dependence on \((t,\mathbf x)\).
It has a distributional extension across
\(\mathbf r=0\). In the class with transverse
scaling degree at most six, the extension
freedom is
\begin{equation}
 \sum_{\abs{\alpha}\le3}
       c_\alpha(t,\mathbf x)
                      \partial_{\mathbf r}^\alpha\delta(\mathbf r).
 \label{eq:contacts}
\end{equation}
The coefficients are not claimed to form a
finite-dimensional space without further
restrictions.
\end{proposition}

\begin{proof}
Choose \(\theta\in\Dc(\R^3)\), equal to one
near zero. For a compact test \(\eta\), put
\[
 T_3\eta
   =\eta-\theta(\mathbf r)
       \sum_{\abs{\alpha}\le3}
          \frac{\mathbf r^\alpha}{\alpha!}
              \partial_{\mathbf r}^\alpha
                     \eta(t,\mathbf x,0).
\]
The remainder is \(O(\abs{\mathbf r}^4)\).
Thus
\[
 \int\beta\,T_3\eta\,
                \dd t\,\dd^3x\,\dd^3r
\]
converges absolutely near zero:
the radial majorant is a constant times
\((1+\abs{\log r})^N\dd r\).
It defines a continuous extension after
specifying values on the subtracted jets.

The finite scaling-degree extension principle
of \cite[Theorems 5.2--5.3 and Section 6]{BF}
gives the scaling-degree bound.
A difference of such extensions is supported
on \(\mathbf r=0\). Its allowed normal
derivative order is at most \(6-3=3\),
giving \eqref{eq:contacts}.
\end{proof}

This is a numerical extension, not automatically
a compatible quantum insertion prescription.
Exchange symmetry, reality, field-equation
relations, and any gauge conditions must be
imposed on it separately.

For the insertion \eqref{eq:IR}, the chosen
route is the Hadamard-subtracted smooth
remainder, not an unannounced identification
with every raw Taylor extension.

There is no additional off-diagonal light-cone
extension in this internal rod contraction:
equal-time distinct points are spacelike.
Other graphs may have light-cone products and
require their own analysis.

%=====================================================================
\section{Native infrared conditions and the cutoff-free theorem}
\label{sec:IR}
%=====================================================================

\subsection{Internal contraction at spatial infinity}

The diagonal subtraction removes a local
singularity. It does not control the
noncompact smooth remainder in
\eqref{eq:drod}.

\begin{assumption}[Internal rod infrared bound]
\label{ass:internal}
For every compact output region \(K\Subset\mathcal M\)
there is \(\epsilon>0\) such that, outside a
fixed ball, the differentiated entries needed
in \eqref{eq:drod} satisfy
\[
 \abs{
   \partial_{t,\mathbf x}^{\beta}
           d^{ia}(t,\mathbf y,\mathbf x)}
       \le C_{K,\beta}
                   (1+\abs{\mathbf y})^{-1-\epsilon},
 \qquad (t,\mathbf x)\in K.
 \label{eq:internalIR}
\]
The constants are uniform for the derivative
orders required for the announced output
regularity.
\end{assumption}

Because \(P_i=O(\abs{\mathbf y}^{-2})\) for
compact \(\mathbf x\), this bound makes the
radial tail integrable.

\begin{lemma}[Cutoff-independent internal correction]
\label{lem:internal}
Under Assumption~\ref{ass:internal},
\(\gamma_{\mathrm{rod},R}\) converges with the
stated output derivatives to
\[
 \gamma_{\mathrm{rod}}
       =-2\int P_i(\mathbf x-\mathbf y)
                      d^{ia}(t,\mathbf y,\mathbf x)\dd^3y.
\]
The limit is independent of the source cutoff
profile and
\[
 \Gamma_f=\int f(\gamma_{\mathrm{loc}}
                           +\gamma_{\mathrm{rod}})\dd V_{\barg}
\]
is a distribution in \(f\).
\end{lemma}

\begin{proof}
Local convergence uses smoothness of \(d\)
and local integrability of \(P\).
For large \(\abs{\mathbf y}\), the product
is bounded by
\(C(1+\abs{\mathbf y})^{-3-\epsilon}\).
Dominated convergence applies to the output
derivatives, and the same estimate makes the
difference of any two admissible cutoff
families tend to zero.
\end{proof}

This is a separately stated hypothesis.
Compact-test smoothness of \(W(\,\cdot\,,g)\)
does not imply \eqref{eq:internalIR}.

