% XX-E.tex
% One self-contained pdfLaTeX source; no external inputs or BibTeX.
%
% Suggested build:
%   pdflatex -interaction=nonstopmode -halt-on-error XX-E.tex
%   pdflatex -interaction=nonstopmode -halt-on-error XX-E.tex
%
% No successful compilation, machine verification, or independent
% symbolic or numerical execution is asserted.
%
% IMPORTANT RELEASE QUALIFICATION:
% The supplied XX-A--XX-D package does not contain the native Q3
% successor package assumed by the revised Q4--Q5 plan. Accordingly,
% an actual native quantum breaking term and its admissible cancellation
% cannot honestly be announced as calculated here.
%
% This manuscript supplies new, self-contained proofs of:
%   (i) the coupled classical Ward calculation, including the rod term;
%  (ii) the exact compact-source-cutoff Ward residual;
% (iii) the distinction between the full compact gauge group and its
%       trivial gauge-fixed compact residual subgroup;
%  (iv) failure of ordinary compact gauge translations to act on the
%       gauge-fixed free equation quotient;
%   (v) the limitation of centrally parametrized second jets for a
%       quantum Ward operation;
%  (vi) explicit Newton images of spatial contact distributions and
%       Ward images of a declared comparison counterterm family;
% (vii) simultaneous actual-image criteria for insertion and subsidiary
%       repair, only after all application data have been constructed;
%(viii) an identical-prescription escaping-source example showing that
%       local convergence does not control Newton composition.
%
% Native Q4 and Q5 remain unentered/unclosed, not universally obstructed.
%
% Debugging corrections to XX-D and the revised plan:
% 1. XX-A uses fixed label density dt dx, whereas XX-D uses a^3 dt dx.
%    The same smeared target is obtained only with f_D = a^{-3} f_A.
% 2. Second jets must transform in the coupled classical Ward identity.
%    Holding them fixed generally destroys that identity.
% 3. A fixed-background scalar Wick correction has zero ordinary field
%    variation; that calculation is not an interacting quantum Ward test.
% 4. Central second jets do not automatically admit a quantum gauge
%    derivation because their proposed gauge images need not be central.
% 5. Residual gauges satisfying Q xi = 0 and compact spacetime support
%    are zero; invariance under them is a vacuous certificate.
% 6. The nonlinear compact geometric gauge group is not automatically
%    an automorphism group of the gauge-fixed free equation quotient.
% 7. After a common s^2 truncation, a product of two lapse insertions
%    needs only their compact linear-linear contraction. Two-rod terms
%    occur at higher total s-order or in coefficientwise products.
% 8. A contact's Newton image is calculated before any diagonal readout;
%    its resulting singular kernel cannot be evaluated at coincidence.
% 9. An arbitrary choice C = -breaking does not establish admissibility.
% 10. The retrieved XIV characteristic-graph continuation is not the
%     cosmological original-rod target and is not substituted for it.

\documentclass[11pt,a4paper]{article}

\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage[margin=25mm]{geometry}
\usepackage{amsmath,amssymb,amsthm,mathtools}
\usepackage{booktabs,tabularx,longtable,array}
\usepackage{enumitem}
\usepackage{microtype}
\usepackage{xurl}
\usepackage[hidelinks]{hyperref}

\hypersetup{
  pdftitle={XX-E: Ward-Domain Audit and Newton Counterterm Images
  for the Original-Rod Lapse},
  pdfauthor={},
  pdfsubject={Track A; prerequisite audit for Gates Q4 and Q5}
}

\setlength{\emergencystretch}{3em}
\setlength{\parskip}{3pt}
\numberwithin{equation}{section}
\setlist[enumerate]{leftmargin=*,itemsep=3pt}
\setlist[itemize]{leftmargin=*,itemsep=3pt}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}[theorem]{Assumption}
\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\newcommand{\R}{\mathbb R}
\newcommand{\C}{\mathbb C}
\newcommand{\dd}{\mathrm d}
\newcommand{\id}{\operatorname{id}}
\newcommand{\supp}{\operatorname{supp}}
\newcommand{\WF}{\operatorname{WF}}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\im}{\operatorname{im}}
\newcommand{\Ker}{\operatorname{ker}}
\newcommand{\Lie}{\mathcal L}
\newcommand{\Dc}{C_c^\infty}
\newcommand{\Sch}{\mathcal S}
\newcommand{\barg}{\bar g}
\newcommand{\barphi}{\bar\phi}
\newcommand{\calA}{\mathcal A}
\newcommand{\calB}{\mathcal B}
\newcommand{\calC}{\mathcal C}
\newcommand{\calF}{\mathcal F}
\newcommand{\calH}{\mathcal H}
\newcommand{\calL}{\mathcal L}
\newcommand{\calR}{\mathcal R}
\newcommand{\Gsp}{\mathsf G_{\mathrm{sp}}}
\newcommand{\pair}[2]{\langle #1,#2\rangle}
\newcommand{\abs}[1]{\lvert #1\rvert}
\newcommand{\norm}[1]{\lVert #1\rVert}
\newcommand{\sid}[1]{\texttt{\detokenize{#1}}}
\newcommand{\ret}{\mathrm{ret}}
\newcommand{\adv}{\mathrm{adv}}
\newcommand{\Ward}{\mathcal V}
\newcommand{\Def}{\mathscr V_{\mathrm{def}}}
\newcommand{\CT}{\mathscr C_{\mathrm{adm}}}
\newcolumntype{Y}{>{\raggedright\arraybackslash}X}

\title{\textbf{XX-E}\\[4pt]
Ward-Domain Audit and Newton Counterterm Images\\
for the Original-Rod Relational Lapse\\[4pt]
\large A Prerequisite-Corrected Q4 Artifact\\
with Explicit Classical Residuals and Conditional Quantum Interfaces}
\author{}
\date{}

\begin{document}
\maketitle

\begin{abstract}
The revised Q4--Q5 plan assumes a native Q3 insertion and a
compatible quantum gauge operation. Those application data are
not present in the supplied XX-A--XX-D package. This manuscript
therefore does not manufacture an actual quantum breaking term
or claim cancellation of an undefined coefficient.

We reproduce the frozen original-rod lapse and calculate its
coupled classical Ward identity. An exact formula identifies
the classical breaking introduced by a compact source cutoff.
We correct the output-density mismatch between XX-A and XX-D
and distinguish the full compact geometric gauge group from
the trivial compact residual subgroup of the gauge-fixed
free operator.

Two further domain tests explain why classical symbol
invariance is not a quantum Ward theorem. Ordinary compact
gauge translations do not descend to the gauge-fixed free
equation quotient. Centrally parametrized second jets also
do not automatically support the proposed quantum gauge
derivation.

