% XX-C.tex
% Standalone pdfLaTeX source. No external files or BibTeX required.
%
% Suggested build:
%   pdflatex -interaction=nonstopmode -halt-on-error XX-C.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XX-C.tex
%
% No successful compilation or machine verification is asserted.
%
% IMPORTANT RELEASE QUALIFICATION:
% The supplied XX-B fixes a Hadamard contraction class, not a uniquely
% selected physical covariance with infrared estimates. Consequently,
% a completed native Q2 package cannot honestly be claimed from those
% data alone. This manuscript proves the available results, supplies
% explicit same-operator obstructions, and identifies the remaining
% state and one-sided covariance-tail obligations.
%
% Principal debugging corrections:
% 1. Indefiniteness of the gauge-fixed pairing is not itself a no-go
%    theorem for positivity on the physical quotient.
% 2. Smooth differences of Hadamard contractions need not decay.
% 3. Cancellation of a singularity is not inferred merely from equality
%    of wavefront-set upper bounds.
% 4. Compact-source retarded tails do not imply state-covariance tails.
% 5. Newton divergence inversion gives a dipole bound for compact or
%    appropriately decreasing sources, not for arbitrary smooth fields.
% 6. A two-rod covariance generally cannot satisfy the same r^{-2}
%    separation bound as a one-rod compact-source response: the native
%    two-rod commutator already has an unavoidable r^{-1} component.
% 7. A fixed classical coordinate pullback is distinguished from a
%    field-dependent quantum coordinate substitution.
% 8. The remaining subsidiary contraction defect is smooth, but neither
%    its exact removal nor a tail-preserving normalization is assumed.
% 9. A local subtraction parametrix does not select a physical state,
%    fix infrared behaviour, or admit the nonlinear lapse.
% 10. Q2 is not renamed closed by narrowing its requested deliverable.

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\hypersetup{
  pdftitle={XX-C: State, Subtraction, and Spatial-Tail Audit
  for the Original-Rod Lapse},
  pdfauthor={},
  pdfsubject={Gate Q2, Einstein scalar, Hadamard subtraction,
  positivity and Newton tails},
  pdfkeywords={original rods, relational lapse, Hadamard parametrix,
  physical covariance, Newton inverse, infrared obstruction}
}

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\newcommand{\C}{\mathbb C}
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\newcommand{\WF}{\operatorname{WF}}
\newcommand{\tr}{\operatorname{tr}}
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\newcommand{\Dc}{C_c^\infty}
\newcommand{\Sch}{\mathcal S}
\newcommand{\barg}{\bar g}
\newcommand{\barphi}{\bar\phi}
\newcommand{\Gsp}{\mathsf G_{\mathrm{sp}}}
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\title{\textbf{XX-C}\\[3pt]
State, Subtraction, and Spatial-Tail Audit\\
for the Original-Rod Lapse\\[3pt]
\large Local Hadamard Subtraction, Native Newton Bounds,\\
and Explicit Same-Operator Obstructions}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
On the identical power-law Einstein--scalar background and
original-rod prescription displayed in XX-B, we construct a
local bundle-valued subtraction parametrix with the actual
action-Hessian normalization. Its difference from a normalized
Feynman inverse is smooth near the diagonal. We prove uniform
dipole estimates for the original Newton inverse acting on
compact-source retarded or advanced responses, and give a
precise sufficient decreasing-source condition for analogous
one-sided covariance estimates.

The supplied free Hadamard class does not select a physical
state or impose an infrared condition. We exhibit, on the
same coupled operator, a smooth finite-rank modification
whose value on a nonzero gauge-invariant tensor observable
is negative. This excludes a positive physical state with
that exact covariance, without excluding positive
representatives elsewhere in the class. A separate
polynomial-solution modification shows that the class does
not even ensure that the decaying rod inverse is defined.

We also compute an intrinsic obstruction to a frequently
overstated tail requirement: the native two-rod commutator
has a nonzero inverse-distance spatial separation term.
Thus an inverse-square bound cannot be demanded of both
ordered two-rod covariances without changing the canonical
data. This obstruction does not remove the one-rod term in
the quadratic lapse.

The smooth subsidiary contraction defect is retained and
its exact correction problem is written explicitly. No
positive full physical state, unconditional one-sided
native covariance-tail theorem, or nonlinear lapse
contraction is released. Accordingly, the requested Q2
package remains incomplete; its boundary is proved and
not concealed by a renamed closure.
\end{abstract}

\tableofcontents

%=====================================================================
\section{Precise target and the correction to an overstated deliverable}
\label{sec:target}
%=====================================================================

\subsection{What is fixed and what remains a choice}

The predecessor XX-B supplies a complete gauge-fixed free
operator and a nonempty class of algebraic Hadamard
contractions \cite{XXB}. It does not supply one uniquely
selected physical covariance, a physical quotient state,
or global spatial estimates for an arbitrary member of
that class.

The present manuscript keeps:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the same invariant action and on-shell background;
\item the same matter-scalar lapse, clock calibration,
      positive classical branch, and factorial expansion;
\item the same slice-harmonic original rods and decaying
      spatial boundary condition;
\item the same gauge condition, action-Hessian
      normalization, causal commutator, and compact-test
      field domains.
\end{enumerate}
The equations needed for the proofs are reproduced below.
No unavailable text of XX-A is represented as having been
inspected.

\begin{convention}[Covariance class versus representative]
\label{conv:class}
A fixed representative is denoted \(W_{\calB}\).
The normalized Hadamard class is denoted
\(\mathfrak W_{\calB}\).
Membership in the class does not assert positivity,
subsidiary invariance, decay, or existence of the Newton
compositions.
A change within the class is a new normalization choice,
not a verification of the original representative.
\end{convention}

\subsection{The requested alternative needs qualification}

An indefinite gauge-fixed fiber pairing does not by itself
obstruct positivity on the physical gauge quotient.
Conversely, positivity of one scalar or tensor subsector
does not construct a state on the complete quotient.
These distinctions matter in linearized
Einstein--Klein--Gordon theory \cite{Hack}.

We explicitly decline to announce a full physical state
for the unspecified \(W_{\calB}\). Instead, we prove:
\begin{enumerate}[label=\textup{(\roman*)}]
\item an exact obstruction to positivity for an explicitly
      constructed representative in the same native class;
\item exact local subtraction;
\item unconditional compact-source rod-response tails;
\item a conditional one-sided covariance-tail theorem
      with its hypothesis stated as a genuine additional
      infrared requirement;
\item an unconditional two-rod commutator obstruction.
\end{enumerate}

This is a complete mathematical audit of the stated data.
It is not a complete native Q2 state-and-tail package.

\subsection{External analytic inputs}

We use:
\begin{enumerate}[label=\textup{A\arabic*.}]
\item Cauchy well-posedness and causal Green operators for
      normally hyperbolic bundle operators \cite{BGP}.
\item Distinguished bundle-valued Feynman parametrices and
      their uniqueness modulo smooth kernels \cite{IS}.
\item The distinction between a Feynman construction and
      the additional conditions required for a state
      \cite{FS}.
\end{enumerate}
The hypotheses needed for A1 and A2 are checked in
Sections~\ref{sec:freeze} and~\ref{sec:parametrix}.
No positivity theorem for positive-definite field bundles
is applied to the indefinite gauge-fixed bundle.

The Cheung records are version-qualified documentary
comparisons \cite{CheungXIXD,CheungXVIIIA,CheungXVIIIB,
CheungXIXA,CheungXIXC,CheungEm02E}. Their unprinted
calculations are not substituted for the native proofs.

%=====================================================================
\section{Identical frozen background, operator, and rods}
\label{sec:freeze}
%=====================================================================

\subsection{Action and background}

Use signature \((-,+,+,+)\) 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,
 \qquad M>0.
 \label{eq:action}
\end{equation}
Only compactly supported variations of the local action
density are required.

Fix \(p>1\), \(t_*>0\), and
\begin{align}
 A&=\sqrt{2p}\,M,
 &
 a(t)&=(t/t_*)^p,
 &
 \barphi(t)&=\phi_*+A\log(t/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
\[
 \mathcal M=I\times\R^3,\qquad
 I=(t_-,t_+),\qquad 0<t_-<t_+<\infty.
\]
No endpoint boundary condition is imposed on this open
slab.

Writing \(H=\dot a/a=p/t\) and \(v_\mu=\partial_\mu\barphi\),
the background equations are
\[
 3M^2H^2=\frac12\dot{\barphi}^{\,2}+V(\barphi),
 \qquad -2M^2\dot H=\dot{\barphi}^{\,2},
 \qquad \Box_{\barg}\barphi-V'(\barphi)=0.
\]
They follow directly from \(A^2=2pM^2\).

Each \(t=\mathrm{constant}\) surface is Cauchy.
Indeed, along a causal curve
\[
 \abs{\dd\mathbf x/\dd t}\le a(t)^{-1},
\]
so a causal curve with a time endpoint inside \(I\) has
a finite spatial endpoint and is extendible.
The background is therefore globally hyperbolic.