\subsection{External one-contraction products}

For \(g\in\Dc(\mathcal M;\mathcal V)\), define
\[
 L_g[z]=(z,g)_0,\qquad w_g=W(\,\cdot\,,g).
\]
The Hadamard wavefront relation implies that
\(w_g\) is smooth: compact testing eliminates
the singular second-slot covector.
It does not imply decay.

\begin{assumption}[External one-rod infrared bound]
\label{ass:external}
For the compact tests \(g\) in the announced
probe class,
\begin{equation}
 \calCw_g\in C^\infty(J;\Sch(\R^3))
 \quad\text{for every }J\Subset I,
 \label{eq:externalIR}
\end{equation}
with uniform seminorm bounds on bounded
families of probes.
\end{assumption}

This is the sufficient one-sided covariance
condition used in XX-C. It is rechecked here
when used.

For polynomials whose contraction is defined,
the order-\(\hbar\) product is
\begin{equation}
 F\star_W G
     =FG+\hbar\pair{F^{(1)}}{W G^{(1)}}
                 \pmod{\hbar^2}.
 \label{eq:star}
\end{equation}

\begin{theorem}[Conditional native insertion and regular-probe products]
\label{thm:native}
Fix an actual representative \(W\) satisfying
\eqref{eq:W}, a declared subtraction germ,
and Assumptions~\ref{ass:internal}
and~\ref{ass:external}.
Then, on \(\mathscr X\) with central smooth
second jets,
\begin{equation}
 \boxed{\;
 \mathcal I_f
    =F_f+\frac{\hbar s^2}{2}\Gamma_f
          \pmod{(s^3,\hbar^2)}
 \;}
 \label{eq:nativeI}
\end{equation}
is a cutoff-independent native Wick symbol.
Its products on either side with compact
regular probe polynomials exist through the
retained contraction order.

For a linear probe, the external contraction
is explicitly
\begin{equation}
 \pair{F_f^{(1)}}{Wg}
    =s\int f\,\ell w_g\,\dd V_{\barg}
         +s^2\int f\,b(z,w_g)\dd V_{\barg}.
 \label{eq:external}
\end{equation}
Every rod in this formula is the original
spatial Newton inverse.
\end{theorem}

\begin{proof}
The classical limit of \(F_{f,R}\) is \(F_f\)
in the seminorm topology of
Theorem~\ref{thm:classicaldomain}.
Lemma~\ref{lem:internal} supplies the scalar
normalization limit.

For \eqref{eq:external}, differentiate
\eqref{eq:Fclass} in the free first jet.
The symmetric polarization gives the displayed
formula. Local entries of \(w_g\) are smooth
and only sampled on \(\supp f\).
The term \(\calRz\) is defined by
Lemma~\ref{lem:Newton}. Assumption
\eqref{eq:externalIR} gives the same lemma
for \(\calRw_g\).
Therefore each term in \(b(z,w_g)\) is a
smooth integrable compact-output expression.

A compact regular polynomial has smooth
compact derivative kernels. After fixing
the remaining compact field variables,
its first derivative is a finite or
continuously compact-tested superposition
of probes of the above type. The uniform
bounds in Assumption~\ref{ass:external}
justify that integration.
The reverse ordered product uses the
transposed contraction. Its causal
difference, compact tested in one slot,
has uniformly compact spatial support
on compact time intervals, so its rod
composition satisfies the compact-source
version of Lemma~\ref{lem:Newton}.

Thus both product orders exist.
Cutoff independence follows from the same
bounds. No infrared bound on a two-rod
state covariance has been used.
\end{proof}

\begin{remark}[Status of this theorem]
The theorem has a noncompact native conclusion,
but it is conditional on two concrete estimates.
The supplied XX-C class does not verify those
estimates for a selected representative.
Consequently \eqref{eq:nativeI} is not announced
as an unconditional native quantum object.
\end{remark}

\subsection{Products needing two noncompact legs}

Even for smooth \(z\), the first derivative
\(F_f^{(1)}[z]\) generally has a noncompact
dipole component. In adjoint notation its
test profiles include
\[
 \calR^\dagger g,\qquad g\in\Dc,
\]
whose spatial decay is \(O(r^{-2})\).

Therefore
\[
 \pair{F_f^{(1)}}{W F_{f'}^{(1)}}
\]
requires an extension of the covariance to
pairs of such noncompact profiles.
Assumption~\ref{ass:external} concerns one
compact-tested endpoint and does not supply
that extension.