We calculate the spatial Newton images of finite contact
families, including their derivative coefficients, and the
classical Ward images of a specified comparison counterterm
family. These are explicit images, not a claim that the family
exhausts admissible quantum counterterms. A simultaneous
actual-image criterion is proved for insertion and subsidiary
repair after their defects and domains have been constructed.

An escaping-source example on the identical classical
prescription shows that compact-local convergence alone does
not control Newton composition. The native quantum Q3
prerequisites remain unverified, so native Q4 and Q5 are
unentered and unclosed. The results are domain and
non-determination theorems, not universal impossibility
theorems for a future completion.
\end{abstract}

\tableofcontents

%=====================================================================
\section{Target, numbering, and the missing application data}
\label{sec:scope}
%=====================================================================

The question requests completion of Gate Q4 using a revised
joint Q4--Q5 plan. Its own prerequisite states that these
gates cannot be entered before a native Q3 successor package
has been supplied. The attached XX-D explicitly does not
supply that package.

This is a substantive mathematical restriction:
one cannot apply a quantum Ward operation to a native
insertion that has not been admitted on its required domain.

\paragraph{Gate numbering.}
In XX-A and the supplied XIX-D inventory, Q4 concerns
complete quantum remainder control and Q5 concerns the
retained-order quantum identity. The revised plan organizes
the breaking calculation and its repair jointly under
Q4--Q5. We follow that requested organization without
silently declaring the original remainder slot completed.

\paragraph{What is proved here.}
The new results concern the printed classical prescription,
compact free-field relations, specified contact images,
and exact conditional repair interfaces. They do not
evaluate a native interacting lapse anomaly.

\paragraph{What is not treated as an input.}
The retrieved XIV-H--XIV-J characteristic-graph continuation
has a different action, background, kernel target, and
admission problem. Its internal amplitude and
off-characteristic references are not the cosmological
original-rod insertion. They are not substituted for Q3.

The relevant documentary separations and actual-image
controls are recorded by Cheung
\cite{CheungXVIIIA,CheungXIXD,CheungXIXE}.
Their records motivate the checks; the mathematical
arguments below use displayed equations.

\subsection{The exact absent successor package}

\begin{definition}[Native prerequisite package]
\label{def:package}
For the requested quantum Q4--Q5 application, a native
prerequisite package must specify:
\begin{enumerate}[label=\textup{(\roman*)}]
\item an actual normalized covariance representative,
      not only its Hadamard class;
\item proved infrared and product estimates for the
      native insertion and every operation used by the
      proposed Ward transformation;
\item the admitted insertion, its output space,
      normalization, and allowed contact images;
\item the treatment of the second jets:
      prescribed transforming classical parameters or
      constructed quantum responses;
\item a quantum Ward operation on that same domain,
      with its action on field relations, products,
      normalizations, and any subsidiary defect;
\item the complete contribution inventory at the
      claimed perturbative and quantum orders.
\end{enumerate}
A particular product involving two noncompact rod legs
additionally needs the relevant two-endpoint covariance
domain.
\end{definition}

The supplied XX-B closes only its stated compact-test free
input. XX-C does not select a positive physical covariance
or prove its native infrared estimates. XX-D's native
one-contraction theorem is conditional. None constructs
the operation in item (v).

\begin{remark}[No cancellation by definition]
Defining a candidate correction to be the negative of a
formal defect does not establish that it belongs to the
allowed counterterm space. Membership in the actual
image of the constructed counterterm map must be proved.
\end{remark}

%=====================================================================
\section{The identical freeze and a necessary smearing correction}
\label{sec:freeze}
%=====================================================================

\subsection{Action, background, and rods}

Use signature \((-,+,+,+)\) and the local action density
\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}
Only compact variations of this density are needed.

Fix \(M>0\), \(p>1\), \(t_*>0\), and
\begin{align}
 A&=\sqrt{2p}\,M,
 &
 a(t)&=(t/t_*)^p,
 &
 H(t)&=p/t,
 \notag\\
 \barphi(t)&=\phi_*+A\log(t/t_*),
 &
 v(t)&=A/t,
 \notag\\
 \barg&=-\dd t^2+a(t)^2\delta_{ij}\dd x^i\dd x^j,
 &
 V(\phi)&=\frac{M^2p(3p-1)}{t_*^2}
             e^{-2(\phi-\phi_*)/A}.
 \label{eq:background}
\end{align}
Work on \(\mathcal M=I\times\R^3\), with
\[
 I=(t_-,t_+),\qquad 0<t_-<t_+<\infty.
\]
Substitution of \(A^2=2pM^2\) verifies the
Einstein--scalar background equations. No limit
toward \(t=0\) is taken.

Retain
\begin{equation}
 g_s=\barg+s h+\frac{s^2}{2}k,\qquad
 \phi_s=\barphi+s\varphi+\frac{s^2}{2}\psi,
 \qquad z=(h,\varphi),\quad \zeta=(k,\psi).
 \label{eq:jets}
\end{equation}
The clock and raw lapse are
\begin{equation}
 T_s=t_*e^{(\phi_s-\phi_*)/A},
 \qquad
 N_s=\left(-g_s^{-1}(\dd\phi_s,\dd\phi_s)\right)^{-1/2}
 \label{eq:clocklapse}
\end{equation}
on the positive timelike branch.

On the actual \(T_s=t\) slice, let \(\gamma_s\) be
the induced metric. The original rods 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:rods}
\end{equation}
No lapse weight is inserted.

The inverse clock displacement is
\begin{equation}
 u_1=-\frac tA\varphi,\qquad
 u_2=-\frac tA\psi
        +\frac{2t^2}{A^2}\varphi\dot\varphi
        +\frac t{A^2}\varphi^2.
 \label{eq:u}
\end{equation}
In particular, the first induced metric is
\[
 (\gamma_1)_{ij}=h_{ij}+2a^2Hu_1\delta_{ij}.
\]
Its original rod-source map is
\begin{equation}
 (\calCz)^{ia}
  =\frac{1}{2a^2}\delta^{ia}\delta^{jk}h_{jk}
       -a^{-2}h_{ia}-\frac pA\delta^{ia}\varphi.
 \label{eq:C}
\end{equation}

Use the unchanged spatial Newton inverse
\begin{align}
 (\Gsp b)(\mathbf x)
   &=-\frac1{4\pi}\int
          \frac{b(\mathbf y)}{\abs{\mathbf x-\mathbf y}}
                                                \dd^3y,
 &
 \Delta\Gsp b&=b,
 \notag\\
 (\calRz)^a
   &=-\Gsp\partial_i(\calCz)^{ia}
     =\int P_i(\mathbf x-\mathbf y)
                    (\calCz)^{ia}(t,\mathbf y)\dd^3y,
 \notag\\
 P_i(\mathbf r)&=-\frac{r_i}{4\pi\abs{\mathbf r}^3}.
 \label{eq:R}
\end{align}
It is not replaced by a retarded spacetime inverse.