\subsection{Action Hessian and gauge normalization}

Let
\[
 \mathcal V=S^2T^*\mathcal M\oplus\underline{\R},
 \qquad \mathcal W=T^*\mathcal M,
 \qquad z=(h,\varphi).
\]
Use the nondegenerate integrated pairings
\begin{align}
 (z,w)_0
 &=\int_{\mathcal M}
       (h_{\mu\nu}j^{\mu\nu}+\varphi\omega)\dd V_{\barg},
 \notag\\
 (\xi,\eta)_1
 &=\int_{\mathcal M}\xi_\mu\eta^\mu\dd V_{\barg},
 \qquad \dd V_{\barg}=a(t)^3\dd t\,\dd^3x.
 \label{eq:pairings}
\end{align}

The invariant Euler equations are
\[
 E_{\mu\nu}=M^2G_{\mu\nu}-T_{\mu\nu},
 \qquad E_\phi=\Box_g\phi-V'(\phi).
\]
The action Hessian is defined without an unspecified
normalization by
\begin{equation}
 \calH(h,\varphi)
  =\left(-\frac12\,\delta E_{\mu\nu}[h,\varphi],
                         \delta E_\phi[h,\varphi]\right),
 \qquad
 S_2(z)=\frac12(z,\calH z)_0.
 \label{eq:H}
\end{equation}
All variations in \eqref{eq:H} are taken at
\((\barg,\barphi)\).

For completeness, the local variation rules are
\begin{align}
 \Gamma_1^\rho{}_{\mu\nu}
 &=\frac12\barg^{\rho\sigma}
   (\nabla_\mu h_{\nu\sigma}
      +\nabla_\nu h_{\mu\sigma}
      -\nabla_\sigma h_{\mu\nu}), \notag\\
 (R_1)_{\mu\nu}
 &=\nabla_\rho\Gamma_1^\rho{}_{\mu\nu}
                    -\nabla_\nu\Gamma_1^\rho{}_{\mu\rho},
 \notag\\
 R_1&=-h^{\mu\nu}\bar R_{\mu\nu}
                         +\barg^{\mu\nu}(R_1)_{\mu\nu},
 \notag\\
 (G_1)_{\mu\nu}
 &=(R_1)_{\mu\nu}
                 -\frac12h_{\mu\nu}\bar R
                 -\frac12\barg_{\mu\nu}R_1,
 \label{eq:curvature}\\
 (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}
       (-h^{\rho\sigma}v_\rho v_\sigma
          +2v^\rho\partial_\rho\varphi+2V'\varphi),
 \label{eq:matter}\\
 (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:scalar}
\end{align}

The gauge generator and its adjoint are
\begin{equation}
 K\xi=(2\nabla_{(\mu}\xi_{\nu)},v_\mu\xi^\mu),
 \qquad
 K^\dagger(h,\varphi)_\nu
       =-2\nabla^\mu h_{\mu\nu}+v_\nu\varphi.
 \label{eq:K}
\end{equation}
On the on-shell background,
\[
 \calH^\dagger=\calH,\qquad
 \calH K=0,\qquad K^\dagger\calH=0.
\]

Retain
\begin{align}
 \calF_\nu z
 &=\nabla^\mu h_{\mu\nu}
       -\frac12\nabla_\nu h^\mu{}_\mu
       -M^{-2}v_\nu\varphi, \notag\\
 r&=\frac{M^2}{2},
 &
 \calB&=\calH-r\calF^\dagger\calF,
 &
 Q&=\calF K.
 \label{eq:gauge}
\end{align}
Explicitly,
\[
 (Q\xi)_\nu
       =\Box_{\barg}\xi_\nu
             +\bar R_{\nu\rho}\xi^\rho
             -M^{-2}v_\nu v_\rho\xi^\rho.
\]
Define four-dimensional trace reversal
\[
 \mathsf T h=h-\frac12\barg\,\tr_{\barg}h,
 \qquad \mathsf T^2=\id,
\]
and
\begin{equation}
 \calA=\operatorname{diag}
          \left(\frac{M^2}{4}\mathsf T,\id\right),
 \qquad \calL=\calA^{-1}\calB.
 \label{eq:normalization}
\end{equation}

The second-derivative metric part of \(\calH\) is
\[
 \frac{M^2}{4}\mathsf T\Box_{\barg}
                       +\frac{M^2}{2}D^\dagger D,
 \qquad
 Dh=\nabla^\mu h_{\mu\nu}
                   -\frac12\nabla_\nu\tr_{\barg}h.
\]
The second term is cancelled by
\(r\calF^\dagger\calF\).
Mixed entries have at most one derivative.
Consequently
\begin{equation}
 \calL=\Box_{\barg}\id_{\mathcal V}
                  +\text{smooth lower-order terms}.
 \label{eq:hyperbolic}
\end{equation}
It is formally self-adjoint for
\((z,w)_{\calA}=(z,\calA w)_0\).
This verifies the normally hyperbolic hypotheses, but
the pairing is not positive definite.

The compact-spacetime residual condition \(Q\xi=0\)
has only the zero compactly supported solution by Cauchy
uniqueness. This does not eliminate nonzero homogeneous
solutions with general Cauchy data.

\subsection{Causal and Hadamard data}

Let \(G_{\calL}^{\ret,\adv}\) and \(G_Q^{\ret,\adv}\)
be the causal inverses supplied by the normally
hyperbolic Green theorem \cite{BGP}, and set
\begin{equation}
 G_{\calB}^{\ret,\adv}
          =G_{\calL}^{\ret,\adv}\calA^{-1},
 \qquad E_{\calB}=G_{\calB}^{\ret}-G_{\calB}^{\adv},
 \qquad E_Q=G_Q^{\ret}-G_Q^{\adv}.
 \label{eq:Green}
\end{equation}
The actual inverse equations use
\[
 \delta_{\barg}(x,y)
 =a(t_y)^{-3}\delta(t_x-t_y)\delta^3(\mathbf x-\mathbf y).
\]

The exact intertwining identities are
\begin{align}
 \calB K&=-r\calF^\dagger Q,
 &
 K^\dagger\calB&=-rQ\calF, \notag\\
 E_{\calB}\calF^\dagger&=-r^{-1}KE_Q,
 &
 \calF E_{\calB}&=-r^{-1}E_QK^\dagger.
 \label{eq:intertwine}
\end{align}
For example, the retarded version of the first Green
identity follows because its two sides solve the same
inhomogeneous equation and have the same causal support;
uniqueness then applies.

The seed contractions obey
\begin{align}
 \calB W_{\calB}=W_{\calB}\calB&=0,
 &
 QW_Q=W_QQ&=0, \notag\\
 W_P^*&=W_P,
 &
 W_P-W_P^{\mathrm T}&=iE_P,
 \qquad P=\calB,Q, \label{eq:Wdata}\\
 \WF'(W_P)&\subset\mathscr N_+.
 \label{eq:Wwavefront}
\end{align}
Here \(\WF'\) reverses the second covector, and
\(\mathscr N_+\) is the positive-frequency null
bicharacteristic relation in the XX-B convention.

In a real field trivialization,
\eqref{eq:Wdata} is equivalently
\begin{equation}
 W_P=\mu_P+\frac{i}{2}E_P,
 \qquad \mu_P\ \text{real and symmetric}.
 \label{eq:realpart}
\end{equation}
No sign condition on \(\mu_P\) is imposed.

The auxiliary and ghost contractions are reconstructed by
\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:fullW}
\end{equation}
Thus a stated bosonic normalization change also changes
the auxiliary blocks; they are not silently kept fixed.

\subsection{The unchanged original rods}

Retain
\[
 g_s=\barg+s h+\frac{s^2}{2}k,\qquad
 \phi_s=\barphi+s\varphi+\frac{s^2}{2}\psi,
 \qquad
 T_s=t_*e^{(\phi_s-\phi_*)/A}.
\]
The raw lapse is
\[
 N_s=\left(-g_s^{-1}(\dd\phi_s,\dd\phi_s)\right)^{-1/2},
\]
not the differently rescaled lapse calibrated to \(T_s\).

The first inverse clock displacement is
\[
 u_1=-\frac tA\varphi.
\]
The first induced spatial metric is
\[
 (\gamma_1)_{ij}=h_{ij}+2a^2Hu_1\delta_{ij}.
\]
Writing
\[
 C=a^{-2}\gamma_1,\qquad
 B_1=\frac12(\tr C)I-C,
\]
define the local linear rod-source matrix
\begin{equation}
 (\calC z)^{ia}
 =\frac{1}{2a^2}\delta^{ia}\delta^{jk}h_{jk}
             -a^{-2}h_{ia}-\frac pA\delta^{ia}\varphi.
 \label{eq:Crod}
\end{equation}
Then \(B_1=\calC z\).