A sufficient additional domain statement is
a continuous bilinear covariance pairing on
\[
 \mathscr T_{\mathrm{rod}}
   =\Dc(\mathcal M;\mathcal V)
                   +\calR^\dagger\Dc(\mathcal M;\R^3),
\]
with cutoff-independent pairings and the
native antisymmetric part. This property
must be proved for the chosen representative;
it is not imposed here as an authenticated
fact.

Thus products of two native lapse insertions
are left undefined outside a separately
verified two-endpoint domain.

%=====================================================================
\section{Why the missing infrared hypotheses cannot be inferred}
\label{sec:failure}
%=====================================================================

\subsection{Compact-source Green tails are available}

For compact \(g\), the retarded and advanced
responses \(G_{\calB}^{\ret,\adv}g\) are smooth
and have uniformly compact spatial support
on \(J\Subset I\).
Indeed their support lies in the causal
future or past of \(\supp g\), and the
coordinate propagation radius is bounded by
\[
 \abs{\int_{t'}^t a(u)^{-1}\dd u}
\]
on compact time intervals.
Their rod sources meet
Lemma~\ref{lem:Newton}.

This proves a Green-response tail. It is not
a proof of a state-covariance tail.

\subsection{Smooth covariance ambiguity can leave the rod domain}

\begin{proposition}[Same-operator infrared non-determination]
\label{prop:IRfailure}
The conditions \eqref{eq:W} alone do not
ensure a decaying rod composition after
compact testing.
\end{proposition}

\begin{proof}
In a spatially homogeneous frame all coefficients
of \(\calL\) depend only on time. Spatial
polynomials of degree at most one are preserved
by the operator, reducing the Cauchy problem
on that subspace to smooth finite-dimensional
ordinary differential equations.

Choose a real solution \(U\) with data at
\(t_0\in I\)
\[
 h_{ij}(t_0,\mathbf x)
       =a(t_0)^2x^1\delta_{ij},
 \qquad h_{00}=h_{0i}=\varphi=0
\]
and any fixed polynomial first time-derivative
data. Then \(\calB U=0\), and
\[
 (\calCU)^{ia}(t_0,\mathbf x)
                       =\frac12x^1\delta^{ia}.
\]
A decaying first rod in component \(a=1\)
would solve
\[
 \Delta Y^1=-\frac12.
\]
Its spherical mean is
\[
 \overline Y^1(r)=\overline Y^1(0)-r^2/12,
\]
which cannot tend to zero.

Both \(W\) and \(W+U\otimes U\) satisfy
\eqref{eq:W}: their difference is a smooth
real symmetric bisolution.
Choose compact \(g\) with \((U,g)_0\ne0\).
If both compact-tested covariances had linear
decaying rod compositions, subtracting them
would yield a decaying rod for the displayed
multiple of \(U\), a contradiction.
At least one representative lies outside
that composition domain.
\end{proof}

This is not a claim that the polynomial
configuration is an admissible decaying
classical rod perturbation. It is an allowed
smooth ambiguity of the compact-test
Hadamard contraction class.

\subsection{The native two-rod inverse-distance term}

The two-rod causal kernel is more weakly
decaying than the compact-source one-rod
response. Its leading term can be checked
from the native normalization alone.

In orthonormal spatial components
\(\widehat h_{ij}=a^{-2}h_{ij}\), the map
\(\calC\) is time independent.
Translation invariance makes
\(E_{\calB}\) a kernel
\(e(t,t';\mathbf x-\mathbf y)\).
For fixed times it has compact spatial
support. Its spatial moment satisfies
\[
 m(t',t')=0,\qquad
 \partial_t m(t,t')\big|_{t=t'}
                     =-a(t')^{-3}\calA^{-1}.
\]

Define
\[
 K_{ij}(\mathbf r)
  =\frac{\delta_{ij}-n_in_j}{8\pi r},
 \qquad r=\abs{\mathbf r},\quad n=\mathbf r/r.
\]
Its Fourier multiplier is
\(k_i k_j/\abs{\mathbf k}^4\).
The double-rod causal kernel is the convolution
of \(K_{ij}\) with the compact spatial
distribution \(\calC^{ia}e(\calC^{jb})^\dagger\).

