% XX-B.tex
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% No successful compilation or machine verification is asserted.
%
% Principal debugging corrections:
% 1. Rods are derived elliptic coordinate functionals, not extra
%    propagating fields of the Einstein--scalar Hessian.
% 2. The original-rod Newton inverse is not a retarded spacetime inverse.
% 3. The Euler linearization must be converted to the action Hessian
%    before its inverse is used to normalize contractions.
% 4. The gauge parameter is fixed so that the action Hessian has a
%    scalar normally hyperbolic principal part after an explicit
%    invertible fiber normalization.
% 5. Retarded gauge reachability does not imply reachability inside
%    XX-A's compactly supported spacetime gauge group.
% 6. A Hadamard contraction on an indefinite gauge-fixed field space
%    is not automatically a positive physical state.
% 7. The classical-limit statement is a principal-symbol statement,
%    not an expectation-value limit of an unconstructed composite.
% 8. Free contraction data do not admit the nonlocal lapse Hessian
%    or prove its interacting Ward identity.

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\hypersetup{
  pdftitle={XX-B: Coupled Gauge-Fixed Free Input for the Original-Rod Lapse},
  pdfauthor={},
  pdfsubject={Track A, Gate Q1, Einstein scalar, Green distributions},
  pdfkeywords={relational lapse, original rods, gauge fixing,
    normally hyperbolic operator, Hadamard contractions}
}

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\title{\textbf{XX-B}\\[3pt]
Coupled Gauge-Fixed Free Input for the Original-Rod Lapse\\
\large A Q1 Construction on the Frozen XX-A Background}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We construct the coupled gauge-fixed free input for the original-rod
lapse variant explicitly frozen in XX-A. The background, scalar
calibration, positive lapse branch, slice-harmonic rods, decaying
spatial boundary condition, and factorial perturbation convention
are retained and reproduced.

The Einstein--scalar Euler linearization is converted to its
correctly normalized action Hessian. A generalized de Donder gauge,
with a specified auxiliary-field normalization, gives a reduced
operator whose principal part is an invertible fiber matrix times
the background wave operator. After fiber normalization it is
normally hyperbolic. Retarded and advanced inverses are constructed
on the open finite-time slab, and the complete auxiliary--ghost
inverse is obtained by an exact block calculation. Causal
intertwining identities and wavefront bounds are proved.

A nonempty class of normalized Hadamard contraction matrices is
constructed using exact Feynman inverses for the resulting bundle
operators. The construction is algebraic: positivity on the
physical quotient, spatial-tail estimates for the lapse, and an
exact BRST-invariant symmetric covariance are not claimed.
The complete second-order classical field equations and the
embedding, induced-metric, clock, and rod contributions to the
lapse are displayed. The second spatial rod is constructed and
its absence from the quadratic lapse is retained.

Q1 is closed for the compact-test, gauge-fixed free-input target
defined in this paper. This does not close Q2--Q5 for the nonlocal
relational lapse, identify the variant with every native Em02-E
convention, or admit an inherited characteristic quantum
coefficient. No quantum evolution without external time,
regulator removal, bulk symmetry restoration, or compilation is
asserted.
\end{abstract}

\tableofcontents

%=====================================================================
\section{Target, mathematical inputs, and scope}
\label{sec:scope}
%=====================================================================

\subsection{The target actually constructed}

XX-A froze an explicit original-rod lapse variant on a power-law
Einstein--scalar background \cite{XXA}. This paper constructs
its \emph{free gauge-fixed field input}. It does not reconstruct
an unavailable interacting calculation.

The target consists of:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the coupled metric--scalar action Hessian and a fixed
      gauge-fixing realization;
\item the auxiliary and ghost sectors, including their signs
      and normalization;
\item retarded and advanced inverses on compact spacetime tests;
\item a normalized Hadamard contraction matrix on that same
      free field system;
\item the classical principal-symbol comparison with the
      independently displayed original-rod lapse.
\end{enumerate}

Here ``normalized contraction'' means a specified field-equation
and graded canonical-commutation normalization with a stated
microlocal singularity class. It does not mean a positive physical
state or a chosen cosmological vacuum.

The original rods are derived coordinate functionals. They do not
appear as additional propagating variables in the invariant bulk
action. Their spatial inverse is supplied separately.
Consequently, ``original-rod Green distributions'' must not mean
that the Newton inverse is replaced by a retarded spacetime inverse.

\subsection{Declared analytic inputs}

We use two published analytic results as explicit mathematical
inputs, after checking their hypotheses:
\begin{enumerate}[label=\textup{A\arabic*.}]
\item The Cauchy and Green-operator theorems for normally hyperbolic
      bundle operators on globally hyperbolic spacetimes:
      \cite[Theorem 3.2.11 and Corollary 3.4.3]{BGP}.
\item The bundle-valued Feynman-parametrix and exact
      Hadamard-bisolution construction:
      \cite[Theorems 1.2 and 1.4]{IS}.
      Only the existence and microlocal conclusions are used.
      No positivity conclusion for an indefinite bundle pairing
      is imported.
\end{enumerate}

Hack's Einstein--Klein--Gordon construction and the BFHPR
relational framework are comparison templates
\cite{Hack,BFHPR}. Their particular gauges, state choices,
and observable definitions are not substituted for the
definitions below.

The supplied Cheung records provide version-qualified comparisons
and audit shapes \cite{CheungEm02E,CheungXVIIIA,CheungXIXC,
CheungXIXD}. Their unprinted proofs and documentary relationship
edges are not premises.

\subsection{Standing nontransfer convention}

For all predecessor-series comparisons,
\[
 \boxed{\text{Declared path: null.}}
\]
This convention does not prohibit using the equations explicitly
frozen and proved in the present manuscript. It prohibits treating
a documentary relationship as a theorem or a prescription
identification.

The construction remains on a fixed background with external
cosmic time. No field-dependent coordinate map removes that
external structure from the free quantum construction.

%=====================================================================
\section{The complete background and classical prescription}
\label{sec:freeze}
%=====================================================================

\subsection{Action and on-shell background}

Use signature \((-,+,+,+)\), a reduced Planck mass \(M>0\), and
\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}
On noncompact slices, this notation specifies the local action
density and its compactly supported variations. No finite total
background action is assumed.

Fix \(p>1\), \(t_*>0\), \(\phi_*\in\R\), and
\begin{align}
 A&=\sqrt{2p}\,M,
 &
 a(t)&=(t/t_*)^p,
 &
 \barphi(t)&=\phi_*+A\log(t/t_*),
 \label{eq:background}\\
 \barg&=-\dd t^2+a(t)^2\delta_{ij}\dd x^i\dd x^j,
 \label{eq:metric}\\
 V(\phi)&=\frac{M^2p(3p-1)}{t_*^2}
           \exp\left(-\frac{2(\phi-\phi_*)}{A}\right).
 \label{eq:potential}
\end{align}
The spacetime is
\[
 \mathcal M=I\times\R^3,\qquad
 I=(t_-,t_+),\qquad 0<t_-<t_+<\infty.
\]
There are no timelike or spacelike boundary conditions at the
missing endpoints \(t_\pm\). The interval is open.

Write
\[
 E_{\mu\nu}=M^2G_{\mu\nu}-T_{\mu\nu},
 \qquad
 E_\phi=\Box_g\phi-V'(\phi).
\]
The background solves \(E_{\mu\nu}=E_\phi=0\):
\[
 H=\frac pt,\qquad
 \dot{\barphi}=\frac At,\qquad
 3M^2H^2=\frac12\dot{\barphi}^{\,2}+V(\barphi),
 \qquad
 -2M^2\dot H=\dot{\barphi}^{\,2}.
\]
The scalar equation follows from \(A^2=2pM^2\).
Also
\[
 -\barg^{-1}(\dd\barphi,\dd\barphi)=A^2/t^2>0.
\]

\begin{lemma}[Global hyperbolicity of the open slab]
\label{lem:GH}
Every surface \(t=t_0\), \(t_0\in I\), is a Cauchy surface
of \((\mathcal M,\barg)\).
\end{lemma}

\begin{proof}
Conformal time \(\eta\), defined by
\(\dd\eta=\dd t/a(t)\), identifies the metric conformally with
Minkowski space on an open time interval.
Along a causal curve,
\(\abs{\dd\mathbf x/\dd t}\le a(t)^{-1}\).
If an inextendible causal curve had a time endpoint strictly
inside \(I\), this estimate would give a finite spatial endpoint,
and the curve could be extended. Its time range therefore
crosses every \(t_0\in I\), exactly once for a future-directed
causal curve after the usual harmless removal of constant
parameter intervals.
\end{proof}

\subsection{Perturbations, clock, and lapse}

Retain the factorial convention
\begin{equation}
 g_s=\barg+s h+\frac{s^2}{2}k,\qquad
 \phi_s=\barphi+s\varphi+\frac{s^2}{2}\psi
 \pmod{s^3}.
 \label{eq:jet}
\end{equation}
The classical coefficient fields are smooth and have a common
compact spatial support on each compact time interval used in
the observable calculation. They need not solve the perturbed
Euler equations.