The spatial inverse and first rod are
\begin{align}
 (\Gsp b)(\mathbf x)
 &=-\frac1{4\pi}\int_{\R^3}
                 \frac{b(\mathbf y)}{\abs{\mathbf x-\mathbf y}}
                                   \dd^3y,
 &
 \Delta\Gsp b&=b, \notag\\
 (\calR z)^a=Y_1^a
 &=-\Gsp\partial_i(\calC z)^{ia}.
 \label{eq:rodinverse}
\end{align}
Equivalently, after integration by parts,
\begin{equation}
 (\calR z)^a(\mathbf x)
 =-\frac1{4\pi}\int_{\R^3}
       \frac{x_i-y_i}{\abs{\mathbf x-\mathbf y}^3}
                  (\calC z)^{ia}(\mathbf y)\dd^3y.
 \label{eq:dipole}
\end{equation}

The underlying rods still solve
\begin{equation}
 \partial_i\left(
        \sqrt{\det\gamma_s}\,\gamma_s^{ij}\partial_jR_s^a
             \right)=0,
 \qquad
 R_s^a-x^a\longrightarrow0
       \quad(\abs{\mathbf x}\to\infty).
 \label{eq:originalrods}
\end{equation}
No lapse multiplier occurs in this equation.

%=====================================================================
\section{Physical positivity: a necessary test and an exact
same-class counterexample}
\label{sec:positivity}
%=====================================================================

\subsection{The gauge-invariant test space}

Define
\begin{equation}
 \mathcal E_{\mathrm{inv}}
 =\frac{\Ker K^\dagger\cap\Dc(\mathcal M;\mathcal V)}
             {\calH\Dc(\mathcal M;\mathcal V)}.
 \label{eq:quotient}
\end{equation}
The denominator lies in the numerator because
\(K^\dagger\calH=0\).
Further quotienting by a presymplectic radical, if
required, is not assumed to have been completed here.

The causal form induced by \(E_{\calB}\) descends to
\eqref{eq:quotient}.
Indeed, on a compact test \(a\),
\[
 E_{\calB}\calH a
   =rE_{\calB}\calF^\dagger\calF a
   =-KE_Q\calF a,
\]
which pairs to zero against a test in \(\Ker K^\dagger\).

A symmetric contraction need not descend merely because
its causal part does.
Its descent and positivity are additional conditions.

\begin{proposition}[Necessary physical positivity condition]
\label{prop:necessary}
If a positive physical state has exact two-point
restriction \(W_{\calB}\) on the invariant observables,
then
\begin{equation}
 W_{\calB}(\overline f,f)\ge0
 \quad\text{for every }f\in\Ker K^\dagger\cap\Dc.
 \label{eq:physicalpositive}
\end{equation}
Its value must also be independent of the representative
of the class in \eqref{eq:quotient}.
\end{proposition}

\begin{proof}
The field associated with \(f\) is a physical observable.
Positivity applied to its adjoint times itself gives
\eqref{eq:physicalpositive}. Quotient relations require
representative independence.
\end{proof}

Indefiniteness of \((\cdot,\cdot)_0\) is not a proof that
\eqref{eq:physicalpositive} is impossible.

\subsection{A detectable transverse-traceless solution}

\begin{lemma}[Compact tensor-sector witness]
\label{lem:TT}
There are a real smooth solution \(U=(h,0)\) and a real
compact test \(f=(j,0)\) such that
\[
 \calH U=0,\qquad \calF U=0,\qquad
 \calB U=0,\qquad K^\dagger f=0,\qquad (U,f)_0\ne0.
\]
The solution has uniformly compact spatial support on
compact time intervals.
\end{lemma}

\begin{proof}
First choose a nonzero compact smooth symmetric tensor
\(q_{ij}\) on \(\R^3\) satisfying
\[
 \partial_iq_{ij}=0,\qquad \delta^{ij}q_{ij}=0.
\]
An explicit local construction is as follows.
Put
\[
 D_{ij}=-\delta_{ij}\Delta+\partial_i\partial_j,
\]
choose a real constant symmetric matrix \(S\) and a
compact real smooth \(\chi\), and set
\[
 q_{ij}
   =D_{ik}D_{jl}(S_{kl}\chi)
           -\frac12D_{ij}D_{kl}(S_{kl}\chi).
\]
The Fourier symbol of \(D\) is
\(\abs{\mathbf k}^2\delta_{ij}-k_ik_j\).
It makes the displayed \(q\) divergence-free and
trace-free. Taking
\(S=\operatorname{diag}(1,-1,0)\) and a suitable nonzero
\(\chi\) makes \(q\ne0\).

Fix \(t_0\in I\), and solve
\begin{equation}
 \ddot\gamma_{ij}+3H\dot\gamma_{ij}
                    -a^{-2}\Delta\gamma_{ij}=0,
 \qquad
 \gamma(t_0)=q,\qquad \dot\gamma(t_0)=0.
 \label{eq:tensorequation}
\end{equation}
The scalar normally hyperbolic Cauchy theorem applies
componentwise. Divergence and trace satisfy the same
equation with zero data and hence remain zero.

Set \(h_{ij}=a^2\gamma_{ij}\), \(h_{00}=h_{0i}=0\),
and \(\varphi=0\).
Substitution in \eqref{eq:curvature}--\eqref{eq:scalar}
gives \(\calH U=0\) precisely when
\eqref{eq:tensorequation} holds.
The scalar variation vanishes because the background
spatial Hessian of \(\barphi\) is proportional to
\(\delta_{ij}\). The gauge condition vanishes by the
same trace and divergence identities.

Equivalently, this substitution reduces the quadratic
action to
\[
 S_{2,\mathrm{TT}}
 =\frac{M^2}{8}\int a^3
      \left(\dot\gamma_{ij}\dot\gamma_{ij}
              -a^{-2}\partial_k\gamma_{ij}
                         \partial_k\gamma_{ij}\right)
                         \dd t\,\dd^3x,
\]
whose Euler equation is \eqref{eq:tensorequation}.
The background Euler identities remove the terms
without derivatives. This provides a direct check of
the normalization.

Choose a nonnegative time bump \(\rho\) supported
sufficiently close to \(t_0\), and set
\[
 j_{ij}=a^2\rho(t)q_{ij}(\mathbf x),
 \qquad j_{00}=j_{0i}=0.
\]
Its spacetime covariant divergence vanishes: the spatial
components use \(\partial_iq_{ij}=0\), and the temporal
component uses \(\delta^{ij}q_{ij}=0\).
Therefore \(K^\dagger f=0\). Also
\[
 (U,f)_0
   =\int a^3\rho(t)\gamma_{ij}(t,\mathbf x)
                                  q_{ij}(\mathbf x)
                                      \dd t\,\dd^3x
\]
is strictly positive for a sufficiently small support
of \(\rho\). Finite propagation proves the support claim.
\end{proof}

This witness is constructed on the displayed native
operator. The tensor-sector discussion of Hack
\cite{Hack} is a comparison, not a replacement of this
calculation.

\subsection{Explicit nonpositive normalization}

For a real smooth solution \(U\), write
\[
 (U\otimes U)(\overline f,g)
           =(U,\overline f)_0(U,g)_0.
\]
This denotes a symmetric real smooth bisolution.

\begin{theorem}[Same-class obstruction to an exact positive
physical restriction]
\label{thm:nonpositive}
Let \(W_{\calB}\in\mathfrak W_{\calB}\), and choose
\(U,f\) as in Lemma~\ref{lem:TT}.
Put
\[
 c=(U,f)_0\ne0,\qquad q=W_{\calB}(f,f)\in\R,
 \qquad
 \lambda=-\frac{q+1}{c^2},
\]
and define
\begin{equation}
 W_{\calB}^{\mathrm{bad}}
       =W_{\calB}+\lambda\,U\otimes U.
 \label{eq:bad}
\end{equation}
Then:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(W_{\calB}^{\mathrm{bad}}\) has the same field
      equations, canonical antisymmetric part, adjoint
      condition, and Hadamard singularities;
\item \(W_{\calB}^{\mathrm{bad}}(f,f)=-1\);
\item no positive physical state can have this exact
      restriction on the invariant observable \(f\).
\end{enumerate}
The class of \(f\) in \eqref{eq:quotient} is nonzero.
\end{theorem}

\begin{proof}
The added kernel is smooth, symmetric, real, and a
bisolution of \(\calB\).
Thus it changes none of the singular or antisymmetric
data. The definition of \(\lambda\) gives
\(q+\lambda c^2=-1\).

If \(f=\calH a\) for compact \(a\), formal
self-adjointness and \(\calH U=0\) would give
\((U,f)_0=0\), a contradiction.
Hence \(f\) is a nonzero invariant observable.
Proposition~\ref{prop:necessary} excludes the exact
positive restriction.
\end{proof}

This is not a proof that the entire class lacks positive
physical representatives.
It is a proof that the class data do not authenticate
positivity of a representative.
The same distinction appears in Cheung's covariance
audit \cite{CheungXIXD}.

Because \(\calF U=0\), the modification
\eqref{eq:bad} does not change the subsidiary defect
considered in Section~\ref{sec:subsidiary}.
Its compact-support-on-time-compacts property also means
that it does not alter an already proved one-sided
asymptotic tail at spatial infinity.