The native normalization gives
\begin{equation}
 k_i k_j\calC^{ia}\calA^{-1}
                    (\calC^{jb})^\dagger
   =\frac2{M^2}\abs{\mathbf k}^2\delta_{ab}
                       +\frac{p+1}{2M^2}k_ak_b.
 \label{eq:matrix}
\end{equation}
For verification, the metric coefficient of
\(k_i\calC^{ia}\) is
\[
 B^a_{kl}=\frac12k_a\delta_{kl}
                      -k_{(k}\delta_{l)a}.
\]
It has trace \(k_a/2\) and contraction
\[
 B^a:B^b
       =\frac12\abs{\mathbf k}^2\delta_{ab}
                              +\frac14k_ak_b.
\]
Four-dimensional trace reversal subtracts
\(k_ak_b/8\), and its normalization is
\(4/M^2\). The scalar contribution is
\(p^2k_ak_b/A^2\). This proves
\eqref{eq:matrix}.

Consequently, for \(t=t_0+\delta\) and
small nonzero \(\delta\),
\begin{align}
 E_{RR}^{ab}(t_0+\delta,t_0;\mathbf r)
 &=-\frac{\delta}{16\pi M^2a(t_0)^3r}
       \left((p+9)\delta_{ab}-(p+1)n_an_b\right)
 \notag\\
 &\quad+O(\delta^2r^{-1})+O_\delta(r^{-2}).
 \label{eq:RRtail}
\end{align}
The calculation uses
\[
 \mathcal F^{-1}(\abs{\mathbf k}^{-2})
           =\frac1{4\pi r},\qquad
 \mathcal F^{-1}(k_ak_b\abs{\mathbf k}^{-4})
           =\frac{\delta_{ab}-n_an_b}{8\pi r}
\]
and the compact-distribution moment expansion
of the convolution.

For \(n\) along axis \(a\), the \(aa\)
coefficient in parentheses is eight.
Hence the inverse-distance component is
nonzero.

\begin{corollary}[No universal inverse-square two-rod covariance bound]
If both ordered two-rod covariances obeyed
\(O(r^{-2})\) uniformly at these unequal times
and retained their native canonical difference,
their difference would contradict
\eqref{eq:RRtail}.
\end{corollary}

An \(r^{-1}\) correlation still tends to zero.
The result does not exclude a suitable weaker
two-endpoint infrared domain. It also does
not erase the single-rod term in
\eqref{eq:ncomplete}.

%=====================================================================
\section{Regularized Wick substitution and operation ledger}
\label{sec:ledger}
%=====================================================================

\subsection{What can legitimately be substituted}

Let \(S_\epsilon\) be a smooth compact-kernel
regularization of the field arguments.
For finite \(\epsilon,R\), the Gaussian
identity for a quadratic polynomial is
\begin{equation}
 \exp\left(\frac{\hbar}{2}
                   \pair{C}{\delta^2}\right)
   F_{f,R}[S_\epsilon z,\zeta]
 =
 F_{f,R}[S_\epsilon z,\zeta]
   +\frac{\hbar s^2}{2}
       \pair{K_{f,R}}{
                   S_\epsilon C S_\epsilon^{\mathrm T}}
       \pmod{\hbar^2}.
 \label{eq:regularized}
\end{equation}
It is an algebraic identity once its displayed
pairing exists.

This reproduces the quadratic Gaussian
content needed for a regularized Wick lemma.
The exact unprovided wording of an XX-A lemma
is not invented.

Taking \(\epsilon\to0\) requires either the
subtraction proof of Section~\ref{sec:Wick}
or a specified extension theorem.
Taking \(R\to\infty\) additionally requires
the estimates of Section~\ref{sec:IR}.
The two limits are not identified merely
because every finite expression is defined.

A joint limit needs joint uniform bounds.
None are supplied by the free commutator,
by positivity of a comparison sector, or
by a finite approximation.

\subsection{Complete domain ledger}

\small
\begin{longtable}{@{}p{.25\linewidth}p{.41\linewidth}p{.26\linewidth}@{}}
\toprule
Operation & Sufficient condition or construction & Status here\\
\midrule
\endfirsthead
\toprule
Operation & Sufficient condition or construction & Status here\\
\midrule
\endhead
Classical local products in
\(S_2,u_2,\gamma_2,B_2,q_{\mathrm{loc}}\)
&
Smooth spatial Schwartz coefficients; ordinary
differentiation and multiplication
&
Defined on \(\mathscr X\)
\\[4pt]

Distributional local products
&
No opposite covectors at the same base point
under D1
&
Defined only when D1 holds; otherwise no raw
product is assigned
\\[4pt]

Local free quadratic fields
&
Compact microcausal kernel and declared Wick
subtraction; smooth diagonal remainder
&
Defined in the compact free Wick realization
\\[4pt]