\subsection{The complete quadratic lapse}

Define
\begin{equation}
 \ell z=-\frac t{2A}h_{00}
          -\frac{t^2}{A^2}\dot\varphi
          -\frac t{A^2}\varphi.
 \label{eq:ell}
\end{equation}
The complete second coefficient is
\begin{equation}
 n_1=\ell z,\qquad
 n_2=\ell\zeta+q(z),\qquad
 q(z)=q_{\mathrm{loc}}(z)
                   -2\calRz\cdot\nabla\ell z,
 \label{eq:n}
\end{equation}
where
\begin{align}
 q_{\mathrm{loc}}(z)
 &=-\frac t{4A}h_{00}^2
      +\frac t{Aa^2}h_{0i}h_{0i}
      +\frac{t^2}{A^2}h_{00}\dot\varphi
      +\frac{2t^3}{A^3}\dot\varphi^2
 \notag\\
 &\quad
      +\frac{2t^2}{A^2a^2}h_{0i}\partial_i\varphi
      +\frac{t^3}{A^3a^2}
                     \partial_i\varphi\partial_i\varphi
 \notag\\
 &\quad
      +\frac{t^2}{A^2}\varphi\dot h_{00}
      +\frac t{A^2}\varphi h_{00}
      +\frac{2t^3}{A^3}\varphi\ddot\varphi
 \notag\\
 &\quad
      +\frac{6t^2}{A^3}\varphi\dot\varphi
      +\frac t{A^3}\varphi^2.
 \label{eq:qloc}
\end{align}
This is the expansion in the supplied XX-D.

For clarity, the underlying composition identity is
\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,\quad Y_1=\calRz.
 \label{eq:composition}
\end{equation}
The second spatial rod is constructed by its elliptic
equation but multiplies \(\partial_iN_0=0\) in this
composition. Thus \(Y_2\) is absent for a proved
reason, not by a tensor or source truncation.

\subsection{Restoring the identical smeared target}

\begin{proposition}[Output-density identification]
\label{prop:density}
Let \(f_A\) be the label-space test in XX-A and
\(f_D\) the test used with the background measure
in XX-D. The two smeared functionals agree if
\begin{equation}
 f_D(t,\mathbf x)=a(t)^{-3}f_A(t,\mathbf x).
 \label{eq:density}
\end{equation}
Keeping the same numerical test function while
changing the integration measure does not in
general keep the same smeared target.
\end{proposition}

\begin{proof}
XX-A uses \(\dd t\,\dd^3x\).
XX-D uses
\(\dd V_{\barg}=a(t)^3\dd t\,\dd^3x\).
Equation \eqref{eq:density} makes their integrands
and pairings equal. The multiplier is smooth
and nowhere zero on \(I\), so it preserves the
compact test space. Without this conversion,
the two pairings have different weights.
\end{proof}

Henceforth \(f\) is a fixed XX-A label-density test.
Write
\[
 L_f(z)=\int f\,\ell z\,\dd t\,\dd^3x.
\]
Let \(b(z,w)\) be the symmetric polarization of \(q\):
\begin{equation}
 b(z,w)=b_{\mathrm{loc}}(z,w)
          -\calRz\cdot\nabla\ell w
          -\calRw\cdot\nabla\ell z.
 \label{eq:b}
\end{equation}
Define
\[
 B_f(z,w)=\int f\,b(z,w)\,\dd t\,\dd^3x.
\]
The full classical functional is
\begin{equation}
 O_f
 =O_{0,f}+sL_f(z)
       +\frac{s^2}{2}\left(L_f(\zeta)+B_f(z,z)\right)
       \pmod{s^3}.
 \label{eq:O}
\end{equation}

%=====================================================================
\section{The evaluated classical Ward identity and cutoff breaking}
\label{sec:classicalWard}
%=====================================================================

\subsection{The actual compact geometric gauge group}

Let \(\xi\in\Dc(\mathcal M;T\mathcal M)\), and let
the infinitesimal admitted diffeomorphism be generated
by \(s\xi\). Put
\begin{equation}
 K\xi=(\Lie_\xi\barg,\Lie_\xi\barphi),\qquad
 A_\xi z=(\Lie_\xi h,\Lie_\xi\varphi).
 \label{eq:K}
\end{equation}
The coefficientwise jet transformation is
\begin{equation}
 \delta_\xi z=K\xi,\qquad
 \delta_\xi\zeta=2A_\xi z.
 \label{eq:jetgauge}
\end{equation}
The corresponding off-shell coefficient operation is
\[
 \Ward_\xi F
   =\pair{F_z^{(1)}}{K\xi}
            +\pair{F_\zeta^{(1)}}{2A_\xi z}.
\]
It acts on classical configurations, not on an
already assumed physical quantum quotient.

Compact diffeomorphisms preserve the spatial
asymptotic labels. They transform the geometric
rod construction covariantly; they need not
leave each unpulled rod function numerically
unchanged in a fixed coordinate chart.

\begin{lemma}[Gauge displacement of the original first rod]
\label{lem:rodgauge}
For every admitted compact \(\xi\),
\begin{equation}
 \ell(K\xi)=0,\qquad
 \calR(K\xi)^a=\xi^a.
 \label{eq:rodgauge}
\end{equation}
\end{lemma}

\begin{proof}
The metric and scalar gauge coefficients are
\[
 (K\xi)_{00}=-2\dot\xi^0,\qquad
 (K\xi)_\phi=\frac At\xi^0,
\]
\[
 (K\xi)_{ij}
    =2a^2H\xi^0\delta_{ij}
        +a^2(\partial_i\xi^j+\partial_j\xi^i).
\]
Substitution in \eqref{eq:ell} gives zero.

The clock displacement \(u_1=-\xi^0\) cancels
the \(2a^2H\xi^0\) term in the induced metric.
Consequently
\[
 (\calC K\xi)^{ia}
    =\delta^{ia}\partial_j\xi^j
               -\partial_i\xi^a-\partial_a\xi^i,
\]
and
\[
 \partial_i(\calC K\xi)^{ia}=-\Delta\xi^a.
\]
Equation \eqref{eq:R} gives
\(\calR(K\xi)^a=\Gsp\Delta\xi^a=\xi^a\).
The equality uses compact spatial support and
the unchanged decaying Newton boundary condition.
\end{proof}

\begin{theorem}[Complete coupled classical Ward calculation]
\label{thm:classicalWard}
On the smooth spatially compact domain, and on the
spatial Schwartz extension of XX-D,
\begin{equation}
 L_f(K\xi)=0,\qquad
 B_f(z,K\xi)+L_f(A_\xi z)=0.
 \label{eq:coupled}
\end{equation}
Thus
\begin{equation}
 \Ward_\xi O_f=0\pmod{s^3}.
 \label{eq:classicalzero}
\end{equation}
Holding \(\zeta\) fixed would instead leave
\(-s^2L_f(A_\xi z)\), which is not generally zero.
\end{theorem}

\begin{proof}
The first identity is Lemma~\ref{lem:rodgauge}.