The field-dependent cosmic label is
\begin{equation}
 T_s=t_*\exp\left(\frac{\phi_s-\phi_*}{A}\right).
 \label{eq:clock}
\end{equation}
The raw \emph{matter-scalar} lapse, on the positive branch, is
\begin{equation}
 N_s=
 \left(-g_s^{\mu\nu}\partial_\mu\phi_s\partial_\nu\phi_s
       \right)^{-1/2}.
 \label{eq:raw-lapse}
\end{equation}
It is not silently replaced by the lapse calibrated to \(T_s\).
The two normalizations satisfy
\[
 N_s=(T_s/A)\left(-g_s^{-1}(\dd T_s,\dd T_s)\right)^{-1/2}.
\]

Let \(t=\theta_s(\tau,\mathbf x)\) invert the clock at fixed
spatial coordinate. Then
\begin{align}
 T_s&=t+s\chi+\frac{s^2}{2}\zeta,
 &
 \chi&=\frac tA\varphi,
 &
 \zeta&=\frac tA\psi+\frac t{A^2}\varphi^2,
 \label{eq:clock-jet}\\
 \theta_s&=\tau+s u_1+\frac{s^2}{2}u_2,
 &
 u_1&=-\chi,
 &
 u_2&=2\chi\dot\chi-\zeta.
 \label{eq:theta}
\end{align}
All inverse coefficients are evaluated at \(t=\tau\).

\subsection{Original rods and their separate spatial inverse}

On each clock slice, use the induced metric
\begin{equation}
 \gamma_{s,ij}
 =(g_s)_{ij}(\theta_s,\mathbf x)
 +(g_s)_{0i}(\theta_s,\mathbf x)\partial_j\theta_s
 +(g_s)_{0j}(\theta_s,\mathbf x)\partial_i\theta_s
 +(g_s)_{00}(\theta_s,\mathbf x)
                       \partial_i\theta_s\partial_j\theta_s.
 \label{eq:induced}
\end{equation}
The original rods are defined by
\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:rods}
\end{equation}
There is no lapse weight in this equation.

Write
\[
 \gamma_s=a^2\delta+s\gamma_1+\frac{s^2}{2}\gamma_2.
\]
The complete embedding contributions are
\begin{align}
 (\gamma_1)_{ij}
 &=h_{ij}+2a^2H u_1\delta_{ij},
 \label{eq:gamma1}\\
 (\gamma_2)_{ij}
 &=k_{ij}+2u_1\dot h_{ij}
    +u_2\partial_t(a^2\delta_{ij})
    +u_1^2\partial_t^2(a^2\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}
Define
\[
 C=a^{-2}\gamma_1,\qquad D=a^{-2}\gamma_2,\qquad
 a^{-1}\sqrt{\det\gamma_s}\,\gamma_s^{-1}
 =I+s B_1+\frac{s^2}{2}B_2.
\]
Then
\begin{align}
 B_1&=\frac12(\tr C)I-C,
 \label{eq:B1}\\
 B_2&=
 \left(\frac12\tr D+\frac14(\tr C)^2-\frac12\tr(C^2)\right)I
       -D-(\tr C)C+2C^2.
 \label{eq:B2}
\end{align}
The spatial inverse is
\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}
For \(R_s=x+sY_1+s^2Y_2/2\),
\begin{align}
 Y_1^a&=-\Gsp\partial_iB_1^{ia},
 \label{eq:Y1}\\
 Y_2^a&=-\Gsp\partial_i
          \left(B_2^{ia}+2B_1^{ij}\partial_jY_1^a\right).
 \label{eq:Y2}
\end{align}

\begin{lemma}[Spatial rod realization]
\label{lem:rod-realization}
Equations \eqref{eq:Y1}--\eqref{eq:Y2} give unique smooth
decaying rod coefficients on the stated classical domain.
They are \(O(\abs{\mathbf x}^{-2})\) on compact time intervals.
\end{lemma}

\begin{proof}
Both sources are divergences of compact smooth spatial vectors.
For the second source, \(B_1\partial Y_1\) is compact because
\(B_1\) is compact. Newton convolution gives smooth solutions.
Integration by parts transfers one derivative to the Newton
kernel and yields the \(1/r^2\) bound.

The difference of two decaying solutions is entire harmonic
and tends to zero at infinity. The maximum principle on
successively larger balls makes it zero.
\end{proof}

This elliptic realization is retained unchanged. No
hyperbolic Green theorem removes its spatial boundary condition.

%=====================================================================
\section{The coupled Euler operator and the normalized action Hessian}
\label{sec:hessian}
%=====================================================================

\subsection{Bundles, pairings, and gauge generator}

Let
\[
 \mathcal V=S^2T^*\mathcal M\oplus\underline{\R},
 \qquad
 \mathcal W=T^*\mathcal M.
\]
Use \(\barg\) to identify vector and covector gauge descriptors.
For \(z=(h,\varphi)\), \(w=(j,\omega)\), and \(a,b\in\mathcal W\),
the integrated pairings are
\begin{align}
 (z,w)_0
 &=\int_{\mathcal M}
       \left(h_{\mu\nu}j^{\mu\nu}+\varphi\omega\right)
                             \dd V_{\barg},
 \label{eq:pair0}\\
 (a,b)_1
 &=\int_{\mathcal M}a_\mu b^\mu\,\dd V_{\barg},
 \qquad
 \dd V_{\barg}=a(t)^3\dd t\,\dd^3x.
 \label{eq:pair1}
\end{align}
These pairings are nondegenerate, not positive definite.
Complexification uses their Hermitian versions.

Set \(v_\mu=\partial_\mu\barphi\). The gauge generator is
\begin{equation}
 K\xi=\left(2\nabla_{(\mu}\xi_{\nu)},\,v_\mu\xi^\mu\right),
 \label{eq:K}
\end{equation}
where \(\nabla\) is the background covariant derivative.
Its adjoint is
\begin{equation}
 K^\dagger(h,\varphi)_\nu
       =-2\nabla^\mu h_{\mu\nu}+v_\nu\varphi.
 \label{eq:Kadj}
\end{equation}

\subsection{Full linearization}

The first curvature variations are
\begin{align}
 \Gamma_1^\rho{}_{\mu\nu}
 &=\frac12\barg^{\rho\sigma}
     \left(\nabla_\mu h_{\nu\sigma}
          +\nabla_\nu h_{\mu\sigma}
          -\nabla_\sigma h_{\mu\nu}\right),
 \label{eq:Gamma1}\\
 (R_1)_{\mu\nu}
 &=\nabla_\rho\Gamma_1^\rho{}_{\mu\nu}
            -\nabla_\nu\Gamma_1^\rho{}_{\mu\rho},
 \label{eq:R1}\\
 R_1&=-h^{\mu\nu}\bar R_{\mu\nu}
                           +\barg^{\mu\nu}(R_1)_{\mu\nu},
 \label{eq:Rscalar1}\\
 (G_1)_{\mu\nu}
 &=(R_1)_{\mu\nu}
                -\frac12h_{\mu\nu}\bar R
                -\frac12\barg_{\mu\nu}R_1.
 \label{eq:G1}
\end{align}
The matter variations are
\begin{align}
 (T_1)_{\mu\nu}
 &=\partial_\mu\varphi\,v_\nu+v_\mu\partial_\nu\varphi
        -\frac12h_{\mu\nu}(v^\rho v_\rho+2V) \notag\\
 &\quad
        -\frac12\barg_{\mu\nu}
            \left(-h^{\rho\sigma}v_\rho v_\sigma
                   +2v^\rho\partial_\rho\varphi+2V'\varphi\right),
 \label{eq:T1}\\
 (E_\phi)_1
 &=\Box_{\barg}\varphi-V''\varphi
       -h^{\mu\nu}\nabla_\mu\nabla_\nu\barphi \notag\\
 &\quad
       -\left(\nabla_\mu h^{\mu\nu}
                         -\frac12\nabla^\nu h^\rho{}_\rho\right)v_\nu.
 \label{eq:scalar1}
\end{align}
All potentials in these expressions are evaluated at \(\barphi\).

The coupled action Hessian is the operator
\begin{equation}
 \calH z=
 \left(-\frac12\bigl(M^2G_1-T_1\bigr)_{\mu\nu},
                           (E_\phi)_1\right).
 \label{eq:H}
\end{equation}
The metric factor \(-1/2\) is essential. It follows from
\[
 \delta S
 =\int\left(-\frac12E^{\mu\nu}\delta g_{\mu\nu}
                         +E_\phi\delta\phi\right)\dd V_g.
\]
At an on-shell background, the variations of the volume and of
the indices multiplying \(E\) contribute no extra term to
the second variation. Therefore
\begin{equation}
 S_2(z)=\frac12(z,\calH z)_0
 \label{eq:S2}
\end{equation}
is the actual quadratic action.

\begin{lemma}[Self-adjointness and invariant gauge kernel]
\label{lem:H}
The operator \(\calH\) is formally self-adjoint with respect
to \((\cdot,\cdot)_0\), and
\begin{equation}
 \calH K=0,\qquad K^\dagger\calH=0.
 \label{eq:HK}
\end{equation}
\end{lemma}

\begin{proof}
Self-adjointness is symmetry of the second variation on
compact variations. Covariance of the Euler equations and the
on-shell background give \(\calH K=0\).
Taking the formal adjoint gives the second identity.
\end{proof}

The raw Euler map, the action Hessian, and its fiber-normalized
wave operator are different maps. Their inverses must not be
identified without the factors displayed below.