First original rod
&
Compact or spatial Schwartz \(\calCz\);
decaying Newton solution
&
Native classical map defined
\\[4pt]

Second original rod
&
Schwartz \(B_2\) and
\(B_1\partial Y_1\)
&
Defined classically; absent from \(n_2\)
\\[4pt]

Fixed classical coordinate pullback
&
Diffeomorphism, or D2 for a more general map
&
Defined on its stated classical domain
\\[4pt]

Field-dependent quantum pullback
&
Each Taylor-product coefficient must already
be admitted
&
Only its displayed retained jet is represented;
no exact operator substitution
\\[4pt]

Equal-time restriction of \(W\)
&
Null wavefront avoids the purely temporal
conormal of the restriction map
&
Defined as a spatial distribution, not a
smooth diagonal value
\\[4pt]

Raw diagonal contraction of \(W\)
&
Ordinary diagonal pullback fails; use subtraction
or a separately specified extension
&
No unrenormalized value assigned
\\[4pt]

Internal rod contraction near coincidence
&
Smooth equal-time \(D\) times locally integrable
\(P\), or Proposition~\ref{prop:Taylor}
&
Subtracted local operation constructed
\\[4pt]

Internal rod contraction at infinity
&
Assumption~\ref{ass:internal}
&
Conditional for the native representative
\\[4pt]

Compact-source Green response with one rod
&
Causal support and Lemma~\ref{lem:Newton}
&
Defined, with inverse-square tail
\\[4pt]

Native insertion times a compact regular probe,
through one contraction
&
Assumptions~\ref{ass:internal}
and~\ref{ass:external}
&
Theorem~\ref{thm:native}; hypotheses unverified
for a selected representative
\\[4pt]

Two native lapse insertions through one contraction
&
Continuous covariance on the required
two-endpoint noncompact test domain
&
Undefined here outside a separately verified domain
\\[4pt]

Two-rod covariance separation
&
Must preserve the canonical
inverse-distance term
&
Universal inverse-square bound excluded
\\[4pt]

Time-ordered products against compact regular probes
&
Use the already normalized Feynman inverse
as the single contraction, with its own
noncompact estimates if needed
&
Compact single-line operation available;
native infrared extension not inferred
\\[4pt]

Products of several propagator distributions,
higher Wick contractions, or interacting graphs
&
Joint wavefront, extension, and infrared
analysis for the actual graph
&
Not constructed by the one-contraction theorem
\\
\bottomrule
\end{longtable}
\normalsize

The normalized Feynman kernel is already a
distribution across its light cone and
diagonal. This does not authorize multiplying
two Feynman kernels or constructing a
time-ordered hierarchy for nonlocal insertions.
The local time-ordered framework
\cite{HWexistence} is an analytic template,
not a replacement for that graph analysis.

%=====================================================================
\section{Classical symbol, normalization changes, and pure-trace tests}
\label{sec:symbol}
%=====================================================================

\subsection{The exact formal classical limit}

The term ``classical symbol'' here refers to
the quotient by the \(\hbar\)-adic filtration.
It is not the pseudodifferential principal
symbol of the Newton operator.

\begin{theorem}[Classical symbol recovery]
\label{thm:symbol}
For finite \(R\),
\[
 \mathcal I_{f,R}\bmod\hbar=F_{f,R}.
\]
Under the hypotheses of
Theorem~\ref{thm:native},
\[
 \mathcal I_f\bmod\hbar=F_f,
 \qquad
 (F\star_W G)\bmod\hbar=FG.
\]
The coefficient of \(s^2/2\) in the native
classical symbol is exactly
\eqref{eq:ncomplete}, including
\(-2Y_1^i\partial_i n_1\) and no \(Y_2\).
\end{theorem}

\begin{proof}
All declared Wick corrections have an explicit
factor of \(\hbar\).
The same is true of the contraction in
\eqref{eq:star}. Reduction modulo \(\hbar\)
therefore yields the ordinary polynomial and
its ordinary product.
The classical coefficient is the one proved
in Theorem~\ref{thm:jet}.
\end{proof}

No expectation-value limit is concluded.
Such a limit would require a constructed
physical state or representation and control
of the corresponding family of observables.

\subsection{Smooth reordering is still a domain-sensitive operation}

If \(W'\) has the same canonical part and
\(W'-W=d\) is a smooth real symmetric
bisolution, compact Wick coordinates change by
\[
 \alpha_d
    =\exp\left(\frac{\hbar}{2}
                         \pair{d}{\delta^2}\right).
\]
For the quadratic insertion, this is exactly
one contraction.