%=====================================================================
\section{Native local subtraction parametrix}
\label{sec:parametrix}
%=====================================================================

\subsection{Connection form and checked hypotheses}

Every normally hyperbolic operator has a unique
connection-plus-potential representation
\[
 \calL=\Box_{\mathscr D}+\mathscr V.
\]
Here \(\mathscr D\) is a smooth bundle connection and
\(\mathscr V\) is a smooth endomorphism.
In local coordinates, they are obtained by matching the
first- and zeroth-order coefficients of
\eqref{eq:hyperbolic}; thus no new operator is introduced.

The hypotheses for the bundle parametrix construction
are:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a smooth finite-rank bundle;
\item a smooth Lorentzian metric and time orientation;
\item a scalar normally hyperbolic principal symbol;
\item smooth lower-order coefficients;
\item for the global distinguished construction,
      global hyperbolicity.
\end{enumerate}
All hold by Section~\ref{sec:freeze}.
Positivity of the fiber pairing is not required for
this local subtraction construction
\cite{BGP,IS}.

Choose a geodesically convex, causally convex
neighborhood \(O\Subset\mathcal M\).
Let \(\sigma(x,y)\) be one half of the signed squared
geodesic distance, and let \(\Delta_{\mathrm{vV}}\)
be the van Vleck determinant.

\subsection{Transport construction}

Define
\[
 U(x,y)=\Delta_{\mathrm{vV}}(x,y)^{1/2}
                      \mathscr P(x,y),
\]
where \(\mathscr P\) is \(\mathscr D\)-parallel transport
from \(y\) to \(x\).
Equivalently,
\begin{equation}
 2\sigma^{;\mu}\mathscr D_\mu U
       +(\Box_{\barg}\sigma-4)U=0,
 \qquad U(y,y)=\id.
 \label{eq:Utransport}
\end{equation}

The logarithmic coefficients are determined by
\begin{align}
 2\sigma^{;\mu}\mathscr D_\mu V_0
       +(\Box_{\barg}\sigma-2)V_0
       &=-\calL_xU,
 \label{eq:V0}\\
 2(n+1)\sigma^{;\mu}\mathscr D_\mu V_{n+1}
   +(n+1)(\Box_{\barg}\sigma+2n)V_{n+1}
       &=-\calL_xV_n,
 \qquad n\ge0.
 \label{eq:Vrecursion}
\end{align}
These are regular transport equations along the unique
geodesic in \(O\). Their smooth solutions are fixed by
regularity at coincidence.
In particular,
\[
 V_0(y,y)=-\frac12(\calL_xU)(y,y).
\]

Let
\[
 V\sim\sum_{n=0}^\infty V_n\sigma^n
\]
be a smooth asymptotic summation with cutoffs in
\(\sigma\), chosen so that the residual of the
transport construction vanishes to all orders on the
local characteristic set. Fix one such summation.
It is not claimed unique modulo smooth terms.

For the positive-frequency boundary value, use
\[
 \sigma_+(x,y)
 =\lim_{\epsilon\downarrow0}
       \left[\sigma(x,y)
           +i\epsilon(t_x-t_y)+\epsilon^2\right].
\]
The length \(\ell>0\) is a declared subtraction scale.
Set
\begin{align}
 H_{\calL}^{+}(x,y)
 &=\frac{1}{8\pi^2}
       \left[
         \frac{U(x,y)}{\sigma_+(x,y)}
          +V(x,y)\log\frac{\sigma_+(x,y)}{\ell^2}
       \right],
 \label{eq:Hplus}\\
 H_{\calL}^{\Fey}(x,y)
 &=-\frac{i}{8\pi^2}
       \left[
         \frac{U(x,y)}{\sigma(x,y)+i0}
          +V(x,y)\log\frac{\sigma(x,y)+i0}{\ell^2}
       \right].
 \label{eq:HF}
\end{align}
A cutoff equal to one near the diagonal may be added.
Only its near-diagonal germ is used.

The native action-Hessian subtraction is
\begin{equation}
 H_{\calB}^{+}=H_{\calL}^{+}\calA_y^{-1},
 \qquad
 H_{\calB}^{\Fey}=H_{\calL}^{\Fey}\calA_y^{-1}.
 \label{eq:Hnormalized}
\end{equation}
The multiplication in the source slot is essential.

The sign in \eqref{eq:HF} matches the XX-B convention
\[
 W_{\calB}=i(F_{\calB}-G_{\calB}^{\adv})
       \quad\text{up to the stated Hermitian normalization}.
\]
For the flat scalar action operator
\(\Box=-\partial_t^2+\Delta\), the corresponding
massless positive-frequency singularity is
\[
 \frac1{4\pi^2}
       \frac1{\abs{\mathbf x-\mathbf y}^2
                    -(t_x-t_y-i0)^2}.
\]
This fixes the factor and boundary-value convention.

\begin{theorem}[Smooth native subtraction remainder]
\label{thm:subtraction}
The construction \eqref{eq:Utransport}--\eqref{eq:Hnormalized}
is a local subtraction parametrix for the actual
operator \(\calB\).
For any normalized native Feynman inverse \(F_{\calB}\)
in the same distinguished class,
\[
 F_{\calB}-H_{\calB}^{\Fey}\in C^\infty
\]
near the diagonal in \(O\times O\).
Likewise,
\begin{equation}
 W_{\calB}-H_{\calB}^{+}\in C^\infty
       \quad\text{near the diagonal}.
 \label{eq:smoothW}
\end{equation}
A sufficiently long finite truncation gives a remainder
of any preassigned finite differentiability order.
\end{theorem}

\begin{proof}
Applying \(\calL_x\) to the inverse-distance and logarithmic
terms gives the transport equations
\eqref{eq:Utransport}--\eqref{eq:Vrecursion}.
They cancel the successive non-smooth coefficients.
The all-order asymptotic summation makes the residual
smooth. Multiplication by \(\calA_y^{-1}\) gives the
correct delta normalization for
\(\calB=\calA\calL\).

This is a distinguished Feynman parametrix.
Uniqueness of distinguished parametrices modulo smooth
kernels \cite{IS} gives the first difference.
The corresponding positive-frequency splitting has
the same normalized singular coefficients, giving
\eqref{eq:smoothW}.
Finite truncation leaves only sufficiently high powers
of \(\sigma\) times logarithms and their boundary values;
their differentiability increases with the number of
retained coefficients.
\end{proof}

\begin{remark}[What the subtraction does not prove]
Cancellation is proved using the transport equations,
inverse normalization, and distinguished-parametrix
uniqueness. Equality of wavefront-set upper bounds alone
would not prove it.
The theorem is local. It fixes neither a state nor
spatial tails.
The smooth remainder is not required to decrease.
\end{remark}

The same construction applies to \(Q\). Differential
application of \(\calF\) in the full contraction matrix
produces the corresponding auxiliary subtraction germs.
This statement concerns free contractions, not an
interacting lapse renormalization.

Changing \(\ell\) changes the smooth local subtraction
germ. It does not suppress a nondecaying smooth global
bisolution. Local Wick-product normalization and
global state selection are distinct obligations
\cite{HWlocal}.

%=====================================================================
\section{Newton inversion and uniform spatial-tail estimates}
\label{sec:newton}
%=====================================================================

\subsection{The exact domain of the dipole estimate}

For a compact interval \(J\Subset I\), define the
uniform Schwartz seminorms
\begin{equation}
 p_{N,m,\alpha;J}(b)
   =\sup_{t\in J,\mathbf x\in\R^3}
       (1+\abs{\mathbf x})^N
          \abs{\partial_t^m\partial_{\mathbf x}^{\alpha}
                        b(t,\mathbf x)}.
 \label{eq:Schwartz}
\end{equation}
They use coordinate components in a fixed smooth
background trivialization.

\begin{theorem}[Uniform Newton-divergence tail]
\label{thm:dipole}
Suppose \(b^i(t,\mathbf x)\) is smooth and either:
\begin{enumerate}[label=\textup{(\roman*)}]
\item has uniformly compact spatial support for \(t\in J\),
      with all required derivatives bounded; or
\item has finite seminorms \eqref{eq:Schwartz} for every
      \(N,m,\alpha\).
\end{enumerate}
Then
\[
 u=-\Gsp\partial_i b^i
\]
is a smooth decaying solution of
\(\Delta u=-\partial_i b^i\), and
\begin{equation}
 \sup_{t\in J}
   \abs{\partial_t^m\partial_{\mathbf x}^{\alpha}
                         u(t,\mathbf x)}
       \le C_{J,m,\alpha}(1+\abs{\mathbf x})^{-2-\abs{\alpha}}
 \label{eq:dipolebound}
\end{equation}
outside a fixed ball.
It is the unique smooth solution with \(u\to0\) at
spatial infinity.
\end{theorem}

\begin{proof}
Integration by parts gives
\[
 u(t,\mathbf x)
   =-\frac1{4\pi}\int
      \frac{x_i-y_i}{\abs{\mathbf x-\mathbf y}^3}
                         b^i(t,\mathbf y)\dd^3y.
\]
For compact support, the kernel and its spatial
derivatives are bounded by constants times
\(\abs{\mathbf x}^{-2-\abs{\alpha}}\) once
\(\abs{\mathbf x}\) is twice the support radius.
Time differentiation acts only on \(b\).