%=====================================================================
\section{The complete second-order classical field equations}
\label{sec:quadratic}
%=====================================================================

This section supplies a finite coefficient algorithm for the
quadratic Einstein--scalar terms. It also fixes what ``through
quadratic order'' means. No nonlinear differential operator is
being assigned a linear Green inverse.

Define
\[
 L(h)_{\sigma\mu\nu}
 =\nabla_\mu h_{\nu\sigma}
      +\nabla_\nu h_{\mu\sigma}
      -\nabla_\sigma h_{\mu\nu}.
\]
For the jet \eqref{eq:jet}, the inverse metric coefficients are
\begin{equation}
 J_0^{\mu\nu}=\barg^{\mu\nu},\quad
 J_1^{\mu\nu}=-h^{\mu\nu},\quad
 J_2^{\mu\nu}=2h^\mu{}_\rho h^{\rho\nu}-k^{\mu\nu}.
 \label{eq:J}
\end{equation}
The connection difference coefficients are
\begin{equation}
 C_1^\rho{}_{\mu\nu}
       =\frac12\barg^{\rho\sigma}L(h)_{\sigma\mu\nu},
 \qquad
 C_2^\rho{}_{\mu\nu}
       =\frac12\barg^{\rho\sigma}L(k)_{\sigma\mu\nu}
                        -h^{\rho\sigma}L(h)_{\sigma\mu\nu}.
 \label{eq:C}
\end{equation}
Thus
\begin{align}
 (R_2)_{\mu\nu}
 &=\nabla_\rho C_2^\rho{}_{\mu\nu}
       -\nabla_\nu C_2^\rho{}_{\mu\rho} \notag\\
 &\quad
       +2C_1^\rho{}_{\rho\lambda}C_1^\lambda{}_{\mu\nu}
       -2C_1^\rho{}_{\nu\lambda}C_1^\lambda{}_{\mu\rho},
 \label{eq:R2}\\
 R_2
 &=J_2^{\mu\nu}\bar R_{\mu\nu}
             +2J_1^{\mu\nu}(R_1)_{\mu\nu}
             +J_0^{\mu\nu}(R_2)_{\mu\nu},
 \label{eq:Rscalar2}\\
 (G_2)_{\mu\nu}
 &=(R_2)_{\mu\nu}
       -\frac12 k_{\mu\nu}\bar R
       -h_{\mu\nu}R_1
       -\frac12\barg_{\mu\nu}R_2.
 \label{eq:G2}
\end{align}

Put \(a_\mu=\partial_\mu\varphi\), \(b_\mu=\partial_\mu\psi\).
For \(U_s=g_s^{-1}(\dd\phi_s,\dd\phi_s)\),
\begin{align}
 U_0&=J_0^{\mu\nu}v_\mu v_\nu, \notag\\
 U_1&=J_1^{\mu\nu}v_\mu v_\nu
                         +2J_0^{\mu\nu}v_\mu a_\nu, \notag\\
 U_2&=J_2^{\mu\nu}v_\mu v_\nu
       +4J_1^{\mu\nu}v_\mu a_\nu
       +2J_0^{\mu\nu}a_\mu a_\nu
       +2J_0^{\mu\nu}v_\mu b_\nu.
 \label{eq:U}
\end{align}
Write \(V_0=V(\barphi)\), \(V_1=V'\varphi\), and
\(V_2=V'\psi+V''\varphi^2\). Then
\begin{align}
 (T_2)_{\mu\nu}
 &=v_\mu b_\nu+b_\mu v_\nu+2a_\mu a_\nu
       -\frac12k_{\mu\nu}(U_0+2V_0) \notag\\
 &\quad
       -h_{\mu\nu}(U_1+2V_1)
       -\frac12\barg_{\mu\nu}(U_2+2V_2).
 \label{eq:T2}
\end{align}
For the scalar equation, define
\begin{align}
 A_{0,\mu\nu}&=\nabla_\mu v_\nu, \notag\\
 A_{1,\mu\nu}&=\nabla_\mu a_\nu-C_1^\rho{}_{\mu\nu}v_\rho,
 \notag\\
 A_{2,\mu\nu}&=\nabla_\mu b_\nu-C_2^\rho{}_{\mu\nu}v_\rho
                                  -2C_1^\rho{}_{\mu\nu}a_\rho.
 \label{eq:A}
\end{align}
Its complete second coefficient is
\begin{equation}
 (E_\phi)_2
 =J_2^{\mu\nu}A_{0,\mu\nu}
        +2J_1^{\mu\nu}A_{1,\mu\nu}
        +J_0^{\mu\nu}A_{2,\mu\nu}
        -V''\psi-V'''\varphi^2.
 \label{eq:scalar2}
\end{equation}

\begin{proposition}[Complete two-jet Euler expansion]
\label{prop:Euler2}
Equations \eqref{eq:J}--\eqref{eq:scalar2}, together with
\eqref{eq:Gamma1}--\eqref{eq:scalar1}, give every coefficient
of the coupled Euler equations through \(s^2\).
The second coefficient separates uniquely into the linear
operator applied to \((k,\psi)\) and a homogeneous quadratic
source in \((h,\varphi)\).
\end{proposition}

\begin{proof}
The exact connection-difference formula is
\[
 \Gamma[g_s]^\rho{}_{\mu\nu}
       -\Gamma[\barg]^\rho{}_{\mu\nu}
 =\frac12g_s^{\rho\sigma}
       \left(\nabla_\mu(g_s-\barg)_{\nu\sigma}
            +\nabla_\nu(g_s-\barg)_{\mu\sigma}
            -\nabla_\sigma(g_s-\barg)_{\mu\nu}\right).
\]
Its Taylor expansion gives \eqref{eq:C}.
The exact curvature-difference formula is its covariant
derivative plus the two connection products in \eqref{eq:R2}.
Contracting with the inverse metric gives the scalar curvature
coefficients.

Expanding the stress tensor gives \eqref{eq:U} and
\eqref{eq:T2}. Finally,
\[
 \nabla^{g_s}_\mu\partial_\nu\phi_s
       =\nabla_\mu\partial_\nu\phi_s
            -C_s^\rho{}_{\mu\nu}\partial_\rho\phi_s
\]
gives \eqref{eq:A} and \eqref{eq:scalar2}.
Every occurrence of \(k,\psi\) is linear; setting them to zero
leaves the homogeneous quadratic source.
\end{proof}

Rods do not add terms to these bulk equations. Their complete
contributions belong to the insertion, and are retained in
Sections~\ref{sec:freeze} and~\ref{sec:lapse}.

%=====================================================================
\section{Gauge fixing and the normally hyperbolic realization}
\label{sec:gauge-fixing}
%=====================================================================

\subsection{Gauge condition and auxiliary normalization}

Retain XX-A's gauge-condition candidate:
\begin{equation}
 \calF_\nu(h,\varphi)
 =\nabla^\mu h_{\mu\nu}
           -\frac12\nabla_\nu h^\mu{}_\mu
           -M^{-2}v_\nu\varphi.
 \label{eq:F}
\end{equation}
Fix
\begin{equation}
 r=\frac{M^2}{2},\qquad \beta=r^{-1}=\frac2{M^2}.
 \label{eq:beta}
\end{equation}
Thus a previously unspecified positive gauge parameter is now
fixed; the rod prescription is not changed.

The free graded fields are
\[
 z=(h,\varphi),\qquad B_\nu,\qquad c_\nu,\qquad\bar c_\nu,
\]
with \(z,B\) even and \(c,\bar c\) odd. Set
\begin{equation}
 s_0z=Kc,\quad s_0c=0,\quad
 s_0\bar c=B,\quad s_0B=0.
 \label{eq:BRST}
\end{equation}
The gauge fermion and quadratic action are
\begin{align}
 \Psi&=(\bar c,\calF z+\tfrac1{2r}B)_1,
 \label{eq:Psi}\\
 S_{2,\mathrm{gf}}
 &=\frac12(z,\calH z)_0
       +(B,\calF z)_1+\frac1{2r}(B,B)_1
       -(\bar c,Qc)_1,
 \label{eq:Sgf}\\
 Q&=\calF K.
 \label{eq:Qdef}
\end{align}
The odd Leibniz rule gives the ghost sign in \eqref{eq:Sgf}.
Since \(\calH K=0\), \(s_0S_{2,\mathrm{gf}}=0\) and
\(s_0^2=0\) on the off-shell free multiplet.

Let
\[
 D_\nu h=\nabla^\mu h_{\mu\nu}
                -\frac12\nabla_\nu h^\mu{}_\mu.
\]
Its adjoint and the adjoint of \(\calF\) are
\begin{align}
 (D^\dagger a)_{\mu\nu}
 &=-\nabla_{(\mu}a_{\nu)}
                       +\frac12\barg_{\mu\nu}\nabla^\rho a_\rho,
 \notag\\
 \calF^\dagger a&=(D^\dagger a,-M^{-2}v^\nu a_\nu).
 \label{eq:Fadj}
\end{align}
The ghost operator is explicitly
\begin{equation}
 (Q\xi)_\nu
 =\Box_{\barg}\xi_\nu+\bar R_{\nu\rho}\xi^\rho
                            -M^{-2}v_\nu v_\rho\xi^\rho.
 \label{eq:Q}
\end{equation}
It is normally hyperbolic and formally self-adjoint for
\((\cdot,\cdot)_1\).