For the second, the raw lapse is a scalar, the
clock slices transform by pullback, and the
slice-harmonic equation is geometrically covariant.
The transformed original rods have the same
asymptotic labels. Uniqueness of the decaying
coefficientwise Newton solution identifies them
with the pullbacks of the old rods.

If \(\mathcal C\) is the full clock-and-rod map,
this gives
\[
 \mathcal C[\rho^*g,\rho^*\phi]
               =\mathcal C[g,\phi]\circ\rho,
 \qquad
 \alpha[\rho^*g,\rho^*\phi]
               =\rho^{-1}\circ\alpha[g,\phi].
\]
Therefore \(N\circ\alpha\) is invariant.
Expanding this equality under the flow generated
by \(s\xi\), with \eqref{eq:jetgauge}, gives
\[
 \Ward_\xi O_f
 =sL_f(K\xi)
      +s^2\bigl(B_f(z,K\xi)+L_f(A_\xi z)\bigr).
\]
Both coefficients vanish.

This argument includes the original rod term.
Indeed, its polarized gauge contribution is
\[
 -\calR(K\xi)\cdot\nabla\ell z
       -\calRz\cdot\nabla\ell(K\xi)
       =-\xi^i\partial_i\ell z,
\]
using \eqref{eq:rodgauge}.
It cannot be omitted from the coefficient identity.

If the second jet is held fixed, the
\(s^2L_f(A_\xi z)\) contribution is lost.
The surviving variation is the stated residual.

It is not identically zero: choose a scalar
\(\varphi\) locally constant and nonzero and a
compact time component \(\xi^0\) with nonzero
time derivative in that region. The metric
part of \(A_\xi z\) can be taken zero.
A compact output test detects the resulting
nonzero \(\ell(A_\xi z)\).
\end{proof}

The same coupled identity is printed in XX-A.
Its independence from an interacting quantum
identity is essential.

\subsection{An exact evaluated compact-source-cutoff residual}

Choose \(\chi_R\in\Dc(\R^3)\), and define
\[
 \calR_Rz=-\Gsp\partial_i\bigl(\chi_R(\calCz)^{ia}\bigr).
\]
Let \(O_{f,R}\) replace \(\calR\) by \(\calR_R\)
in \eqref{eq:O}, leaving every local term unchanged.

\begin{theorem}[Cutoff Ward breaking]
\label{thm:cutoff}
The complete classical breaking of this compact-source
comparison is
\begin{equation}
 \boxed{
 \Ward_\xi O_{f,R}
   =s^2\int f\,
        \left(\xi^a-\calR_R(K\xi)^a\right)
                   \partial_a\ell z\,\dd t\,\dd^3x
          \pmod{s^3}.}
 \label{eq:cutoffbreaking}
\end{equation}
It vanishes when \(\chi_R=1\) on the spatial support
of \(\xi\). It is not a universal identity for a
fixed finite cutoff and the full compact gauge group.
\end{theorem}

\begin{proof}
The difference from the native functional is
\[
 O_{f,R}-O_f
   =s^2\int f\,(\calR-\calR_R)z\cdot\nabla\ell z.
\]
Its variation is
\[
 s^2\int f\left[
   (\calR-\calR_R)K\xi\cdot\nabla\ell z
       +(\calR-\calR_R)z\cdot\nabla\ell(K\xi)\right].
\]
The second term vanishes, and the first is
\eqref{eq:cutoffbreaking} by Lemma~\ref{lem:rodgauge}.

If the cutoff is one on the spatial support of
\(\xi\), it is one on the source
\(\calC K\xi\), giving equality of the rod maps.

For a failure witness at fixed finite cutoff,
choose a nonzero purely spatial \(\xi\) supported
outside \(\supp\chi_R\). Then
\(\calR_RK\xi=0\).
Choose a compact \(h_{00}\) so that
\(\xi^a\partial_a\ell z\) is nonzero on an open
set, and choose \(f\) to detect it.
\end{proof}

This is an evaluated breaking term for a stated
classical comparison. It is not the absent native
quantum breaking term.

%=====================================================================
\section{Why the free input does not supply a quantum Ward operation}
\label{sec:quantumdomain}
%=====================================================================

\subsection{Residual gauge invariance can be vacuous}

Use the fixed free operators
\begin{align}
 \calH z
  &=\left(-\frac12\delta E_{\mu\nu}[z],
                              \delta E_\phi[z]\right),
 \notag\\
 \calF_\nu z
  &=\nabla^\mu h_{\mu\nu}
       -\frac12\nabla_\nu\tr_{\barg}h
       -M^{-2}\partial_\nu\barphi\,\varphi,
 \notag\\
 r&=M^2/2,\qquad
 \calB=\calH-r\calF^\dagger\calF,\qquad
 Q=\calF K,
 \notag\\
 \calA&=\operatorname{diag}
          \left(\frac{M^2}{4}\mathsf T,\id\right),
 \qquad
 \calL=\calA^{-1}\calB.
 \label{eq:free}
\end{align}
Here \(\mathsf T\) is four-dimensional trace reversal.
The differential principal part of \(\calL\) is
\(\Box_{\barg}\id\), and \(Q\) is normally hyperbolic.
These are the reproduced XX-B operators.

\begin{proposition}[Compact gauge-fixed residual subgroup]
\label{prop:residual}
The condition
\[
 Q\xi=0,\qquad \xi\in\Dc(\mathcal M;T^*\mathcal M)
\]
has only the zero solution. Hence invariance tested only
against compact gauge-fixed residual parameters does not
certify the full compact geometric gauge identity.
\end{proposition}

\begin{proof}
Compact spacetime support lies to the future of a
Cauchy surface on which both Cauchy data vanish.
Uniqueness for a normally hyperbolic operator gives
\(\xi=0\).
This uses the checked globally hyperbolic slab and the
normally hyperbolic Cauchy theorem \cite{BGP}.
Testing an identity only at the zero parameter is
vacuous.
\end{proof}

Retarded gauge reachability may produce parameters
not compact in spacetime. It does not enlarge the
frozen compact gauge group by implication.