For the native insertion the change requires
the noncompact pairing
\[
 \pair{K_f}{d}.
\]
Smoothness alone does not ensure its existence.
Proposition~\ref{prop:IRfailure} supplies the
relevant counterexample mechanism.
Hence compact ordering transport does not
automatically give native ordering transport.

Likewise, a defect can be repaired by a declared
normalization class only when the actual defect
lies in the actual image of its normalization
map. Finite tests or membership in an image
closure are not that statement
\cite[locator \sid{thm:ward}]{CheungXVIIIA}.
No complete physical counterterm image is
constructed here.

\subsection{Pure-trace non-determination}

\begin{proposition}[A smooth blind sector]
\label{prop:traceblind}
Pure-trace metric evaluations do not determine
a general quadratic insertion kernel, even
within the smooth compact class.
\end{proposition}

\begin{proof}
Choose a nonzero smooth compact symmetric
contravariant tensor density \(A^{\mu\nu}\)
with
\[
 A^{\mu\nu}\barg_{\mu\nu}=0.
\]
For example, in the background coordinates
take only compactly supported \(12\) and
\(21\) entries.

The nonzero linear functional
\[
 L_A(h)=\int A^{\mu\nu}h_{\mu\nu}\dd^4x
\]
vanishes for every \(h_{\mu\nu}=\barg_{\mu\nu}\rho\).
Thus the smooth compact quadratic kernel
\[
 B_A(z,w)=L_A(h)L_A(j)
\]
vanishes on every pair of pure-trace metric
inputs but is not zero.
Adding \(B_A\) to a kernel leaves all such
probe values unchanged.
\end{proof}

This is a local proof of the probe
non-determination shape recorded in
\cite[locators \sid{thm:witness},
\sid{cor:smoothwitness}]{CheungXVIIIA}.
It does not declare \(B_A\) an authorized
physical counterterm.

There is also a direct tensor sensitivity
in the actual native rod term.
Choose a spatial trace-free constant matrix
\(S\), for example \(S_{12}=S_{21}=1\), and
\[
 h_{ij}=a^2\rho(t)\chi(\mathbf x)S_{ij},
 \qquad h_{00}=h_{0i}=\varphi=0,
\]
with compact smooth \(\rho,\chi\).
Then
\[
 \ell z=0,\qquad
 \calRz=\rho(t)\Gsp S\nabla\chi.
\]
Choose \(\chi\) so that \(S\nabla\chi\ne0\);
then \(\calRz\ne0\).

For a second input \(w\) with only
\[
 j_{00}=-\frac{2A}{t}\eta(t)g(\mathbf x),
\]
one has \(\ell w=\eta g\).
Its polarized local term with \(z\) vanishes,
but
\[
 b(z,w)=-\calRz\cdot\nabla(\eta g).
\]
A compact \(g\) and output test \(f\) can
detect this expression wherever \(\calRz\ne0\).
Thus the complete native Hessian includes
a mixed shear--lapse contribution that
cannot be authenticated by a pure-trace
calculation alone.

%=====================================================================
\section{Subsidiary defect and exact release}
\label{sec:release}
%=====================================================================

\subsection{The retained free defect}

Retain the XX-B/XX-C smooth contraction defect
\begin{equation}
 \mathscr D=-KW_Q-rW\calF^\dagger.
 \label{eq:subsidiary}
\end{equation}
Its causal imaginary part vanishes by the
native intertwining identity; its remaining
real smooth part is not assumed zero.

Nothing in the Wick construction above
cancels \(\mathscr D\).
Consequently neither the free symbol nor its
classical recovery is promoted to a quantum
BRST identity or a state on the physical
quotient.

\subsection{Release theorem}

\begin{theorem}[XX-D release and Q3 status]
\label{thm:release}
On the identical printed power-law background
and original-rod prescription, this manuscript
establishes:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the full second classical jet
      \eqref{eq:inverse}--\eqref{eq:qloc},
      including all embedding and induced-metric
      contributions and the proved absence of \(Y_2\);
\item the native unsmoothed classical insertion
      on the spatial Schwartz domain;
\item compact microcausal Hessian admission
      after a source cutoff, without mistaking
      opposite-sign equal-time covectors for a
      microcausal exclusion;
\item explicit compact one-contraction Wick
      symbols and an alternative transverse
      numerical extension;
\item a conditional native cutoff-free symbol
      and compact regular-probe products, with
      distinct internal and external infrared
      hypotheses;
\item native same-class infrared
      non-determination and the unavoidable
      inverse-distance two-rod canonical term;
\item exact recovery of the classical formal
      symbol and a smooth pure-trace blind sector.
\end{enumerate}
The data supplied in XX-C do not verify the
infrared hypotheses for a selected representative
or supply the covariance on the required
two-endpoint noncompact domain.
Accordingly the requested native quantum Q3
gate is not closed by the present release.
\end{theorem}