\subsection{Reduced bosonic operator}

The auxiliary equation is \(B=-r\calF z\).
Eliminating \(B\) gives
\[
 S_{2,\mathrm{red}}
 =\frac12(z,\calH z)_0-\frac r2(\calF z,\calF z)_1.
\]
Its operator is
\begin{equation}
 \calB=\calH-r\calF^\dagger\calF.
 \label{eq:Bbulk}
\end{equation}
Define the four-dimensional trace reversal
\[
 (\mathsf T h)_{\mu\nu}
       =h_{\mu\nu}-\frac12\barg_{\mu\nu}h^\rho{}_\rho,
 \qquad \mathsf T^2=\id,
\]
and the parallel invertible fiber map
\begin{equation}
 \calA=\diag\left(\frac{M^2}{4}\mathsf T,\id\right),
 \qquad
 \calA^{-1}=\diag\left(\frac4{M^2}\mathsf T,\id\right).
 \label{eq:Amap}
\end{equation}

\begin{theorem}[Normally hyperbolic coupled realization]
\label{thm:NH}
The operator
\begin{equation}
 \calL=\calA^{-1}\calB
 \label{eq:L}
\end{equation}
has differential principal part \(\Box_{\barg}\id_{\mathcal V}\).
All its lower-order coefficients are smooth on \(\mathcal M\).
It is formally self-adjoint for the nondegenerate pairing
\[
 (z,w)_{\calA}=(z,\calA w)_0.
\]
\end{theorem}

\begin{proof}
The second-derivative part of the linearized Einstein tensor is
\[
 G_1=-\frac12\mathsf T\Box_{\barg}h-D^\dagger Dh
                                      +\text{lower differential order}.
\]
Therefore the metric block of \(\calH\) has second-derivative
part
\[
 \frac{M^2}{4}\mathsf T\Box_{\barg}
                    +\frac{M^2}{2}D^\dagger D.
\]
The second-derivative part of
\(r\calF^\dagger\calF\) is \(rD^\dagger D\).
Equation \eqref{eq:beta} cancels it exactly.

The scalar block has principal part \(\Box_{\barg}\).
Mixed metric--scalar entries have at most one derivative:
the background scalar gradient and its derivatives are fixed
coefficients. The scalar term in \(\calF\) likewise creates
only first- and zeroth-order mixed entries.

Thus
\(\calB=\calA\Box_{\barg}+\text{lower differential order}\).
All coefficients are smooth because \(t\) stays away from zero.
Finally, \(\calB\) is self-adjoint for \((\cdot,\cdot)_0\);
using \(\calA\calL=\calB\) proves the asserted weighted
self-adjointness.
\end{proof}

This proves hyperbolicity for the actual displayed gauge and
normalization. It does not rely on identifying this operator
with Hack's differently presented operator.

\subsection{Residual gauges and boundary discipline}

\begin{proposition}[Residual transformations in the frozen group]
\label{prop:residual}
Within XX-A's compactly supported spacetime gauge group,
the linear residual gauge condition is
\[
 Q\xi=0,\qquad \xi\in\Dc(\mathcal M;\mathcal W),
\]
and its only solution is \(\xi=0\).
\end{proposition}

\begin{proof}
A compact spacetime support lies strictly to the future of
some Cauchy surface in the slab. Both Cauchy data of \(\xi\)
vanish there. Uniqueness for the normally hyperbolic operator
\(Q\) gives zero.
\end{proof}

There are nonzero homogeneous \(Q\)-solutions with general
spacelike-compact Cauchy data. They are not compactly supported
spacetime transformations of the frozen group.

Likewise, the retarded solution of
\(Q\xi=-\calF z\) need not be compact in time.
It does not prove that every configuration has a gauge-fixed
representative inside XX-A's compact spacetime group.
The gauge-fixed action is defined independently of that
unproved reachability claim.

Compact admitted diffeomorphisms preserve the asymptotic rod
labels. Their pullbacks preserve the timelike character of a
classical scalar gradient and its chosen branch locally around
the background. The gauge condition does not itself enforce a
pointwise positive quantum lapse.

%=====================================================================
\section{Retarded and advanced Green distributions}
\label{sec:greens}
%=====================================================================

\subsection{Reduced inverses}

Write \(G_{\calL}^\pm\), \(G_Q^\pm\) for the retarded and advanced
Green operators, respectively. The superscripts \(+\) and \(-\)
mean retarded and advanced throughout.

\begin{theorem}[Existence, normalization, and support]
\label{thm:greens}
There are unique continuous operators
\[
 G_{\calL}^\pm:\Dc(\mathcal M;\mathcal V)
                 \longrightarrow C^\infty(\mathcal M;\mathcal V),
 \qquad
 G_Q^\pm:\Dc(\mathcal M;\mathcal W)
                 \longrightarrow C^\infty(\mathcal M;\mathcal W)
\]
with the left and right inverse identities and
\[
 \supp(G_P^\pm f)\subset J^\pm_{\barg}(\supp f),
 \qquad P=\calL,Q.
\]
The correctly normalized reduced action inverse is
\begin{equation}
 G_{\calB}^\pm=G_{\calL}^\pm\calA^{-1},
 \qquad
 \calB G_{\calB}^\pm f=f,\qquad
 G_{\calB}^\pm\calB f=f.
 \label{eq:GB}
\end{equation}
Moreover,
\[
 (G_{\calB}^+)^\dagger=G_{\calB}^-,
 \qquad (G_Q^+)^\dagger=G_Q^-.
\]
\end{theorem}

\begin{proof}
Lemma~\ref{lem:GH} verifies global hyperbolicity.
The bundles have finite rank and smooth coefficients.
Theorem~\ref{thm:NH} and \eqref{eq:Q} verify normal
hyperbolicity. No boundary or positivity assumption is needed.
The Green theorem \cite[Corollary 3.4.3]{BGP} therefore applies.

The inverse identities in \eqref{eq:GB} follow from
\(\calB=\calA\calL\). The fiber map is local and invertible,
so it changes neither support nor wavefront relations.
Formal self-adjointness, integration by parts, and uniqueness
give the adjoint identities.
\end{proof}

The corresponding distribution kernels use the background
volume normalization:
\[
 \calB_xG_{\calB}^\pm(x,y)
            =\delta_{\barg}(x,y)\id_{\mathcal V},
 \qquad
 Q_xG_Q^\pm(x,y)=\delta_{\barg}(x,y)\id_{\mathcal W}.
\]
In the displayed coordinates,
\[
 \delta_{\barg}(x,y)
     =a(t_y)^{-3}\delta(t_x-t_y)\delta^3(\mathbf x-\mathbf y).
\]

\subsection{An explicit continuous-momentum representation}

Choose the global coframe \(\dd t,a(t)\dd x^i\).
Spatial homogeneity writes either normalized operator as
\[
 P=-\partial_t^2+a(t)^{-2}\Delta
       +B_0(t)\partial_t+B_i(t)\partial_i+C(t),
 \qquad P=\calL,Q.
\]
The matrices are obtained directly from
\eqref{eq:H}, \eqref{eq:Fadj}, \eqref{eq:Bbulk},
\eqref{eq:Amap}, and \eqref{eq:Q}; this is an exact
coordinate representation, not a new operator.

For each \(\mathbf q\in\R^3\), let \(S_P(t,t';\mathbf q)\)
be the matrix solution of
\begin{equation}
 \left[-\partial_t^2+B_0(t)\partial_t
          -a(t)^{-2}\abs{\mathbf q}^2
          +i q_iB_i(t)+C(t)\right]S_P=0,
 \quad
 S_P(t',t';\mathbf q)=0,\quad
 \partial_tS_P(t',t';\mathbf q)=-\id.
 \label{eq:mode}
\end{equation}
Then
\begin{align}
 G_P^+(t,\mathbf x;t',\mathbf y)
 &=\frac{\theta(t-t')}{(2\pi)^3a(t')^3}
      \int_{\R^3}e^{i\mathbf q\cdot(\mathbf x-\mathbf y)}
                       S_P(t,t';\mathbf q)\,\dd^3q,
 \label{eq:mode-ret}\\
 G_P^-(t,\mathbf x;t',\mathbf y)
 &=-\frac{\theta(t'-t)}{(2\pi)^3a(t')^3}
      \int_{\R^3}e^{i\mathbf q\cdot(\mathbf x-\mathbf y)}
                       S_P(t,t';\mathbf q)\,\dd^3q.
 \label{eq:mode-adv}
\end{align}
The integrals are distributional Fourier integrals.

\begin{proof}[Verification]
The time-derivative jump is \(-\id\), so the leading
\(-\partial_t^2\) produces the correctly normalized delta.
Ordinary matrix ODE uniqueness supplies the mode functions.
On compact time intervals, the usual wave-equation energy
estimate bounds the Fourier-mode solutions and their fixed
derivatives by polynomial momentum weights. Thus the formulas
define distributions on compact tests.
Their time support, inverse equations, and Green uniqueness
identify them with Theorem~\ref{thm:greens}.
\end{proof}

This is an exact continuous-momentum representation.
No finite-mode approximation or finite-register continuum
limit is used.