\subsection{Ordinary gauge translations do not descend to the free quotient}

The action Hessian identities give
\begin{equation}
 \calH K=0,\qquad
 \calB K=-r\calF^\dagger Q.
 \label{eq:BK}
\end{equation}
The free field quotient imposes
\[
 \widehat\Phi(\calB f)=0,\qquad f\in\Dc(\mathcal M;\mathcal V).
\]

\begin{theorem}[Failure of the naive compact gauge translation]
\label{thm:translation}
The ordinary affine transformation
\[
 \widehat\Phi(g)\longmapsto
      \widehat\Phi(g)+(K\xi,g)_0\id
\]
preserves the free equation relations only if
\(\calB K\xi=0\).
For compact \(\xi\) with \(K\xi\ne0\), this condition
fails. Thus the full compact geometric gauge action
is not supplied as an ordinary translation action
on the gauge-fixed free equation quotient.
\end{theorem}

\begin{proof}
Applying the proposed map to
\(\widehat\Phi(\calB f)=0\) leaves the scalar
\[
 (K\xi,\calB f)_0=(\calB K\xi,f)_0.
\]
It vanishes for every compact \(f\) exactly when
\(\calB K\xi=0\).

If that equation held, the compactly supported
section \(K\xi\) would be a homogeneous solution
of \(\calL\). Its zero Cauchy data before its
support force \(K\xi=0\), contradicting the
hypothesis. Such nontrivial compact descriptors
exist, for example time-dependent compact
spatial vector fields.
\end{proof}

This excludes one specific naive action. It does
not exclude an interacting BV construction, an
appropriate gauge comparison, or an off-shell
formulation. Such constructions require their
own operations and identities
\cite{FRBV,BFR}.

\subsection{The central-second-jet issue}

XX-D treats \(\zeta\) as prescribed classical
parameters in its free Wick construction.
That choice is legitimate for a family of
insertions. It does not automatically define
a quantum gauge transformation of those
parameters inside a fixed algebra.

Let
\[
 Z_g=(z,g)_0,\qquad M_a=(\zeta,a)_0
\]
for compact tests. In the central-parameter
extension, all \(M_a\) are central, whereas
\[
 [Z_g,Z_b]_{\star_W}
                   =i\hbar E_{\calB}(g,b)\id.
\]
The classical transformation \eqref{eq:jetgauge}
would give
\[
 \Ward_\xi Z_g=(K\xi,g)_0\id,\qquad
 \Ward_\xi M_a=2Z_{A_\xi^\dagger a}.
\]

\begin{proposition}[Necessary centrality condition]
\label{prop:central}
A derivation implementing these two formulas on the
central-parameter free algebra can exist only if
\begin{equation}
 E_{\calB}(A_\xi^\dagger a,g)=0
               \quad\text{for all relevant }a,g.
 \label{eq:centralcondition}
\end{equation}
Whenever this pairing is nonzero, the proposed
derivation is inconsistent with centrality.
\end{proposition}

\begin{proof}
The relation \([M_a,Z_g]_{\star_W}=0\) must remain zero
under a derivation. Its proposed variation is
\[
 [2Z_{A_\xi^\dagger a},Z_g]_{\star_W}
       +[M_a,(K\xi,g)_0\id]_{\star_W}
   =2i\hbar E_{\calB}(A_\xi^\dagger a,g)\id.
\]
This gives the necessary condition and the stated
incompatibility when it fails.
\end{proof}

The proposition is not a theorem against externally
transforming a family of parameters and algebras.
It identifies the extra comparison that such an
interpretation must supply. Nor does it construct
quantum second-order Einstein--scalar responses.

\subsection{The ordinary symbol test is not an anomaly calculation}

Suppose, conditionally, that the XX-D scalar Wick
correction \(\Gamma_f\) has been defined for a
selected representative, and write its free symbol
as
\begin{equation}
 I_f^{\mathrm{sym}}
       =O_f+\frac{\hbar s^2}{2}\Gamma_f.
 \label{eq:symbol}
\end{equation}
With the background, covariance, subtraction, and
test held fixed, \(\Gamma_f\) is independent of
\(z,\zeta\). Therefore ordinary off-shell coefficient
differentiation gives
\[
 \Ward_\xi I_f^{\mathrm{sym}}=0\pmod{s^3,\hbar^2}
\]
by Theorem~\ref{thm:classicalWard}.

This calculation does not supply:
\begin{itemize}
\item a derivation of the relevant quantum products;
\item preservation of the free equation quotient;
\item transformation of covariance and subtraction choices;
\item an interacting retarded insertion or its anomaly;
\item repair of the retained subsidiary contraction defect.
\end{itemize}

The distinctions are concrete:
Theorem~\ref{thm:translation} and
Proposition~\ref{prop:central} already prevent promoting
the ordinary coefficient test to the requested
quantum identity without additional structure.

In a future admitted domain, a specified operation
could have the expansion
\[
 \mathcal D_\xi=\Ward_\xi+\hbar\mathcal A_\xi
                         \pmod{\hbar^2}.
\]
Then its defect on \eqref{eq:symbol} would be
\[
 \mathcal D_\xi I_f^{\mathrm{sym}}
       =\hbar\mathcal A_\xi O_f
                         \pmod{s^3,\hbar^2}.
\]
Without a constructed \(\mathcal A_\xi\), this is
an unevaluated interface, not a computed breaking.

Anomalous Ward and renormalized BV frameworks
address precisely specified quantum operations
\cite{FRBV,BDFrR}. Their existence as frameworks
does not set the present \(\mathcal A_\xi\) to zero.

%=====================================================================
\section{Explicit Newton-contact images and comparison Ward images}
\label{sec:contacts}
%=====================================================================

\subsection{The operation that can be constructed independently}

For a finite spatial multiindex family, consider the
vector-valued source kernel
\begin{equation}
 b^{ia}(t,\mathbf y;t',\mathbf z)
  =\delta(t-t')
       \sum_{\abs{\alpha}\le m}
          c_\alpha^{ia}(t,\mathbf y,\mathbf z)
                    \partial_{\mathbf y}^{\alpha}
                            \delta^3(\mathbf y-\mathbf z),
 \label{eq:sourcecontact}
\end{equation}
where the coefficients are smooth and compactly
supported in the tangential variables used below.
This is an explicit contact family, not a claimed
exhaustive native quantum counterterm class.

Its rod image is
\[
 J^a=\int P_i(\mathbf x-\mathbf y)
                        b^{ia}(t,\mathbf y;t',\mathbf z)
                                                  \dd^3y.
\]

\begin{theorem}[Complete spatial Newton image of a contact]
\label{thm:contactimage}
The image is the joint distribution
\begin{equation}
 \boxed{
 \begin{aligned}
 J^a(t,\mathbf x;t',\mathbf z)
  ={}&\delta(t-t')
      \sum_{\abs{\alpha}\le m}\;
      \sum_{\beta\le\alpha}
        (-1)^{\abs{\beta}}\binom{\alpha}{\beta}\\
     &\quad\times
       \left(\partial_{\mathbf y}^{\beta}
                c_\alpha^{ia}\right)(t,\mathbf z,\mathbf z)
          \partial_{\mathbf x}^{\alpha-\beta}
                                  P_i(\mathbf x-\mathbf z).
 \end{aligned}}
 \label{eq:contactimage}
\end{equation}
It is well defined across \(\mathbf x=\mathbf z\)
as a distribution, not as a pointwise value.
For compact \(\mathbf z\) support, each displayed
term has a spatial tail
\(O(r^{-2-\abs{\alpha-\beta}})\) away from its
singular support.
\end{theorem}

\begin{proof}
The variables
\[
 (\mathbf x,\mathbf y,\mathbf z)
       \longmapsto(\mathbf x-\mathbf y,
                       \mathbf y-\mathbf z,\mathbf z)
\]
form a diffeomorphism. Thus the product of the
Newton factor and the contact factor before
integration is a tensor-product pullback in
independent relative variables. Smooth coefficient
multiplication is legitimate.