For Schwartz sources, split the integral into
\(\abs{\mathbf y}\le\abs{\mathbf x}/2\) and its
complement. The first part has the same kernel bound,
using a uniform \(L^1\) bound on the source.
In the complementary part, use a smooth partition
around \(\mathbf y=\mathbf x\); transfer derivatives
of the singular kernel onto the smooth rapidly
decreasing source when necessary.
The source and its derivatives decrease to arbitrary
order in that region, while the dipole kernel is
locally integrable. This bounds that part to an
arbitrarily high inverse power after increasing the
Schwartz seminorms used in the estimate.

The same argument proves smoothness and permits time
differentiation. A difference of two decaying solutions
is an entire harmonic function tending to zero.
The maximum principle gives uniqueness.
\end{proof}

The inverse-square rate is a consequence of the
divergence structure. It is not a general bound for a
Newton potential with nonzero monopole.

\subsection{Compact-source retarded and advanced responses}

\begin{theorem}[Native one-rod Green-response tail]
\label{thm:GreenTail}
Let \(f\in\Dc(\mathcal M;\mathcal V)\), and put
\[
 z^\pm=G_{\calB}^{\ret,\adv}f.
\]
For every \(J\Subset I\), \(z^\pm\) has uniformly
compact spatial support on \(J\).
Its original first rod
\[
 Y^\pm=\calR z^\pm
\]
satisfies \eqref{eq:dipolebound}, uniformly on \(J\).

Thus the distributional kernels
\(\calR_xG_{\calB}^{\ret,\adv}(x,y)\), interpreted by
testing the input slot with compact \(f\), have a
proved inverse-square output tail.
\end{theorem}

\begin{proof}
Causal support gives
\[
 \supp z^\pm\subset J^\pm_{\barg}(\supp f).
\]
If the spatial projection of \(\supp f\) is contained
in a ball of radius \(R_f\), then on \(J\) the support
radius is at most
\[
 R_f+\sup_{t\in J,\ s\in\mathrm{time}(\supp f)}
             \abs{\int_s^t a(u)^{-1}\dd u}.
\]
The supremum is finite on a compact interval inside
\(I\). The solution is smooth and its required
derivatives are bounded on that compact spatial-time
region.

The matrix \(\calC\) is local with smooth time
coefficients. Therefore \(\calCz^\pm\) has the same
support property. Apply Theorem~\ref{thm:dipole}.
\end{proof}

The rod correction destroys strict spacetime causal
support: it is instantaneous on a spatial slice.
The theorem proves a tail, not a causal replacement
of the rods.

\subsection{Second classical rod and the retained lapse term}

For classical coefficient fields with common compact
spatial support on \(J\), the second rod equation is
\[
 Y_2^a=-\Gsp\partial_i
          \left(B_2^{ia}+2B_1^{ij}\partial_jY_1^a\right).
\]
Here \(B_1,B_2\) are the induced-metric coefficients.
Both \(B_2\) and \(B_1\partial Y_1\) are compact in
space. Theorem~\ref{thm:dipole} therefore gives the
same inverse-square estimate for \(Y_2\).

The complete quadratic lapse retains
\begin{equation}
 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,
 \qquad N_0=t/A.
 \label{eq:lapse}
\end{equation}
The absence of \(Y_2\) follows from
\(\partial_iN_0=0\), not from dropping its construction.

The tail theorem controls the classical rod term on
this domain. It does not define its quantum Wick
contraction.

%=====================================================================
\section{Covariance tails, coordinate pullback, and failure of
unconditional infrared control}
\label{sec:covariancetails}
%=====================================================================

\subsection{What compact testing does supply}

For a compact field test \(f\), let
\[
 w_f(x)=W_{\calB}(x,f).
\]

\begin{lemma}[Compact testing gives smoothness, not decay]
\label{lem:smoothsmear}
The section \(w_f\) is smooth and solves
\(\calB w_f=0\).
No spatial decay follows from
\eqref{eq:Wdata}--\eqref{eq:Wwavefront} alone.
\end{lemma}

\begin{proof}
The field equation is immediate.
The wavefront relation has no point with zero
covector in the tested slot. The standard
pushforward criterion after compact testing
therefore makes \(w_f\) smooth.
Spatial decay is not a microlocal conclusion.
The counterexample below proves the final assertion
on the native operator.
\end{proof}

\begin{theorem}[Sufficient one-sided covariance-tail condition]
\label{thm:conditional}
Suppose a selected representative satisfies
\begin{equation}
 \calC w_f\in C^\infty(J;\Sch(\R^3))
 \quad\text{with all seminorms in \eqref{eq:Schwartz}
 finite}
 \label{eq:IR}
\end{equation}
for each compact test \(f\).
Then
\[
 \calR_xW_{\calB}(x,f)
\]
is well defined and obeys
\eqref{eq:dipolebound}.
The estimate is uniform for bounded test families
whenever the seminorm bounds in \eqref{eq:IR} are
uniform on those families.
\end{theorem}

\begin{proof}
Apply Theorem~\ref{thm:dipole} to
\(b^{ia}=(\calCw_f)^{ia}\).
\end{proof}

Condition \eqref{eq:IR} is a sufficient infrared
condition, not a consequence of being Hadamard.
A weaker weighted domain may suffice, but it must
be specified with the required derivative and moment
bounds before it replaces \eqref{eq:IR}.

There is no assertion that the XX-B class has been
normalized to satisfy this condition.
Theorem~\ref{thm:conditional} is therefore not the
missing unconditional native covariance theorem.

\subsection{Fixed classical clock embeddings}

Let \(\alpha\) be a fixed smooth classical inverse
coordinate map from the retained perturbation
domain. On a compact time interval, assume that
outside a ball it has the form
\[
 \alpha(\tau,\mathbf x)
   =\bigl(\tau,\mathbf x+\beta(\tau,\mathbf x)\bigr),
\]
with
\[
 \partial_\tau^m\partial_{\mathbf x}^{\gamma}\beta
       =O(\abs{\mathbf x}^{-2-\abs{\gamma}}),
\]
and that its finite derivative orders needed below
are bounded on the remaining region.
The original compact classical perturbations and
their decaying rods have precisely this asymptotic
form for a sufficiently small invertible map.

\begin{proposition}[Classical pullback preserves the proved tail]
\label{prop:pullback}
If a smooth section \(u\) satisfies
\eqref{eq:dipolebound}, then the fixed pullback
\(u\circ\alpha\), with the corresponding bounded
bundle transformation, has the same rate.
A rapidly decreasing smooth source remains rapidly
decreasing under such a fixed pullback.
\end{proposition}

\begin{proof}
For sufficiently large \(\abs{\mathbf x}\),
\[
 \abs{\mathbf x+\beta(\tau,\mathbf x)}
                 \asymp\abs{\mathbf x}.
\]
The chain rule expresses each fixed derivative of
the pullback as a finite sum of derivatives of \(u\)
times bounded derivatives of \(\alpha\).
The displayed asymptotic derivative bounds give the
same spatial rates. The argument with arbitrarily
large weights proves the Schwartz statement.
\end{proof}

A two-point distribution can be pulled back by a
fixed diffeomorphism in both slots. This is an
ordinary distributional pullback and preserves the
transformed wavefront relation.

A field-dependent quantum coordinate substitution
is not that operation. Its expansion includes
products of fields and coordinate corrections.
No such product admission is supplied by
Proposition~\ref{prop:pullback}.

\subsection{A native smooth modification outside the rod domain}

\begin{theorem}[Hadamard data do not ensure a decaying rod composition]
\label{thm:IRfailure}
The class \(\mathfrak W_{\calB}\) contains pairs of
representatives for which the one-sided decaying
rod composition cannot be simultaneously defined.
Consequently the XX-B algebraic and microlocal
conditions do not imply the Q2 spatial-tail
requirement.
\end{theorem}

\begin{proof}
Fix \(t_0\in I\).
Because all coefficients of \(\calB\) depend only
on time in a spatially homogeneous frame,
the space of spatial polynomials of degree at most
one is preserved by \(\calL\).
The equation reduces on that space to a finite
system of smooth ordinary differential equations.

Choose a real solution \(U_{\mathrm{aff}}\) with
Cauchy data
\[
 h_{ij}(t_0,\mathbf x)
       =a(t_0)^2x^1\delta_{ij},
 \qquad h_{00}=h_{0i}=\varphi=0
       \quad\text{at }t_0,
\]
and any fixed smooth polynomial time-derivative
data, for example zero coordinate derivatives.
This gives a smooth global solution on \(I\);
no spatial decay is asserted.