\subsection{Causal intertwining identities}

Define
\[
 E_{\calB}=G_{\calB}^+-G_{\calB}^-,
 \qquad E_Q=G_Q^+-G_Q^-.
\]

\begin{proposition}[Exact gauge--Green normalization]
\label{prop:intertwining}
On compact tests,
\begin{align}
 \calB K&=-r\calF^\dagger Q,
 &
 K^\dagger\calB&=-rQ\calF,
 \label{eq:operator-intertwine}\\
 G_{\calB}^\pm\calF^\dagger
       &=-r^{-1}K G_Q^\pm,
 &
 \calF G_{\calB}^\pm
       &=-r^{-1}G_Q^\pm K^\dagger.
 \label{eq:green-intertwine}
\end{align}
The same identities hold for the causal differences.
\end{proposition}

\begin{proof}
The first line follows from \(\calH K=0\),
\(\calF K=Q\), and self-adjointness.
For a compact test \(a\), both
\(G_{\calB}^\pm\calF^\dagger a\) and
\(-r^{-1}KG_Q^\pm a\) solve the same inhomogeneous
\(\calB\)-equation and have the same retarded or advanced
support. Uniqueness gives the first Green identity.
Taking adjoints gives the second.
\end{proof}

If \(\calBz=j\), then
\[
 Q\calFz=-r^{-1}K^\dagger j.
\]
In particular, invariant sources \(K^\dagger j=0\) and
zero subsidiary Cauchy data preserve \(\calFz=0\).
This is a free subsidiary-equation statement, not an
interacting lapse Ward identity.

\subsection{Complete auxiliary and ghost inverse}

The even quadratic operator on \((z,B)\) is
\begin{equation}
 \mathbb P_{\mathrm{ev}}
 =\begin{pmatrix}
       \calH&\calF^\dagger\\
       \calF&r^{-1}\id
   \end{pmatrix}.
 \label{eq:even-operator}
\end{equation}
With left/right Grassmann derivatives fixed by
\(-(\bar c,Qc)_1\), the odd matrix on \((c,\bar c)\) is
\begin{equation}
 \mathbb P_{\mathrm{odd}}
   =\begin{pmatrix}0&Q\\-Q&0\end{pmatrix}.
 \label{eq:odd-operator}
\end{equation}

\begin{theorem}[Full free-multiplet Green matrices]
\label{thm:full-green}
The complete retarded and advanced inverses are
\begin{align}
 \mathbb G_{\mathrm{ev}}^\pm
 &=\begin{pmatrix}
       G_{\calB}^\pm&K G_Q^\pm\\
       G_Q^\pm K^\dagger&0
   \end{pmatrix},
 \label{eq:even-green}\\
 \mathbb G_{\mathrm{odd}}^\pm
 &=\begin{pmatrix}
       0&-G_Q^\pm\\
       G_Q^\pm&0
   \end{pmatrix}.
 \label{eq:odd-green}
\end{align}
They are left and right inverses on compact tests, with causal
support in every component.
\end{theorem}

\begin{proof}
The Schur complement of the lower-right even block is
\(\calB=\calH-r\calF^\dagger\calF\).
Its formal block inverse is
\[
 \begin{pmatrix}
 G_{\calB}^\pm&-rG_{\calB}^\pm\calF^\dagger\\
 -r\calF G_{\calB}^\pm&
 r\id+r^2\calF G_{\calB}^\pm\calF^\dagger
 \end{pmatrix}.
\]
Proposition~\ref{prop:intertwining} converts the off-diagonal
blocks to those in \eqref{eq:even-green}.
The lower-right block equals
\[
 r\id-r\calF K G_Q^\pm
      =r\id-rQG_Q^\pm=0.
\]
Direct multiplication verifies both inverse identities.
The odd inverse is verified by matrix multiplication.
Differential operators are local, so all blocks retain the
corresponding causal support.
\end{proof}

The vanishing \(BB\) Green block is a result of the exact
calculation. It is not an omission of an auxiliary contact term.

%=====================================================================
\section{Wavefront bounds and the separation of hyperbolic and elliptic
support}
\label{sec:WF}
%=====================================================================

Let \(\WF'\) denote the kernel wavefront relation with the
second covector reversed. Define
\[
 \mathscr N
 =\{(x,\xi;y,\eta):
       (x,\xi)\sim(y,\eta),\
       \xi,\eta\ne0,\
       \barg^{-1}(\xi,\xi)=0\},
\]
where \(\sim\) means transport along a null bicharacteristic.
Let \(\mathscr N^\pm\) restrict \(x\) to the causal
future/past of \(y\), and let
\[
 \Delta^*=\{(x,\xi;x,\xi):\xi\ne0\}.
\]

\begin{proposition}[Bulk and multiplet wavefront bounds]
\label{prop:WF}
For \(P=\calL,\calB,Q\),
\begin{align}
 \WF'(G_P^\pm)&\subset\Delta^*\cup\mathscr N^\pm,
 \label{eq:greenWF}\\
 \WF'(E_P)&\subset\mathscr N.
 \label{eq:causalWF}
\end{align}
Every component of the full Green matrices has the same
corresponding upper bound.
\end{proposition}

\begin{proof}
Away from coincidence, the inverse equation is homogeneous.
The normally hyperbolic principal symbol gives null
propagation. The diagonal inverse singularity is conormal
to the diagonal; the retarded or advanced support chooses
the corresponding directed null relation. This is the
bundle-valued distinguished-parametrix construction
\cite{IS}.

In the causal difference the inhomogeneous diagonal source
cancels. Elliptic regularity excludes non-null covectors.
Multiplication by \(\calA^{-1}\), and application of the
differential operators in the full matrices, do not enlarge
the wavefront set.
\end{proof}

A retarded Green distribution can have a smooth tail inside
the light cone. The wavefront bound does not assert support
only on the null cone.

By contrast, the spatial rod kernel is
\begin{equation}
 K_{\mathrm{sp}}(\tau,\mathbf x;\tau',\mathbf y)
       =-\frac{\delta(\tau-\tau')}
                        {4\pi\abs{\mathbf x-\mathbf y}}.
 \label{eq:spkernel}
\end{equation}
At distinct spatial endpoints,
\begin{equation}
 \WF(K_{\mathrm{sp}})
 =\{(\tau,\mathbf x;\tau,\mathbf y;
             \lambda,0;-\lambda,0):
            \lambda\ne0\},
 \qquad \mathbf x\ne\mathbf y.
 \label{eq:spWF}
\end{equation}
Indeed the nonzero smooth spatial factor preserves the
wavefront set of the time delta.

\begin{corollary}[No retarded replacement of the original rods]
\label{cor:no-ret-rods}
The original rod inverse has instantaneous off-diagonal
spatial support and temporal conormal singularities.
It is not one of the causal inverses in
Theorem~\ref{thm:full-green}.
\end{corollary}

\begin{proof}
Equation \eqref{eq:spkernel} is nonzero for arbitrary distinct
spatial endpoints on the same time slice, whereas distinct
same-time spatial points are spacelike for \(\barg\).
A retarded or advanced bulk kernel vanishes there.
\end{proof}

Thus Q1 supplies the propagating field input and retains the
actual classical elliptic coordinate construction.
It does not falsify that construction by imposing causal
support on its spatial inverse.

%=====================================================================
\section{Normalized Hadamard contraction data}
\label{sec:contractions}
%=====================================================================

\subsection{Exact inverses, not positivity by terminology}

The pairings used for \(\calL\) and \(Q\) are indefinite.
An algebraic Hadamard contraction can nevertheless be
constructed. Positivity on a physical quotient is a
different problem.

For a real self-adjoint action operator \(P=\calB,Q\),
choose an exact Feynman inverse \(F_P\) such that
\[
 P F_P=F_P P=\id
\]
on compact tests. Choose the Feynman orientation for which
\(i(F_P-G_P^-)\) has positive-frequency Hadamard wavefront.
This is the convention appropriate to the action operator
whose scalar principal differential part is
\(\Box_{\barg}\), with signature \((-,+,+,+)\).
Under the opposite operator-sign convention, the familiar
formula has the opposite prefactor.

\begin{lemma}[Existence of the required exact Feynman inverses]
\label{lem:Feynman}
The choices above exist. For \(\calB\), they are obtained
from the normally hyperbolic operator \(\calL\) by
\[
 F_{\calB}=F_{\calL}\calA^{-1}.
\]
They can be symmetrized under the real formal transpose
without changing their exact inverse equations or their
Feynman wavefront class.
\end{lemma}

\begin{proof}
The operators are smooth, normally hyperbolic after the
displayed fiber normalization, and formally self-adjoint
for nondegenerate Hermitian bundle pairings.
The spacetime is globally hyperbolic.
The exact Feynman construction in
\cite[Theorems 1.2 and 1.4]{IS} therefore applies.
The required operator-sign convention is obtained by
multiplying the operator and its inverses by \(-1\) if needed.