The exact distributional identity is
\[
 c(\mathbf y)\partial_{\mathbf y}^{\alpha}
                    \delta^3(\mathbf y-\mathbf z)
 =\sum_{\beta\le\alpha}
       (-1)^{\abs{\beta}}\binom{\alpha}{\beta}
       (\partial_{\mathbf y}^{\beta}c)(\mathbf z)
          \partial_{\mathbf y}^{\alpha-\beta}
                    \delta^3(\mathbf y-\mathbf z).
\]
Pairing a derivative of the delta with
\(P_i(\mathbf x-\mathbf y)\) gives
\(\partial_{\mathbf x}^{\alpha-\beta}
P_i(\mathbf x-\mathbf z)\).
The two differentiation signs cancel.
This proves \eqref{eq:contactimage}.

The explicit formula defines the resulting joint
distribution. Away from its spatial diagonal,
the stated tail follows by differentiating the
homogeneous dipole kernel.
\end{proof}

The contact support \(\mathbf y=\mathbf z\) is
therefore not retained as compact output support
after Newton composition. Conversely, the
distributional formula does not authorize an
additional diagonal evaluation at
\(\mathbf x=\mathbf z\).

For comparison, a contact source acted on by
\(\Gsp\) without the divergence has a
\(1/r\) Newton image. One must not assign every
Newton-contact image the dipole rate regardless
of its source structure.

\subsection{A declared comparison counterterm family}

Let \(\mathcal T\) be a fixed local linear
differential map with smooth coefficients whose
output is a uniformly compact spatial vector
source on compact time intervals. Define
\begin{equation}
 (\calR_{\mathcal T}z)^a
       =-\Gsp\partial_i(\mathcal Tz)^{ia}.
 \label{eq:RT}
\end{equation}
Let \(m(z,w)\) be a symmetric local bilinear
differential expression with smooth coefficients.
Set
\begin{equation}
 C_{m,\mathcal T;f}(z)
    =\frac{s^2}{2}\int f\left(
        m(z,z)-2\calR_{\mathcal T}z\cdot\nabla\ell z
                            \right)\dd t\,\dd^3x.
 \label{eq:Ccandidate}
\end{equation}
This is a specified comparison family of insertion
corrections. If used with an overall factor
\(\hbar\), it does not change the classical symbol.

\begin{proposition}[Evaluated classical Ward image of the family]
\label{prop:counterimage}
Its image under the coupled classical operation,
through the retained order, is
\begin{equation}
 \Ward_\xi C_{m,\mathcal T;f}
  =s^2\int f\left(
       m(z,K\xi)
        -\calR_{\mathcal T}(K\xi)^a
                              \partial_a\ell z
                           \right)\dd t\,\dd^3x
       \pmod{s^3}.
 \label{eq:counterimage}
\end{equation}
When \(\mathcal T\) is presented by contact kernels,
Theorem~\ref{thm:contactimage} supplies its complete
Newton image before this pairing.
\end{proposition}

\begin{proof}
Differentiate the symmetric bilinear term and the
two rod factors in \eqref{eq:Ccandidate}. This gives
\[
 s^2\int f\left[
  m(z,K\xi)
   -\calR_{\mathcal T}(K\xi)\cdot\nabla\ell z
   -\calR_{\mathcal T}z\cdot\nabla\ell(K\xi)\right].
\]
The last term vanishes by Lemma~\ref{lem:rodgauge}.
The source map is fixed and linear, so
\eqref{eq:RT} commutes with this variation on its
declared smooth domain.
\end{proof}

These calculations exhibit actual spatial and
classical variation images for the printed family.
They do not prove:
\begin{enumerate}[label=\textup{(\roman*)}]
\item that every member is admissible in a native
      renormalization prescription;
\item that the family exhausts the allowed changes;
\item that its quantum Ward image equals
      \eqref{eq:counterimage};
\item that it cancels an unevaluated native quantum defect.
\end{enumerate}

Local extension frameworks classify corrections
only after the relevant distribution, locus, degree,
and normalization constraints are specified
\cite{BF,HWlocal}. A Newton image is not
automatically a local action Euler derivative.

%=====================================================================
\section{Post-calculation repair: simultaneous actual-image conditions}
\label{sec:repair}
%=====================================================================

\subsection{The retained subsidiary defect}

Retain the normalized free contraction relations
\[
 \calB W=W\calB=0,\qquad
 W-W^{\mathrm T}=iE_{\calB},\qquad
 W^*=W,
\]
and the corresponding ghost contraction \(W_Q\).
The XX-B/XX-C defect is
\begin{equation}
 \mathscr D=-KW_Q-rW\calF^\dagger.
 \label{eq:subsidiary}
\end{equation}
It is reported and proved smooth in the supplied
package, but not zero.

For real symmetric smooth bisolution corrections
\(S_{\calB},S_Q\), its change is
\begin{equation}
 \mathscr D\longmapsto
   \mathscr D-\mathscr T(S_{\calB},S_Q),
 \qquad
 \mathscr T(S_{\calB},S_Q)
        =KS_Q+rS_{\calB}\calF^\dagger.
 \label{eq:subrepair}
\end{equation}
The allowed correction space must include any
required infrared, quotient, and positivity
restrictions. Smoothness alone supplies none of them.