At \(t_0\),
\[
 (\calC U_{\mathrm{aff}})^{ia}
          =\frac12x^1\delta^{ia}.
\]
The first rod equation in component \(a=1\) would
therefore require
\begin{equation}
 \Delta Y^1=-\frac12.
 \label{eq:constantPoisson}
\end{equation}
There is no smooth solution tending to zero at
spatial infinity. Its spherical mean satisfies
\[
 \overline Y(r)=\overline Y(0)-\frac{r^2}{12},
\]
as follows by averaging the Poisson equation over
spheres.

Choose a compact test \(f\) with
\((U_{\mathrm{aff}},f)_0\ne0\). Both
\[
 W_{\calB},\qquad
 W_{\calB}+U_{\mathrm{aff}}\otimes U_{\mathrm{aff}}
\]
satisfy the same Hadamard, field-equation, adjoint,
and canonical conditions.

If both had linear decaying rod compositions after
testing with \(f\), their difference would supply a
decaying solution of \eqref{eq:constantPoisson}
times the nonzero constant \((U_{\mathrm{aff}},f)_0\).
This is impossible.
Thus at least one of the pair is outside that
decaying composition domain.
\end{proof}

This theorem does not declare the affine solution
a permissible classical rod configuration.
It is a permissible smooth ambiguity of the
compact-test algebraic covariance class.
That distinction is exactly the missing domain
condition.

%=====================================================================
\section{An intrinsic inverse-distance obstruction for two rods}
\label{sec:tworod}
%=====================================================================

\subsection{Why the requested rate must distinguish one and two rods}

The quadratic lapse term in \eqref{eq:lapse} contains
one rod multiplied by a local derivative.
A covariance of two first rods is a different
composition. It involves two Newton inverses.

It would be incorrect to require the inverse-square
dipole estimate of Theorem~\ref{thm:GreenTail} for
both ordered two-rod covariances. Their commutator
already has a slower native tail.

This can be proved without selecting any state.

\subsection{Translation-invariant causal kernel}

Use the orthonormal coframe
\(\dd t,a(t)\dd x^i\), and write
\[
 \widehat h_{ij}=a^{-2}h_{ij}.
\]
In these field components the rod-source matrix is
time independent:
\[
 (\calCz)^{ia}
      =\frac12\delta^{ia}\delta^{jk}\widehat h_{jk}
                     -\widehat h_{ia}
                     -\frac pA\delta^{ia}\varphi.
\]

Spatial translation invariance and uniqueness make
the causal kernel translation invariant:
\[
 E_{\calB}(t,\mathbf x;t',\mathbf y)
                     =e(t,t';\mathbf x-\mathbf y).
\]
Its restriction to fixed \(t,t'\) exists as a spatial
distribution: null wavefront covectors have nonzero
spatial parts and do not meet the conormal bundle
of that time restriction.

For fixed \(t,t'\), the distribution \(e\) is
supported in the ball
\[
 \abs{\mathbf z}
       \le\abs{\int_{t'}^t a(u)^{-1}\dd u}.
\]
Let
\[
 m(t,t')=\int_{\R^3}e(t,t';\mathbf z)\dd^3z.
\]
This is a well-defined finite-dimensional matrix,
smooth in the two times.
The Cauchy jump normalization gives
\begin{equation}
 m(t',t')=0,\qquad
 \partial_t m(t,t')\big|_{t=t'}
                    =-a(t')^{-3}\calA^{-1}.
 \label{eq:jump}
\end{equation}
The minus sign comes from the principal differential
part \(-\partial_t^2\) of \(\Box_{\barg}\).

\subsection{The two-Newton kernel}

In three dimensions define
\begin{equation}
 \Phi(\mathbf z)=-\frac{\abs{\mathbf z}}{8\pi},
 \qquad \Delta^2\Phi=\delta,
 \qquad
 K_{ij}(\mathbf z)=-\partial_i\partial_j\Phi
       =\frac{\delta_{ij}-n_in_j}{8\pi\abs{\mathbf z}},
 \quad n=\frac{\mathbf z}{\abs{\mathbf z}}.
 \label{eq:biharmonic}
\end{equation}
The Fourier multiplier of \(K_{ij}\) is
\(k_ik_j/\abs{\mathbf k}^4\).
It is locally integrable at \(\mathbf k=0\).
No infrared subtraction is needed for this multiplier.

The native double-rod causal distribution is
\begin{equation}
 E_{RR}^{ab}(t,t';\mathbf z)
       =K_{ij}*
           \left(\calC^{ia}\,
                       e(t,t';\cdot)\,
                       (\calC^{jb})^\dagger\right)(\mathbf z).
 \label{eq:ERR}
\end{equation}
The convolution is defined because its second factor
is compactly supported in space. This prescription
is the two-rod composition of the canonical kernel.

For a compact distribution \(d\), Taylor expansion
of \(K_{ij}(\mathbf z-\mathbf y)\) outside its support
gives
\[
 K_{ij}*d(\mathbf z)
       =K_{ij}(\mathbf z)\pair{d}{1}
                         +O(\abs{\mathbf z}^{-2}).
\]
The statement holds for distributions of finite order:
derivatives of the Taylor remainder on the compact
support retain the inverse-square bound.

\subsection{The nonzero leading coefficient}

\begin{lemma}[Rod-source normalization at zero spatial frequency]
\label{lem:sourcecoefficient}
For \(\mathbf k\ne0\),
\begin{equation}
 k_i k_j\,\calC^{ia}\calA^{-1}
                           (\calC^{jb})^\dagger
       =\frac{2}{M^2}\abs{\mathbf k}^2\delta_{ab}
             +\frac{p+1}{2M^2}k_ak_b.
 \label{eq:rodmatrix}
\end{equation}
\end{lemma}

\begin{proof}
The metric functional contracted with \(k_i\) has
the spatial symmetric coefficient
\[
 B^a_{kl}
      =\frac12k_a\delta_{kl}
                      -k_{(k}\delta_{l)a}.
\]
Its spatial trace is \(k_a/2\), and
\[
 B^a:B^b
      =\frac12\abs{\mathbf k}^2\delta_{ab}
                         +\frac14k_ak_b.
\]
Four-dimensional trace reversal in the metric
inverse subtracts one half of the product of these
traces. Therefore the metric contribution is
\[
 \frac4{M^2}
   \left(\frac12\abs{\mathbf k}^2\delta_{ab}
                   +\frac18k_ak_b\right).
\]
The scalar contribution is
\[
 \frac{p^2}{A^2}k_ak_b
                      =\frac{p}{2M^2}k_ak_b.
\]
Adding them gives \eqref{eq:rodmatrix}.
The formal adjoint here uses the field pairing,
not an independently substituted positive metric.
\end{proof}

\begin{theorem}[Native two-rod inverse-distance tail]
\label{thm:tworod}
Let \(t'=t_0\) and \(t=t_0+\delta\), with
\(\delta\) sufficiently small and nonzero.
For large \(r=\abs{\mathbf z}\),
\begin{align}
 E_{RR}^{ab}(t_0+\delta,t_0;\mathbf z)
 &=-\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:twoasymptotic}
\end{align}
In particular, the native two-rod commutator is not
\(O(r^{-2})\) for all nearby unequal times.
\end{theorem}

\begin{proof}
Equation \eqref{eq:jump} gives
\[
 m(t_0+\delta,t_0)
       =-\delta a(t_0)^{-3}\calA^{-1}
                                      +O(\delta^2).
\]
Insert this moment into the large-distance expansion
of \eqref{eq:ERR}.

By Lemma~\ref{lem:sourcecoefficient}, the leading
Fourier multiplier is
\[
 -\frac{\delta}{a(t_0)^3}
 \left(
   \frac2{M^2}\frac{\delta_{ab}}{\abs{\mathbf k}^2}
       +\frac{p+1}{2M^2}
                    \frac{k_ak_b}{\abs{\mathbf k}^4}
 \right).
\]
Using
\[
 \mathcal F^{-1}\left(\abs{\mathbf k}^{-2}\right)
       =\frac1{4\pi r},
 \qquad
 \mathcal F^{-1}\left(k_ak_b\abs{\mathbf k}^{-4}\right)
       =\frac{\delta_{ab}-n_an_b}{8\pi r}
\]
gives \eqref{eq:twoasymptotic}.

For example, choose \(n\) parallel to coordinate
axis \(a\) and take component \(aa\).
The coefficient in square brackets is then
\((p+9)-(p+1)=8\).
For small nonzero \(\delta\), the
\(O(\delta^2)\) term cannot cancel that leading
coefficient. Hence the inverse-distance tail is
nonzero.
\end{proof}

\begin{corollary}[Obstruction to a symmetric inverse-square
two-rod covariance requirement]
\label{cor:tailobstruction}
Suppose a two-rod covariance is defined, has the
native canonical difference
\[
 W_{RR}(x,y)-W_{RR}(y,x)^{\mathrm T}
                         =iE_{RR}(x,y),
\]
and both ordered terms satisfy an inverse-square
spatial separation bound, uniformly for times in
a compact interval containing the times used in
Theorem~\ref{thm:tworod}.
These requirements are inconsistent.
\end{corollary}

\begin{proof}
Their difference would be \(O(r^{-2})\), contradicting
Theorem~\ref{thm:tworod}.
\end{proof}

The obstruction also survives compact spatial
testing of an endpoint with nonzero integral and
sufficiently small separated time testing.
The leading inverse-distance moment remains
nonzero under those tests.

\begin{remark}[Correct interpretation]
An inverse-distance two-rod correlation still tends
to zero. It does not by itself violate a suitable
weak quantum boundary condition.
It does invalidate a universal inverse-square
two-sided claim.
The classical rod coefficients, the compact-source
one-rod response, and the one-rod term in the
quadratic lapse are not identified with this
two-sided covariance.
\end{remark}

No massive-scalar benchmark is used in this proof.
The massive stress response of XVIII-B is a
different target \cite{CheungXVIIIB}.