The formula for \(F_{\calB}\) follows from
\(\calB=\calA\calL\).
Its transpose is also a left and right inverse of \(\calB\).
The Feynman relation is transpose-invariant, so averaging
the inverse and its transpose preserves the required
microlocal class.
\end{proof}

No claim of physical positivity is drawn from this lemma.
Recent analyses of the passage from Feynman inverses to
states also emphasize that positivity is an additional
condition \cite{FS}.

\subsection{Bosonic and ghost contractions}

Take real-transpose-symmetric \(F_{\calB},F_Q\), and set
\begin{equation}
 W_{P,0}=i(F_P-G_P^-),\qquad
 W_P=\frac12\left(W_{P,0}+W_{P,0}^{*}\right),
 \qquad P=\calB,Q,
 \label{eq:WP}
\end{equation}
where star is the Hermitian kernel adjoint.

Let \(\mathscr N_+\subset\mathscr N\) mean that the first
covector is future-directed. In signature \((-,+,+,+)\),
this means its metric-raised vector is future-directed.

\begin{proposition}[Exact contraction normalization]
\label{prop:W}
The bidistributions in \eqref{eq:WP} satisfy
\begin{align}
 P_xW_P&=0,\qquad W_PP=0, \label{eq:Wfield}\\
 W_P^*&=W_P,\qquad
 W_P-W_P^{\mathrm T}=iE_P, \label{eq:WCCR}\\
 \WF'(W_P)&\subset\mathscr N_+.
 \label{eq:WWF}
\end{align}
\end{proposition}

\begin{proof}
The difference of two exact inverses is a bisolution.
Since \(F_P^{\mathrm T}=F_P\) and
\((G_P^-)^{\mathrm T}=G_P^+\),
\[
 W_{P,0}-W_{P,0}^{\mathrm T}
       =i(G_P^+-G_P^-)=iE_P.
\]
Hermitian averaging preserves that identity because \(E_P\)
is real and transpose-antisymmetric.
It also preserves the bisolution equations.

The Feynman orientation fixes the wavefront of
\(i(F_P-G_P^-)\) to the positive-frequency relation.
Hermitian kernel adjunction reverses both frequency and
slot order, preserving that relation.
\end{proof}

The smooth freedom in these choices is declared input
freedom. It is not a cosmological state-selection theorem.

The auxiliary field is realized on the free equations as
\(B=-r\calF z\). Define
\begin{equation}
 \mathbb W_{\mathrm{ev}}
  =\begin{pmatrix}
       W_{\calB}&-rW_{\calB}\calF^\dagger\\
       -r\calF W_{\calB}&
                   r^2\calF W_{\calB}\calF^\dagger
    \end{pmatrix},
 \qquad
 \mathbb W_{\mathrm{odd}}
  =\begin{pmatrix}0&-W_Q\\W_Q&0\end{pmatrix}.
 \label{eq:Wfull}
\end{equation}
Mixed even--odd contractions are zero.
For ghosts use
\[
 c^*=c,\qquad \bar c^*=-\bar c.
\]
This convention makes the displayed odd contractions
compatible with the adjoint.

\begin{theorem}[Normalized complete free contraction matrix]
\label{thm:contractions}
The matrices \eqref{eq:Wfull}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item are bisolutions of the full even and odd free operators;
\item have the graded canonical antisymmetric part
      \(i(\mathbb G^+-\mathbb G^-)\);
\item obey the positive-frequency wavefront upper bound
      componentwise.
\end{enumerate}
They therefore define normalized contractions for compactly
smeared free fields.
\end{theorem}

\begin{proof}
For the even matrix, the lower row equation uses
\(B=-r\calF z\), and the upper row reduces to
\(\calB W_{\calB}=0\).
Right equations follow by adjunction.
The odd field equations follow from \(QW_Q=W_QQ=0\).

The \(zz\) antisymmetric part is \(iE_{\calB}\).
The \(zB\) and \(Bz\) parts follow from
\[
 -rE_{\calB}\calF^\dagger=KE_Q,\qquad
 -r\calF E_{\calB}=E_QK^\dagger.
\]
The \(BB\) part vanishes because
\[
 r^2\calF E_{\calB}\calF^\dagger
        =-rQ E_Q=0.
\]
For odd variables, graded antisymmetrization is ordinary
symmetrization across the two odd slots.
The upper cross entry gives
\[
 -W_Q+W_Q^{\mathrm T}=-iE_Q,
\]
and the lower cross entry gives \(iE_Q\), exactly as in
\eqref{eq:odd-green}.

Differential operators in either slot do not enlarge the
wavefront set.
\end{proof}

\subsection{What is and is not BRST-normalized}

The causal Green identities are exact.
The symmetric choice of \(W_{\calB},W_Q\) need not be
strictly invariant under the undeformed free operation
\(s_0\).

For example, the uncontracted free Ward test for
\(z(x)\bar c(y)\) has defect
\begin{equation}
 \mathcal R_{z\bar c}
       =-K W_Q-rW_{\calB}\calF^\dagger.
 \label{eq:symmetric-defect}
\end{equation}
Its causal antisymmetric part vanishes by
Proposition~\ref{prop:intertwining}.

\begin{lemma}[The remaining contraction Ward defect is smooth]
\label{lem:smooth-defect}
The kernel \eqref{eq:symmetric-defect} is a smooth
mixed bisolution. Its vanishing is not asserted.
\end{lemma}

\begin{proof}
Before Hermitian averaging, the advanced Green contributions
cancel exactly, leaving a constant multiple of
\[
 T=F_{\calB}\calF^\dagger+r^{-1}K F_Q.
\]
The operator identities imply
\[
 \calB T=0,\qquad TQ=0.
\]
Its wavefront relation is contained in the Feynman relation.
A homogeneous two-slot solution cannot have a nonzero
Feynman wavefront: propagation in the first slot along
the full null bicharacteristic would carry a directed
Feynman singularity through coincidence to the forbidden
time orientation. Non-null diagonal directions are
excluded by elliptic regularity.
Thus \(T\) is smooth. This is also the uniqueness-modulo-smooth
part of the distinguished-parametrix construction
\cite{IS}.
Adjunction and averaging preserve smoothness.
\end{proof}

An exact choice killing \eqref{eq:symmetric-defect}, and
positivity on the physical quotient, require additional
smooth normalization and state arguments.
Neither is silently declared solved.

This distinction does not invalidate the compact-test
gauge-fixed contraction construction. It limits the
interpretation of that construction.

\subsection{Compact-test field algebra}

For compact field tests, impose
\[
 [\widehat\Phi(f),\widehat\Phi(g)]_{\mathrm{gr}}
       =i\hbar\,\mathbb E(f,g)\id,
 \qquad
 \mathbb E=\mathbb G^+-\mathbb G^-,
\]
and the full free field equations.
The contractions are \(\hbar\mathbb W\).
Finite Wick polynomials in compactly smeared \emph{linear}
fields are defined by finite pairings with smooth compact
joint tests.

This does not admit an arbitrary singular or noncompact
composite kernel. In particular, it does not yet define
the native pairing of the full lapse Hessian with
\(W_{\calB}\).

%=====================================================================
\section{Complete lapse coefficient and the classical principal-symbol
check}
\label{sec:lapse}
%=====================================================================

\subsection{Raw lapse coefficients}

With
\[
 W_s=-g_s^{-1}(\dd\phi_s,\dd\phi_s)
           =W_0+sW_1+\frac{s^2}{2}W_2,
\]
the complete coefficients are
\begin{align}
 W_0&=A^2/t^2, \notag\\
 W_1&=h^{\mu\nu}v_\mu v_\nu
                    -2\barg^{\mu\nu}v_\mu\partial_\nu\varphi,
 \notag\\
 W_2&=
   \left(k^{\mu\nu}-2h^\mu{}_\rho h^{\rho\nu}\right)v_\mu v_\nu
       +4h^{\mu\nu}v_\mu\partial_\nu\varphi \notag\\
 &\quad
       -2\barg^{\mu\nu}\partial_\mu\varphi\partial_\nu\varphi
       -2\barg^{\mu\nu}v_\mu\partial_\nu\psi.
 \label{eq:Wlapse}
\end{align}
For \(N_s=N_0+sN_1+s^2N_2/2\),
\begin{equation}
 N_0=\frac tA,\qquad
 N_1=-\frac12W_0^{-3/2}W_1,\qquad
 N_2=\frac34W_0^{-5/2}W_1^2-\frac12W_0^{-3/2}W_2.
 \label{eq:Nlapse}
\end{equation}

The forward coordinate map is
\[
 \mathcal C_s(t,\mathbf x)
       =\bigl(T_s(t,\mathbf x),
                   R_s(T_s(t,\mathbf x),\mathbf x)\bigr).
\]
Its inverse \(\alpha_s=\id+s\alpha_1+s^2\alpha_2/2\)
has
\begin{align}
 \alpha_1^0&=u_1,&
 \alpha_1^i&=-Y_1^i, \notag\\
 \alpha_2^0&=u_2-2Y_1^i\partial_i u_1,&
 \alpha_2^i&=2Y_1^j\partial_jY_1^i-Y_2^i.
 \label{eq:alpha}
\end{align}
Define \(n_s=N_s\circ\alpha_s\).