\begin{proof}
Item (i) is Theorem~\ref{thm:jet} and the
explicit expansions.
Item (ii) is Theorem~\ref{thm:classicaldomain}.
Item (iii) is Proposition~\ref{prop:microcausal}.
Item (iv) is Theorem~\ref{thm:compactWick}
and Proposition~\ref{prop:Taylor}.
Item (v) is Theorem~\ref{thm:native}.
Item (vi) is Proposition~\ref{prop:IRfailure}
and \eqref{eq:RRtail}.
Item (vii) is Theorem~\ref{thm:symbol}
and Proposition~\ref{prop:traceblind}.

The hypotheses distinguishing the conditional
native theorem from the unconditional compact
construction have been explicitly retained.
Their verification is not among the conclusions.
\end{proof}

\subsection{What remains necessary for native quantum closure}

A genuine Q3 completion must supply:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a specific normalized native covariance
      representative;
\item proved infrared bounds for the internal
      rod contraction and external compact-probe
      contractions, or another explicitly
      constructed noncompact integration domain;
\item a domain for products with two noncompact
      rod legs, if those products are part of
      the claimed insertion algebra;
\item a complete compatible normalization
      prescription, including any contact
      images under Newton composition;
\item a separately constructed second-jet
      quantum response if \(k,\psi\) are not
      merely prescribed classical jets.
\end{enumerate}
Higher products, interacting graphs, retained-order
gauge identities, and regulator-removal bounds
need additional gates. Their absence is not
a theorem excluding every future extension.

\subsection{Nontransfer boundary}

The relation between original and modified rods
is not altered \cite{CheungXIXA}.
The finite source cutoff is not promoted to
a native continuum object merely because
each approximation exists. Finite scheduled
support and record statements are not
distributional product theorems
\cite{CheungXIXC}.

No claim is made of:
\[
 \begin{gathered}
 \text{a selected positive physical state,}\\
 \text{an unconditional native quantum lapse algebra,}\\
 \text{a complete interacting one-loop lapse coefficient,}\\
 \text{a quantum BRST/BV identity or bulk restoration,}\\
 \text{ultraviolet completion or external-time-free evolution.}
 \end{gathered}
\]
The inherited characteristic-map programme,
XV-B admission exit, XVI-N, and the spatially
compact Route B are not discharged. The
standing O.1 status outside any separately
documented narrow scope is not changed.

%=====================================================================
\section*{Declarations}
\addcontentsline{toc}{section}{Declarations}
%=====================================================================

\paragraph{Source and target discipline.}
The registered Cheung manuscripts are cited as
Cheung, N., by identifier and version.
The supplied companion excerpts delimit their
reported scope; an excerpt or registry summary
is not an independent proof verification.
XX-A and XX-B are not claimed to have been
fully inspected. The target equations used
here are reproduced from the supplied XX-C
freeze and independently expanded.

\paragraph{Authorship.}
The submission is anonymous.
Citation of N.~Cheung does not establish
authorship of this draft. A signed submission
must supply authorized authorship and
affiliation metadata.

\paragraph{Analytic completeness.}
Every native quantum construction here has
its noncompact hypotheses stated.
Failure of a sufficient criterion is marked
as an undefined raw operation, not as a
universal nonexistence theorem.
The native classical construction is
unconditional on its declared smooth domain.

\paragraph{No expectation or operational claim.}
An algebraic contraction on an indefinite
bundle is not renamed a physical state.
A formal insertion symbol is not a positive
probability, measurement effect, destructive
readout instrument, or decoder.

\paragraph{Verification.}
This is one internal-bibliography pdfLaTeX
source, without external figures or input
files. No successful compilation, proof-assistant
check, independent symbolic execution,
numerical run, referee decision, or acceptance
is asserted.

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

\bibitem{XXC}
\emph{XX-C: State, Subtraction, and Spatial-Tail Audit
for the Original-Rod Lapse},
draft supplied in the present manuscript exchange,
\sid{XX-C.tex}, n.d.
Frozen background and normalization, local subtraction,
conditional covariance tails, and native two-rod
commutator calculation.
Its quantum Q2 package is explicitly not closed.