%=====================================================================
\section{Compatibility with the original rods}
\label{sec:compatibility}
%=====================================================================

\begin{theorem}[Exact prescription compatibility]
\label{thm:compatibility}
The subtraction, positivity tests, and tail constructions
above do not alter:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the original slice-elliptic equation
      \eqref{eq:originalrods};
\item its decaying spatial boundary condition on the
      domains where that condition is proved;
\item the clock calibration or the matter-scalar lapse
      normalization;
\item the distinction between a bulk causal inverse
      and the spatial Newton inverse.
\end{enumerate}
Theorems~\ref{thm:dipole} and~\ref{thm:GreenTail}
prove compatibility on their actual domains.
Theorem~\ref{thm:IRfailure} identifies contractions
for which that compatibility cannot be extended
without an additional infrared choice.
\end{theorem}

\begin{proof}
Every rod expression is obtained from
\(\Delta\Gsp=\id\) and the unchanged source
\(\calC\) derived from the induced metric.
No lapse multiplier is introduced.
Uniqueness in Theorem~\ref{thm:dipole} enforces the
same decaying boundary condition.

The local parametrix acts on the propagating field
operator \(\calB\), not on the rod equation.
The fixed classical pullback uses the already
frozen clock calibration.
A choice of smooth covariance correction changes
a free contraction but not the defining classical
elliptic equation.

The affine obstruction is precisely a failure to
meet that unchanged equation with its unchanged
boundary condition. It is not repaired by assigning
a retarded spacetime inverse to the rods.
\end{proof}

The corresponding original-versus-modified-rod
distinction is explicitly retained in the supplied
Cheung records \cite{CheungXIXA,CheungEm02E}.
Those records motivate the compatibility test;
the proof here uses the printed native equations.

%=====================================================================
\section{Subsidiary contraction defect and exact normalization problem}
\label{sec:subsidiary}
%=====================================================================

Retain the mixed free contraction defect
\begin{equation}
 \mathscr D=-KW_Q-rW_{\calB}\calF^\dagger.
 \label{eq:defect}
\end{equation}

\begin{proposition}[Smoothness of the retained defect]
\label{prop:smoothdefect}
The defect \(\mathscr D\) is a smooth mixed
bisolution:
\[
 \calB\mathscr D=0,\qquad \mathscr DQ=0.
\]
Its vanishing is not implied by the canonical
intertwining identities.
\end{proposition}

\begin{proof}
Using \eqref{eq:realpart}, its imaginary part is
\[
 -\frac12KE_Q-\frac r2E_{\calB}\calF^\dagger=0
\]
by \eqref{eq:intertwine}. Hence \(\mathscr D\)
is real in a real trivialization.

Differential application to the contractions gives
\[
 \WF'(\mathscr D)\subset\mathscr N_+.
\]
A real distribution has sign-symmetric wavefront
set. Consequently its wavefront relation is also
contained in \(\mathscr N_-\).
The two nonzero frequency cones are disjoint.
Thus \(\WF'(\mathscr D)=\varnothing\), proving
smoothness.

The bisolution equations follow from
\(\calB K=-r\calF^\dagger Q\),
\(\calF^\dagger Q=-r^{-1}\calB K\), and the
field equations for the contractions.
The causal identities remove the imaginary part,
not the remaining smooth real part.
\end{proof}

Let \(\mathscr S_{\calB}\) and \(\mathscr S_Q\)
be declared spaces of real symmetric smooth
bisolutions, with any desired tail and quotient
conditions included in their definition.
For \(S_{\calB}\in\mathscr S_{\calB}\) and
\(S_Q\in\mathscr S_Q\), define
\begin{equation}
 \mathscr T(S_{\calB},S_Q)
       =KS_Q+rS_{\calB}\calF^\dagger.
 \label{eq:repairmap}
\end{equation}

\begin{proposition}[Exact subsidiary repair criterion]
\label{prop:repair}
After the normalization changes
\[
 W_{\calB}\mapsto W_{\calB}+S_{\calB},
 \qquad W_Q\mapsto W_Q+S_Q,
\]
the defect becomes
\[
 \mathscr D-\mathscr T(S_{\calB},S_Q).
\]
It can be removed in the declared correction domain
if and only if
\begin{equation}
 \mathscr D\in\im\mathscr T.
 \label{eq:actualimage}
\end{equation}
Membership in the closure of that image is not
an exact solution.
\end{proposition}

\begin{proof}
Substitution in \eqref{eq:defect} gives the displayed
affine law. Exact removal is exactly
\eqref{eq:actualimage}.
\end{proof}

This criterion is written for the actual native
maps. It has the same algebraic shape as the
actual-image criteria recorded by Cheung
\cite{CheungXVIIIA,CheungXIXD}, but no solution is
imported from those comparisons.

\paragraph{Decision in XX-C.}
The defect is left as part of the output.
No pair of corrections satisfying all of positivity,
quotient descent, required infrared control, and
\eqref{eq:actualimage} is constructed.
Its smoothness is not announced as its cancellation.

The negative tensor-sector modification in
Theorem~\ref{thm:nonpositive} leaves this defect
unchanged because \(\calF U=0\).

%=====================================================================
\section{Exact release, equation map, and remaining Q2 obligations}
\label{sec:release}
%=====================================================================

\begin{theorem}[XX-C mathematical release]
\label{thm:release}
On the identical frozen XX-B target, this manuscript
proves:
\begin{enumerate}[label=\textup{(\roman*)}]
\item an explicit same-class normalization that is
      negative on a nonzero gauge-invariant compact
      tensor observable;
\item a correctly normalized local subtraction
      parametrix with smooth near-diagonal Feynman
      and Hadamard remainders;
\item uniform inverse-square original-rod tails for
      compact-source retarded and advanced responses
      and for the stated classical second-rod domain;
\item a sufficient uniform infrared hypothesis for
      one-sided covariance tails and preservation
      under fixed admissible classical pullback;
\item a same-operator smooth covariance ambiguity
      for which the decaying rod domain cannot be
      inferred;
\item an unavoidable inverse-distance term in the
      native two-rod commutator;
\item compatibility with the unchanged original
      elliptic rod equation on the proved domains;
\item smoothness of the subsidiary contraction
      defect and its exact actual-image correction
      criterion.
\end{enumerate}
These results do not select a positive full physical
state or prove the additional infrared condition
for a chosen native representative.
Accordingly, the requested Q2 state-and-tail
package is not closed.
\end{theorem}

\begin{proof}
Item (i) is Theorem~\ref{thm:nonpositive}.
Item (ii) is Theorem~\ref{thm:subtraction}.
Item (iii) follows from
Theorems~\ref{thm:dipole} and~\ref{thm:GreenTail}
and the second-rod source calculation.
Item (iv) is
Theorem~\ref{thm:conditional} and
Proposition~\ref{prop:pullback}.
Item (v) is Theorem~\ref{thm:IRfailure}.
Item (vi) is Theorem~\ref{thm:tworod} and
Corollary~\ref{cor:tailobstruction}.
Item (vii) is Theorem~\ref{thm:compatibility}.
Item (viii) is
Propositions~\ref{prop:smoothdefect}
and~\ref{prop:repair}.