\begin{theorem}[Retained classical coefficient]
\label{thm:lapse}
The complete original-rod lapse coefficients are
\begin{align}
 n_1&=N_1+\dot N_0u_1,
 \label{eq:n1}\\
 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:n2}
\end{align}
For the frozen background,
\[
 n_1=-\frac t{2A}h_{00}
        -\frac{t^2}{A^2}\dot\varphi
        -\frac t{A^2}\varphi,\qquad \ddot N_0=0.
\]
The constructed \(Y_2\) does not enter \(n_2\).
\end{theorem}

\begin{proof}
Taylor composition gives
\[
 n_2=N_2+2\alpha_1^\mu\partial_\mu N_1
       +\alpha_2^\mu\partial_\mu N_0
       +\alpha_1^\mu\alpha_1^\nu
                            \partial_\mu\partial_\nu N_0.
\]
The background \(N_0=t/A\) has no spatial derivative.
Substitution of \eqref{eq:alpha} gives \eqref{eq:n2}.
The spatial part of \(\alpha_2\), containing \(Y_2\),
multiplies \(\partial_iN_0=0\).
The first-order explicit expression follows from
\eqref{eq:clock-jet} and \eqref{eq:Nlapse}.
\end{proof}

No metric component or embedding contribution has been
discarded. In particular, \eqref{eq:gamma2} remains part
of the rod construction even though \(Y_2\) drops out of
this specific coefficient.

\subsection{Meaning of the \(\hbar\to0\) comparison}

\begin{proposition}[Classical principal-symbol comparison]
\label{prop:classical-limit}
On the compact-test regular polynomial algebra, reduction
modulo \(\hbar\) returns the classical commuting polynomial.
All Wick-ordering corrections are of positive \(\hbar\)-degree.
Consequently, any later admission of the complete frozen
lapse with this principal symbol must recover
\eqref{eq:n1}--\eqref{eq:n2}, including their rod term and the
absence of \(Y_2\).
\end{proposition}

\begin{proof}
Each contraction carries one factor of \(\hbar\).
A finite polynomial has only finitely many contractions,
and setting \(\hbar=0\) removes all of them.
The resulting commutative symbol is the chosen classical
polynomial.

The lapse coefficients are already fixed by the classical
composition calculation. Their absence or replacement cannot
be supplied by a positive-order quantum correction.
\end{proof}

This is not a numerical limit of field operators or a
state-dependent expectation theorem.
The unsmoothed nonlocal quadratic lapse is not yet an element
of the compact-test regular polynomial algebra.
Its admission remains a Q3 problem.

In particular, the existence of \(W_{\calB}\) does not make
\[
 \frac{\hbar}{2}\,
      \pair{\text{full lapse Hessian}}{W_{\calB}}
\]
a defined distributional pairing without the required
product, pullback, spatial-tail, and extension arguments.

%=====================================================================
\section{Negative controls and the same-target map}
\label{sec:map}
%=====================================================================

\subsection{Pure-trace observations and Green uniqueness}

\begin{proposition}[Pure-trace probes do not authenticate a full kernel]
\label{prop:trace}
Pure-trace evaluations alone do not determine an unrestricted
compact smooth two-metric-slot kernel. They do not, however,
create ambiguity between two retarded inverses of the fully
specified operator \(\calB\).
\end{proposition}

\begin{proof}
Choose a nonzero smooth compact trace-free symmetric tensor
\(a^{\mu\nu}\) in a coordinate patch, and set
\[
 T^{\mu\nu,\rho\sigma}(x,y)
       =a^{\mu\nu}(x)a^{\rho\sigma}(y).
\]
It annihilates all pure-trace metric tests and is nonzero.
Thus selected trace probes cannot authenticate a full kernel.

On the other hand, Theorem~\ref{thm:greens} proves uniqueness
from the complete differential equation and retarded support.
The trace-blind perturbation is not asserted to preserve
those inverse equations and support conditions.
\end{proof}

This retains the non-determination shape of XVIII-A
\cite{CheungXVIIIA} without incorrectly claiming that the
fully frozen causal inverse is nonunique.

\subsection{One-page equation map to XX-A}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.25\linewidth}YY@{}}
\toprule
XX-A object & XX-B realization & Scope retained\\
\midrule
Action and background &
\eqref{eq:action}--\eqref{eq:potential} &
Same power-law variant; \(t_->0\)\\[3pt]
Factorial metric--scalar jet &
\eqref{eq:jet} &
All metric components retained\\[3pt]
Clock calibration and positive lapse &
\eqref{eq:clock}, \eqref{eq:raw-lapse} &
Matter-scalar lapse, not silently cosmic-rescaled\\[3pt]
Clock-slice inverse &
\eqref{eq:clock-jet}--\eqref{eq:theta} &
Same local time inversion\\[3pt]
Embedding and induced metric &
\eqref{eq:induced}--\eqref{eq:gamma2} &
Mixed components and scalar-gradient terms retained\\[3pt]
Original rods and decay &
\eqref{eq:rods}--\eqref{eq:Y2} &
Same elliptic inverse; no causal replacement\\[3pt]
Unfixed coupled equations &
\eqref{eq:Gamma1}--\eqref{eq:scalar1} &
Converted to action Hessian by \eqref{eq:H}\\[3pt]
Gauge-fixing candidate &
\eqref{eq:F}--\eqref{eq:Sgf} &
Unspecified parameter fixed to \(2/M^2\)\\[3pt]
Missing coupled realization &
\eqref{eq:Bbulk}--\eqref{eq:L} &
Normal hyperbolicity proved\\[3pt]
Missing full inverses &
\eqref{eq:GB}, \eqref{eq:even-green}, \eqref{eq:odd-green} &
Exact compact-test causal matrices\\[3pt]
Missing normalized contractions &
\eqref{eq:WP}--\eqref{eq:Wfull} &
Hadamard algebraic contractions; not physical state selection\\[3pt]
Complete lapse coefficient &
\eqref{eq:Wlapse}--\eqref{eq:n2} &
Rod term included; \(Y_2\) absence proved\\[3pt]
Quantum observable and identity &
Not constructed here &
Q2--Q5 not promoted from Q1\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Source map and refused transfers}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.22\linewidth}YY@{}}
\toprule
Comparison & Shape used & Transfer refused\\
\midrule
XX-A &
Frozen variant and Q1 target &
Native Em02-E identification without a full data comparison\\[3pt]
Cheung, Em02-E &
Original rods and incomplete quantum audit &
Native covariance, state, tails, or lapse Ward identity\\[3pt]
Cheung, XIX-D &
Q1--Q5 distinction &
State positivity or nonlocal product admission from a free inverse\\[3pt]
Cheung, XVIII-A &
Trace-blind and formal-order controls &
Kernel authentication from selected contractions\\[3pt]
Cheung, XIX-C &
Finite/continuum and scheduled-support discipline &
A continuum field construction from a finite register\\[3pt]
Hack; BFHPR &
Coupled linear gauge and relational frameworks &
Their particular covariance or rods as the present construction\\[3pt]
XVI-Q and catalogue records &
Standing scope boundary &
Bulk two-point counterterms as lapse counterterms;
catalogue as a proof premise\\
\bottomrule
\end{tabularx}
\end{center}

No finite-mode or lattice approximation is introduced.
Therefore no finite-channel refinement theorem is needed
or inferred.

%=====================================================================
\section{Exact release and remaining gates}
\label{sec:release}
%=====================================================================

\begin{definition}[Q1 closure used in this paper]
\label{def:closure}
Q1 closure means:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a complete gauge-fixed free multiplet and its
      quadratic action;
\item a specified coupled operator domain on compact
      spacetime tests;
\item normalized retarded and advanced inverse matrices,
      including auxiliary and ghost sectors;
\item proved support and wavefront bounds;
\item a nonempty, explicitly normalized Hadamard
      contraction class with its remaining smooth freedom
      declared.
\end{enumerate}
It does not include physical state positivity, nonlocal
lapse-product admission, spatial-tail estimates for that
insertion, or an interacting Ward identity.
\end{definition}

If a stronger use of ``Q1'' includes a selected positive
physical state or strict invariance of the symmetric
two-point function, that stronger target is not closed
by this paper. Equation~\eqref{eq:symmetric-defect}
displays the remaining normalization issue rather than
concealing it.

\begin{theorem}[XX-B Q1 release]
\label{thm:release}
For the frozen XX-A original-rod variant and
Definition~\ref{def:closure}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the complete coupled action Hessian, gauge fermion,
      auxiliary normalization, and ghost operator are
      constructed;
\item the reduced operator is normally hyperbolic after
      the explicit invertible fiber normalization;
\item all retarded and advanced Green matrices are
      constructed with their actual action normalization,
      support, and wavefront bounds;
\item normalized Hadamard contractions for the complete
      free multiplet exist and are supplied by
      \eqref{eq:WP}--\eqref{eq:Wfull};
\item the original elliptic rods and all retained
      classical lapse contributions remain unchanged;
\item the classical principal-symbol check preserves the
      rod term and the proved absence of \(Y_2\).
\end{enumerate}
Thus Q1 is closed for the compact-test gauge-fixed free-input
target. No renormalized nonlinear relational lapse,
positive physical state, spatial-tail theorem, complete
quantum remainder, or interacting Ward identity is released.
\end{theorem}

\begin{proof}
Lemma~\ref{lem:H} and
Section~\ref{sec:gauge-fixing} prove (i).
Theorem~\ref{thm:NH} proves (ii).
Theorems~\ref{thm:greens} and~\ref{thm:full-green},
Proposition~\ref{prop:intertwining}, and
Proposition~\ref{prop:WF} prove (iii).
Lemma~\ref{lem:Feynman},
Proposition~\ref{prop:W}, and
Theorem~\ref{thm:contractions} prove (iv).
Lemma~\ref{lem:rod-realization} and
Theorem~\ref{thm:lapse} prove (v).
Proposition~\ref{prop:classical-limit} proves (vi).