\subsection{A complete conditional affine application}

\begin{assumption}[Constructed quantum repair problem]
\label{ass:repair}
A successor has supplied Definition~\ref{def:package}.
On its admitted domain:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the complete native breaking is a defined
      element \(B\in\Def\);
\item a specified vector space \(\CT\) contains
      the fully constrained corrections and their
      calculated insertion images;
\item their exact leading change in the breaking
      is a linear map \(L:\CT\to\Def\);
\item if subsidiary repair is also required, the
      same correction data have a calculated map
      \(T:\CT\to\mathscr V_{\mathrm{sub}}\) and
      the initial subsidiary defect is
      \(D\in\mathscr V_{\mathrm{sub}}\).
\end{enumerate}
These are constructed application data, not
definitions that authenticate themselves.
\end{assumption}

\begin{theorem}[Simultaneous exact repair]
\label{thm:repair}
Under Assumption~\ref{ass:repair}, simultaneous
insertion and subsidiary cancellation means
\begin{equation}
 B+Lc=0,\qquad D-Tc=0,\qquad c\in\CT.
 \label{eq:jointrepair}
\end{equation}
It is possible exactly when
\begin{equation}
 (-B,D)\in
   \im\left[
       \CT\ni c\longmapsto(Lc,Tc)
       \in\Def\oplus\mathscr V_{\mathrm{sub}}
       \right].
 \label{eq:jointimage}
\end{equation}
If \(c_*\) is one solution, every solution is
\[
 c_*+\Ker L\cap\Ker T.
\]
If the subsidiary defect is explicitly retained
rather than canceled, the insertion-only condition
is \(-B\in\im L\), with the remaining subsidiary
value stated for the chosen correction.
\end{theorem}

\begin{proof}
Equation \eqref{eq:jointrepair} is precisely the
linear equation
\((Lc,Tc)=(-B,D)\).
This proves the image criterion. Subtracting two
solutions gives the stated common kernel.
Dropping the second cancellation requirement gives
the insertion-only criterion, without changing
the need to report \(D-Tc\).
\end{proof}

\begin{corollary}[Separate fits need not give a common correction]
\label{cor:separate}
The separate conditions
\(-B\in\im L\) and \(D\in\im T\) do not generally
imply \eqref{eq:jointimage}.
\end{corollary}

\begin{proof}
Take \(\CT=\R\), \(L=T=\id_\R\),
\(-B=0\), and \(D=1\). Each scalar target is in
the relevant image, but no scalar correction
maps simultaneously to \((0,1)\).
\end{proof}

The criterion uses the actual image, not its
closure. It is proved independently here and
has the actual-image shape of Cheung's
XVIII-A and XIX-D
\cite{CheungXVIIIA,CheungXIXD}.
It is not the cancellation requested by Q4--Q5:
Assumption~\ref{ass:repair} has not been verified
for the supplied native lapse.

\begin{proposition}[Finite probes do not certify exact repair]
\label{prop:probes}
Let \(P\) be a finite probe map on the combined
defect space. A tested residual \(P(R)=0\)
implies \(R=0\) only with an independently proved
injectivity or coverage condition on the allowed
residual space.
\end{proposition}

\begin{proof}
Without that condition a nonzero element of
\(\Ker P\) is invisible. For example,
\(P(x,y)=x\) misses \((0,1)\).
With injectivity on the allowed residual space,
the implication follows immediately.
\end{proof}

Pure-trace probes in particular cannot authenticate
an unrestricted metric-slot kernel. A smooth
trace-free tensor \(a^{\mu\nu}\) produces the
nonzero compact kernel
\(a^{\mu\nu}(x)a^{\rho\sigma}(y)\), invisible
to every pair of pure-trace inputs.
This is the blind-sector control recorded in
Cheung's XVIII-A \cite{CheungXVIIIA}.

%=====================================================================
\section{Formal orders, product orders, and the classical symbol}
\label{sec:orders}
%=====================================================================

\begin{lemma}[Positive quantum orders cannot repair a classical residual]
\label{lem:formal}
In a vector-valued formal series, adding
\(\hbar C\) cannot change the coefficient of
\(\hbar^0\). In particular, it cannot cancel
a nonzero classical residual such as
\eqref{eq:cutoffbreaking}.
\end{lemma}

\begin{proof}
Formal equality is coefficientwise equality.
The added series has no \(\hbar^0\) coefficient.
\end{proof}

Thus a regulator that changes the classical
prescription cannot be declared harmless by
appealing to a quantum counterterm of positive
\(\hbar\)-degree.

\begin{proposition}[Classical symbol recovery]
\label{prop:symbol}
Whenever a quantum insertion with the frozen
classical symbol has actually been admitted,
its reduction modulo \(\hbar\) is
\eqref{eq:O}. The coefficients are exactly
\eqref{eq:n}--\eqref{eq:qloc}, with the original
one-rod term and no \(Y_2\).
\end{proposition}

\begin{proof}
The prescribed quantum and normalization corrections
have positive \(\hbar\)-degree. Reduction modulo
\(\hbar\) removes them. The remaining coefficient
is the already displayed classical polynomial.
\end{proof}

This is the formal classical-symbol statement,
not an expectation-value limit or a limit of
self-adjoint operators in a physical representation.

\subsection{A necessary correction to product-order bookkeeping}

Let the same formal parameter \(s\) be used in
two lapse insertions. Their first derivatives are
\[
 O_f^{(1)}
     =sL_f+s^2B_f(z,\cdot)\pmod{s^3}.
\]
The order-\(\hbar\) product correction is
\[
 \hbar\pair{O_f^{(1)}}{W O_g^{(1)}}.
\]

\begin{proposition}[Which total order needs two rod legs]
\label{prop:productorder}
Through common total order \(s^2\), that contraction
is only
\begin{equation}
 \hbar s^2\,W(L_f,L_g).
 \label{eq:lowproduct}
\end{equation}
One noncompact derivative leg can enter at order
\(s^3\), and two can enter at order \(s^4\).
Coefficientwise products of the separate quadratic
coefficients still require their own noncompact
domains.
\end{proposition}

\begin{proof}
Multiply the derivative expansions.
The linear-linear term has degree two, the
linear-quadratic terms degree three, and the
quadratic-quadratic term degree four.
The linear functional \(L_f\) has a compact
smooth test kernel after integration by parts.
Noncompact rod tails occur in the quadratic
derivatives.
\end{proof}

Thus the two-endpoint obligation is not required
for every product described merely as a
``second-order lapse product.'' Its exact grading
must be declared. It remains required whenever
the actual announced operation contains two
noncompact rod legs.

The native two-rod canonical tail retained from
XX-C is
\begin{align}
 E_{RR}^{ab}(t_0+\delta,t_0;\mathbf r)
 &=-\frac{\delta}{16\pi M^2a(t_0)^3r}
    \left((p+9)\delta_{ab}-(p+1)n_an_b\right)
 \notag\\
 &\quad+O(\delta^2r^{-1})+O_\delta(r^{-2}),
 \qquad n=\mathbf r/r.
 \label{eq:twoRod}
\end{align}
Its nonzero inverse-distance coefficient must be
preserved by a two-rod covariance domain.
It cannot be removed by demanding inverse-square
decay of both ordered terms.

No two-rod quantum product is evaluated in this
manuscript. Equation \eqref{eq:twoRod} is an
unchanged canonical constraint, not a calculation
of a lapse anomaly.

Finally, one free contraction of a quadratic
insertion is not a complete interacting one-loop
coefficient. A complete claim requires the
interactions, ghost and auxiliary contributions,
retarded normalization, and all terms at the
stated coupling order.