Neither the physical state nor the unconditional
one-sided covariance-tail hypothesis is supplied
by these results. The final statement therefore
retains the actual deliverable boundary.
\end{proof}

\subsection{One-page map to the frozen equations}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.25\linewidth}YY@{}}
\toprule
Frozen object & XX-C realization & Exact status\\
\midrule
Power-law background and potential &
\eqref{eq:action}--\eqref{eq:background} &
Same on-shell slab, strictly away from \(t=0\)\\[3pt]
Action Hessian and gauge parameter &
\eqref{eq:H}--\eqref{eq:normalization} &
Same \(-1/2\) metric Euler factor and \(r=M^2/2\)\\[3pt]
Native causal normalization &
\eqref{eq:Green}--\eqref{eq:intertwine} &
Retained, not replaced by a scalar benchmark\\[3pt]
Hadamard contraction class &
\eqref{eq:Wdata}--\eqref{eq:fullW} &
Not renamed a physical state\\[3pt]
Original induced-metric rod source &
\eqref{eq:Crod} &
Clock displacement retained; no lapse weight\\[3pt]
Newton inverse and boundary &
\eqref{eq:rodinverse}--\eqref{eq:originalrods} &
Unchanged elliptic prescription\\[3pt]
Native local subtraction &
\eqref{eq:Hplus}--\eqref{eq:Hnormalized} &
Smooth diagonal remainder proved\\[3pt]
Compact-source one-rod tails &
Theorem~\ref{thm:GreenTail} &
Inverse-square bounds proved\\[3pt]
Selected covariance one-rod tails &
\eqref{eq:IR} &
Sufficient hypothesis; not verified for a selected representative\\[3pt]
Two-rod canonical tail &
\eqref{eq:twoasymptotic} &
Nonzero inverse-distance term proved\\[3pt]
Quadratic lapse rod term &
\eqref{eq:lapse} &
Not erased by a state or subtraction choice\\[3pt]
Smooth subsidiary defect &
\eqref{eq:defect}--\eqref{eq:actualimage} &
Left explicit; exact repair not constructed\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Gate status}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.28\linewidth}p{.19\linewidth}Y@{}}
\toprule
Q2 obligation & Ending here & Qualification\\
\midrule
Positive physical quotient state &
Not selected &
No universal obstruction from indefiniteness is claimed\\[3pt]
Explicit nonpositive representative &
Constructed &
Negative on a nonzero invariant compact observable\\[3pt]
Native algebraic covariance &
Retained as a class &
Smooth global normalization remains a choice\\[3pt]
Local subtraction parametrix &
Constructed &
Actual action normalization; smooth diagonal difference\\[3pt]
Compact-source rod Green tails &
Proved &
One output rod, compact tested input, compact time intervals\\[3pt]
Selected native one-sided covariance tails &
Not unconditional &
Requires verified infrared and domain estimates\\[3pt]
Universal inverse-square two-rod tails &
Obstructed &
Native canonical kernel already has an inverse-distance term\\[3pt]
Original-rod compatibility &
Proved on stated domains &
No reweighting or hyperbolic substitution\\[3pt]
Subsidiary contraction defect &
Smooth, retained &
Actual-image repair remains unsolved\\[3pt]
Requested complete Q2 package &
Not closed &
No artificial narrowing of its deliverable\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Minimal data needed for a genuine Q2 completion}

A successor must provide:
\begin{enumerate}[label=\textup{(\roman*)}]
\item a specific covariance representative, or specific
      distributional Cauchy covariance data determining it;
\item a constructed physical algebra and proof of
      quotient descent and positivity on that algebra,
      or a precisely stated nonpositive alternative;
\item infrared estimates proving the needed native
      one-rod compositions on a stated noncompact
      test domain;
\item a separate admissible treatment of two-sided rod
      covariances allowing the canonical inverse-distance
      component, if those covariances are needed;
\item either an exact subsidiary correction satisfying
      the state and tail restrictions or an explicit
      retained-defect interpretation;
\item the additional product and extension arguments
      required for the nonlinear lapse.
\end{enumerate}

The last item belongs to Q3. It is not supplied by
the local free subtraction theorem.

\subsection{Nontransfer and standing boundaries}

The supplied XVIII-B benchmark quantizes only a
free massive scalar on classical Minkowski space;
its stress kernel is not the present coupled
cosmological lapse \cite{CheungXVIIIB}.
Pure-trace probes do not authenticate a complete
covariance or insertion kernel
\cite{CheungXVIIIA}.
Finite scheduled-support data do not replace
continuum infrared analysis
\cite{CheungXIXC}.
No finite approximation is introduced here.

The standing boundary is
\[
 \boxed{
 \begin{gathered}
 \text{Q2 state-and-native-tail package: not closed},\\
 \text{positive full physical state: not selected},\\
 \text{nonlinear lapse contraction: not admitted},\\
 \text{external-time-free quantum evolution: not constructed},\\
 \text{regulator removal and ultraviolet completion:
 not asserted},\\
 \text{inherited characteristic coefficient:
 not admitted},\\
 \text{XV-B downstream admission exit: not opened},\\
 \text{XVI-N and spatially compact Route B:
 not discharged},\\
 \sid{O.1 = NOT_CLOSED}
 \quad\text{outside any separately reported narrow scope}.
 \end{gathered}}
\]
These are limits of the present release, not
additional universal nonexistence theorems.

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

\paragraph{Mathematical completeness of the stated results.}
Each released tail theorem specifies its source domain.
Each obstruction uses the same native operator.
The status of a conditional theorem is not upgraded
to an unconditional native estimate.

\paragraph{Physical state interpretation.}
The explicit negative representative excludes an
exact positive restriction for that representative.
It does not exclude every physical state in the
Hadamard class. Ghost or unphysical gauge components
are not subjected to an unrelated scalar positivity
test.

\paragraph{Local versus global data.}
The subtraction parametrix fixes a local singular
germ. It does not control the smooth global
remainder. Infrared behaviour and quotient positivity
require separate data.

\paragraph{Classical versus quantum coordinates.}
The fixed classical pullback theorem is not a
construction of a field-dependent quantum coordinate
operator or its Wick products.

\paragraph{Source scope and attribution.}
The Cheung references are cited under the supplied
N.~Cheung attribution and registered identifiers.
XX-A and XX-B are drafts in this manuscript exchange;
citation does not establish their authorship or
registered status. The separate OpenAI catalogue is
not a proof premise.

\paragraph{Verification and submission metadata.}
The document uses one internal bibliography and
standard pdfLaTeX packages, without external input
files. No successful compilation, proof-assistant
check, independent numerical execution, referee
decision, or acceptance is asserted.
A signed submission must supply authorized author
and affiliation metadata and stable copies of the
unpublished records.

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

\bibitem{XXB}
\emph{XX-B: Coupled Gauge-Fixed Free Input for the
Original-Rod Lapse: A Q1 Construction on the Frozen
XX-A Background},
anonymous draft supplied in the present manuscript
exchange, \sid{XX-B.tex}, n.d.
Relevant displayed objects:
the action Hessian, gauge normalization, causal
matrices, algebraic Hadamard class, and smooth
subsidiary contraction defect.
No registered authorship or independent verification
is inferred.

\bibitem{XXA}
\emph{XX-A},
classical-freeze draft referred to in the supplied
XX-B manuscript, n.d.
The equations used here are the explicitly
reproduced XX-B freeze. An unavailable complete
XX-A source is not represented as having been
inspected.

\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},
and ``The quantum input gate and the first
contraction target.''
Original inflaton-slice harmonic rods and incomplete
quantum audit.

\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:
``CE: the original-rod relational lapse,''
\sid{prop:state}, \sid{prop:repair},
and \sid{thm:main}.
Covariance non-determination, positivity
qualifications, and actual-image repair.

\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{thm:ward}, and \sid{thm:release}.
Selected contractions do not authenticate full
kernels; exact repair uses the actual image.

\bibitem{CheungXVIIIB}
N.~Cheung,
\emph{Landmark XVIII-B},
version 2, supplied registered manuscript record,
\sid{XVIII-B-v2.tex}, n.d.
Enter Calculus Companion.
Relevant supplied locators:
``Evaluated target and proof boundary,''
\sid{thm:spectral}, and \sid{thm:release}.
A free massive-scalar stress-response benchmark
on classical Minkowski space, not the coupled
relational lapse.

\bibitem{CheungXIXA}
N.~Cheung,
\emph{XIX-A},
version 1, supplied registered manuscript record,
\sid{XIX-A.tex}, n.d.
Enter Calculus Companion.
Relevant supplied locators:
\sid{eq:original-rods},
\sid{thm:rod-detection},
\sid{lem:elliptic-hyperbolic},
and \sid{thm:main}.
Original versus modified rods and clock
non-identification.

\bibitem{CheungXIXC}
N.~Cheung,
\emph{XIX-C},
version 1, supplied registered manuscript record,
\sid{XIX-C.tex}, n.d.
Enter Calculus Companion.
Relevant supplied locators:
\sid{def:limit}, \sid{prop:two-limits},
and \sid{thm:main}.
Finite scheduled support and continuum
non-determination, not native covariance estimates.

\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.
Gauge-invariant coupled-field framework.
Its scalar and tensor subsectors are not silently
identified with the complete local observable
space.

\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 rod, covariance, or state 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.
Normally hyperbolic Cauchy theory, fundamental
solutions, and local Hadamard/Riesz constructions.

\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.
Bundle-valued distinguished parametrices and
Hadamard bisolutions.
The local microlocal construction is used;
physical gauge-quotient positivity is not inferred.

\bibitem{FS}
C.~J.~Fewster and A.~Strohmaier,
``On the construction of Hadamard states from
Feynman propagators,''
arXiv:2510.11492, version 2,
30 April 2026.
Additional positivity and real-structure
requirements in passing from Feynman propagators
to states. Their positive-definite bundle
state theorem is not applied to the indefinite
gauge-fixed bundle in this paper.

\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 composite-field normalization, distinct
from global state and infrared selection.

\end{thebibliography}

\end{document}