None of these results establishes the missing nonlocal
distributional operations or physical state conditions.
The final sentence therefore retains their explicit gate
boundaries.
\end{proof}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.10\linewidth}p{.23\linewidth}Y@{}}
\toprule
Gate & Ending here & Exact qualification\\
\midrule
Q1 &
Constructed, free compact-test scope &
Full operator, causal inverses, wavefront bounds, and
normalized algebraic Hadamard contractions\\[4pt]
Q2 &
Partial input only &
Hadamard contraction class exists; physical positivity,
state choice, subsidiary symmetric normalization, and
lapse-specific tails remain\\[4pt]
Q3 &
Not closed &
Nonlocal lapse Hessian products, extensions, and
counterterm images not constructed\\[4pt]
Q4 &
Not closed &
Formal orders identified; no complete quantum remainder
or regulator-removal estimate\\[4pt]
Q5 &
Not closed &
Free causal identities proved; complete interacting
lapse breaking and admissible repair not evaluated\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Standing release boundary}

The original-rod variant is not identified with the VI
harmonic scalar, the VIII incidence parameter, or the X-A
density ramp \cite{CheungXIXD,CheungEm02E}.
The independently supplied narrow XVI-Q report is not a
clock certificate or a de Sitter state \cite{CheungXVIQ}.

The standing boundary is
\[
 \boxed{
 \begin{gathered}
 \text{Declared path: null},\\
 \text{Q1 free-input construction: closed in the stated scope},\\
 \text{quantum relational-lapse milestone: not yet closed},\\
 \text{external-time-free quantum evolution: not constructed},\\
 \text{inherited characteristic quantum coefficient: not admitted},\\
 \sid{O.1 = NOT_CLOSED}
       \quad\text{outside the narrow XVI-Q scope},\\
 \text{XVI-N and spatially compact Route B: not discharged},\\
 \text{XV-B downstream quantum-admission exit: not opened},\\
 \text{regulator removal, UV completeness, and compilation:
       not asserted}.
 \end{gathered}}
\]

No change to a native registered manuscript's release is
inferred merely from citing this separately frozen construction.

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

\paragraph{Self-contained prescription.}
The background, action, field normalizations, gauge condition,
spatial rod equations, and classical insertion coefficients
used in the proofs are reproduced explicitly.

\paragraph{Analytic theorem use.}
The normally hyperbolic Green and bundle-valued Feynman
theorems are declared mathematical inputs.
Their global-hyperbolicity, smoothness, principal-symbol,
and self-adjointness hypotheses are checked above.
Hack and BFHPR are not used to supply unidentified native
objects.

\paragraph{State interpretation.}
A gauge-fixed Hadamard contraction on an indefinite field
bundle is not announced as a positive physical state.
The smooth subsidiary contraction defect is displayed.
No physical expectation of the unsmoothed lapse is evaluated.

\paragraph{Source and attribution scope.}
Cheung manuscripts are cited by their supplied registered
identifiers and versions. The current Companion record
attributes its registered manuscripts to Ngo Cheung.
XX-A is cited as the supplied anonymous construction in
this manuscript exchange, not as an authenticated registered
Cheung manuscript. The separate OpenAI catalogue is not
a theorem premise.

\paragraph{Verification.}
No successful pdfLaTeX run, proof-assistant execution,
independent numerical calculation, native programme
compilation, referee decision, or journal acceptance is
asserted. The mathematical arguments are the printed
proofs and the explicitly identified analytic theorem
applications.

\paragraph{Submission metadata.}
The source is anonymous. A signed submission must supply
authorized author and affiliation information and stable
copies or archival locations for unpublished records.
No authorship is inferred from a citation.

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

\bibitem{XXA}
\emph{XX-A: Original-Rod Lapse: Classical Freeze and Quantum
Admission Contract},
anonymous draft supplied in the present manuscript exchange,
\sid{XX-A.tex}, n.d.
The independently frozen power-law original-rod variant.
No registered authorship, compilation, or independent
verification is inferred.

\bibitem{CheungEm02E}
N.~Cheung,
\emph{Em02-E},
version 1, supplied registered manuscript record,
\sid{Em02-E-v1.tex}, n.d.
Enter Calculus Companion.
Relevant supplied locators:
\sid{thm:lapse}, \sid{thm:release},
``The quantum input gate and the first contraction target,''
and ``Checks that prevent false quantum conclusions.''
Original inflaton-slice harmonic rods and incomplete
quantum audit.

\bibitem{CheungXVIIIA}
N.~Cheung,
\emph{Landmark XVIII-A: Prepared Relational Observables:
Non-Determination, Normalization Compatibility, and
Conditional Ultraviolet Matching},
version 2, supplied registered manuscript record,
\sid{XVIII-A-v2.tex}, n.d.
Enter Calculus Companion.
Relevant supplied locators:
\sid{thm:witness}, \sid{cor:smoothwitness},
\sid{prop:hbar}, \sid{thm:ward}, and \sid{thm:release}.
Pure-trace, formal-order, and actual-counterterm-image
controls.

\bibitem{CheungXIXC}
N.~Cheung,
\emph{XIX-C},
version 1, supplied registered manuscript record,
\sid{XIX-C.tex}, n.d.
Enter Calculus Companion.
Finite incidence registers, scheduled support, and
continuum non-determination.
Relevant supplied locators:
\sid{def:limit}, \sid{prop:two-limits},
and ``A local checklist, not an authentication of C1--C7.''

\bibitem{CheungXIXD}
N.~Cheung,
\emph{XIX-D},
version 1, supplied registered manuscript record,
\sid{XIX-D.tex}, n.d.
Enter Calculus Companion.
Relevant supplied locators:
\sid{def:slots}, \sid{def:target},
\sid{prop:repair}, and
``CE: the original-rod relational lapse.''
Same-target quantum-certificate discipline.

\bibitem{CheungXVIQ}
N.~Cheung,
\emph{XVI-Q},
release Q11, supplied registered manuscript record,
\sid{XVI-Q.tex}, n.d.
Enter Calculus Companion.
Relevant supplied locators:
\sid{rem:o1}, \sid{rem:qds}, \sid{rem:insertion},
and \sid{rem:package}.
Narrow bulk two-point closure report and explicit
nontransfer qualifications.

\bibitem{Hack}
T.-P.~Hack,
``Quantization of the linearised Einstein--Klein--Gordon
system on arbitrary backgrounds and the special case of
perturbations in inflation,''
\emph{Classical and Quantum Gravity}
\textbf{31} (2014), 215004.
DOI: \sid{10.1088/0264-9381/31/21/215004}.
arXiv:1403.3957.
Coupled linearized field and gauge framework; its
particular realization is not substituted for this one.

\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-observable context; no native rod or covariance
identification is imported.

\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.
Theorem 3.2.11 and Corollary 3.4.3:
Cauchy well-posedness and unique causal Green operators
for normally hyperbolic bundle operators.

\bibitem{IS}
O.~Islam and A.~Strohmaier,
``On microlocalisation and the construction of Feynman
propagators for normally hyperbolic operators,''
\emph{Communications in Analysis and Geometry}
\textbf{32} (2024), 1811--1883.
DOI: \sid{10.4310/CAG.241204020919}.
arXiv:2012.09767.
Theorems 1.2 and 1.4:
bundle-valued distinguished parametrices and exact
Hadamard bisolutions. Positivity for an indefinite
gauge-fixed pairing is not imported.

\bibitem{FS}
C.~J.~Fewster and A.~Strohmaier,
``On the construction of Hadamard states from Feynman
propagators,''
arXiv:2510.11492.
Primary research account of the additional positivity
conditions in passing from Feynman propagators to states.
No physical-state result for the present gauge theory
is inferred.

\bibitem{BFR}
R.~Brunetti, K.~Fredenhagen, and K.~Rejzner,
``Quantum gravity from the point of view of locally
covariant quantum field theory,''
\emph{Communications in Mathematical Physics}
\textbf{345} (2016), 741--779.
DOI: \sid{10.1007/s00220-016-2676-x}.
arXiv:1306.1058.
Perturbative gravity and BV context, not a native lapse
admission or ultraviolet-completeness theorem.

\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.
Local Wick-product framework; not automatic admission
of an instantaneous nonlocal rod composite.

\end{thebibliography}

\end{document}