\bibitem{XXB}
\emph{XX-B: Coupled Gauge-Fixed Free Input for the
Original-Rod Lapse},
draft referred to in the supplied XX-C record,
n.d.
Only the equations reproduced in the present
manuscript are used. Full-text inspection and
registered authorship are not asserted.

\bibitem{CheungEm02E}
N.~Cheung,
\emph{Em02-E},
version 1, \sid{Em02-E-v1.tex},
registered manuscript, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Relevant supplied locators:
\sid{thm:lapse}, \sid{thm:release},
and ``The quantum input gate and the first
contraction target.''
Original inflaton-slice rods, the classical
second expansion, and incomplete quantum audit.

\bibitem{CheungXIXA}
N.~Cheung,
\emph{XIX-A},
version 1, \sid{XIX-A.tex},
registered manuscript, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Relevant supplied locators:
\sid{eq:original-rods},
\sid{prop:lapse-composition},
\sid{eq:n1}, \sid{eq:n2},
and \sid{thm:rod-detection}.
Original-versus-modified rod distinction and
the second-order composition identity.

\bibitem{CheungXIXD}
N.~Cheung,
\emph{XIX-D},
version 1, \sid{XIX-D.tex},
registered manuscript, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Relevant supplied sections:
``CE: the original-rod relational lapse''
and ``An equation and a singularity condition
do not select a state.''
The five native quantum audit slots are not
filled by the inventory.

\bibitem{CheungXVIIIA}
N.~Cheung,
\emph{Landmark XVIII-A: Prepared Relational
Observables: Non-Determination, Normalization
Compatibility, and Conditional Ultraviolet Matching},
version 2, \sid{XVIII-A-v2.tex},
registered manuscript, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Relevant supplied locators:
\sid{thm:witness}, \sid{cor:smoothwitness},
\sid{thm:ward}, and \sid{thm:release}.
Pure-trace blind sectors and actual-image
normalization criteria.

\bibitem{CheungXIXC}
N.~Cheung,
\emph{XIX-C},
version 1, \sid{XIX-C.tex},
registered manuscript, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Relevant supplied locator: \sid{thm:main}.
Finite scheduled support, residual-versus-record
separation, and continuum non-determination.

\bibitem{BF}
R.~Brunetti and K.~Fredenhagen,
``Microlocal analysis and interacting quantum
field theories: Renormalization on physical
backgrounds,''
\emph{Communications in Mathematical Physics}
\textbf{208} (2000), 623--661.
DOI: \sid{10.1007/s002200050004}.
arXiv:math-ph/9903028.
In particular, Theorems 5.2--5.3 and Section 6
give finite-scaling-degree extension results.

\bibitem{HWlocal}
S.~Hollands and R.~M.~Wald,
``Local Wick polynomials and time ordered products
of quantum fields in curved spacetime,''
\emph{Communications in Mathematical Physics}
\textbf{223} (2001), 289--326.
DOI: \sid{10.1007/s002200100540}.
arXiv:gr-qc/0103074.
Compact distributional Wick kernels, smooth
ordering transport, and local normalization.

\bibitem{HWexistence}
S.~Hollands and R.~M.~Wald,
``Existence of local covariant time ordered
products of quantum fields in curved spacetime,''
\emph{Communications in Mathematical Physics}
\textbf{231} (2002), 309--345.
DOI: \sid{10.1007/s00220-002-0719-y}.
arXiv:gr-qc/0111108.
Local time-ordered-product construction;
not automatic admission of the noncompact
Newton insertion.

\bibitem{BGP}
C.~B\"ar, N.~Ginoux, and F.~Pf\"affle,
\emph{Wave Equations on Lorentzian Manifolds
and Quantization},
ESI Lectures in Mathematics and Physics,
European Mathematical Society, Z\"urich, 2007.
DOI: \sid{10.4171/037}.
arXiv:0806.1036.
Normally hyperbolic Cauchy theory and causal
Green operators.

\bibitem{BFHPR}
R.~Brunetti, K.~Fredenhagen, T.-P.~Hack,
N.~Pinamonti, and K.~Rejzner,
``Cosmological perturbation theory and quantum
gravity,''
\emph{Journal of High Energy Physics}
\textbf{08} (2016), 032.
DOI: \sid{10.1007/JHEP08(2016)032}.
arXiv:1605.02573.
Relational-coordinate context; no prescription
or native insertion identification is imported.

\end{thebibliography}

\end{document}