%=====================================================================
\section{Refinement non-determination and why removal is not entered}
\label{sec:refinement}
%=====================================================================

\subsection{An identical-prescription escaping-source example}

The following example is entirely classical and
uses the same original spatial inverse.

Choose a nonnegative, nonzero
\(\rho\in\Dc((1,2))\) and define
\[
 \eta(\mathbf Y)=Y_1\rho(\abs{\mathbf Y}).
\]
Choose \(\tau\in\Dc(I)\).
For \(R\ge1\), take the metric input
\begin{equation}
 h_{ij,R}(t,\mathbf y)
   =2a(t)^2\tau(t)R^{-1}
                         \eta(\mathbf y/R)\delta_{ij},
 \qquad
 h_{00,R}=h_{0i,R}=\varphi_R=0.
 \label{eq:escaping}
\end{equation}
Then
\[
 (\calCz_R)^{ia}
       =\tau(t)R^{-1}\eta(\mathbf y/R)\delta^{ia},
 \qquad \ell z_R=0.
\]

\begin{theorem}[Local convergence does not determine Newton composition]
\label{thm:escaping}
The inputs \(z_R\) converge to zero in the
compact-local smooth topology, but
\begin{equation}
 \calRz_R(t,\mathbf x)
        \longrightarrow \tau(t)\mathbf v
 \label{eq:escapinglimit}
\end{equation}
smoothly on compact sets, where
\[
 v^a=\int P_a(-\mathbf Y)\eta(\mathbf Y)\dd^3Y,
 \qquad v^1>0,\quad v^2=v^3=0.
\]
Each finite-\(R\) rod is the decaying original
Newton solution.

There are fixed compact \(f\) and a fixed
compact lapse input \(w\) for which
\[
 B_f(z_R,w)\longrightarrow
       -\int f\,\tau(t)v^a\partial_a\ell w\,
                                      \dd t\,\dd^3x\ne0.
\]
Consequently compact-local convergence to the
same classical input does not determine the
limit of this nonlocal Hessian operation.
\end{theorem}

\begin{proof}
The source annulus has radius between \(R\) and
\(2R\), so the input and all derivatives vanish
on each fixed compact set for sufficiently
large \(R\).

Changing variables \(\mathbf y=R\mathbf Y\)
in the rod formula gives
\[
 \calRz_R(t,\mathbf x)
    =\tau(t)\int
         P_a(-\mathbf Y+\mathbf x/R)\eta(\mathbf Y)
                                                   \dd^3Y.
\]
On the fixed source annulus the kernel is smooth
for compact \(\mathbf x\) and large \(R\).
All compact-set derivatives converge by direct
differentiation and dominated convergence.
This proves \eqref{eq:escapinglimit}.

Moreover
\[
 v^1=\frac1{4\pi}\int
       \frac{Y_1^2}{\abs{\mathbf Y}^3}
                          \rho(\abs{\mathbf Y})\dd^3Y>0.
\]
The other entries vanish by oddness.
Each finite source is compact, so its native
rod is smooth and decays at infinity.

Take \(w\) to have only \(j_{00}\ne0\).
The local polarized expression with \(z_R\)
vanishes, and \(\ell z_R=0\), leaving
\[
 b(z_R,w)=-\calRz_R\cdot\nabla\ell w.
\]
Choose \(\ell w\) with nonzero first spatial
derivative on an open region and choose \(f\)
there with a sign and time support detecting
\(\tau v^1\partial_1\ell w\).
The resulting limit is nonzero.
\end{proof}

This does not contradict continuity in XX-D's
spatial Schwartz topology: the escaping inputs
do not converge to zero in that topology.
For example, high spatial weighted seminorms
grow with \(R\).

The example supplies a precise negative control
for a future two-refinement claim. It does
not construct a quantum regulator limit or
invalidate a limit proved with the required
uniform infrared estimates.

\subsection{What an eventual removal theorem must prove}

If regulator removal is later attempted, retain
independent parameters such as
\[
 \lambda=(\epsilon,R,\Lambda_{\mathrm{IR}}).
\]
A sufficient comparison theorem on a fixed
domain would exhibit, for every relevant
continuous seminorm \(p\),
\[
 p(I_\lambda-I_{\lambda'})
       \le e_p(\lambda)+e_p(\lambda'),
 \qquad e_p(\lambda)\longrightarrow0,
\]
together with the completion and uniform
operation estimates needed to define the
limit. To compare two different regulator
families, one additionally needs a cross-family
estimate or an independently proved common
characterization of their limits.

Having a limit within each family is not
cross-family independence. Selected outcome
laws, records, or compact-local field limits
are not substitutes for product convergence
on the native domain
\cite{CheungXIXC}.

No such quantum estimate is proved here.
The identity prerequisite in the revised plan
has also not been supplied. Accordingly,
optional quantum regulator removal is not
entered.

%=====================================================================
\section{Exact application verdict}
\label{sec:verdict}
%=====================================================================

\begin{theorem}[XX-E mathematical release and native Q4 verdict]
\label{thm:release}
For the identical printed original-rod variant,
this manuscript proves:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the output-density identification restoring the
      XX-A smeared target;
\item the complete coupled classical Ward calculation
      and the exact compact-source-cutoff residual;
\item the triviality of compact gauge-fixed residual
      parameters and the failure of the naive compact
      gauge translation on the free equation quotient;
\item the central-second-jet derivation test;
\item the full spatial Newton images of the displayed
      contact family and the classical Ward images
      of the declared comparison correction family;
\item the conditional simultaneous actual-image
      criterion for insertion and subsidiary repair;
\item formal classical-symbol recovery, corrected
      product-order bookkeeping, and the
      identical-prescription refinement counterexample.
\end{enumerate}

The supplied XX-A--XX-D data do not verify
Definition~\ref{def:package} or
Assumption~\ref{ass:repair}. In particular, no
actual native quantum breaking \(B\), complete
admissible quantum image map \(L\), or solution
of their cancellation equation is constructed.

Native Q3 therefore remains open. Under the
revised prerequisite rule, native Q4 and Q5
are unentered and unclosed. No universal
nonexistence theorem for a future completion
is asserted.
\end{theorem}

\begin{proof}
Item (i) is Proposition~\ref{prop:density}.
Item (ii) is Theorem~\ref{thm:classicalWard}
and Theorem~\ref{thm:cutoff}.
Item (iii) is Proposition~\ref{prop:residual}
and Theorem~\ref{thm:translation}.
Item (iv) is Proposition~\ref{prop:central}.
Item (v) is Theorem~\ref{thm:contactimage}
and Proposition~\ref{prop:counterimage}.
Item (vi) is Theorem~\ref{thm:repair}.
Item (vii) follows from
Lemma~\ref{lem:formal},
Propositions~\ref{prop:symbol} and
\ref{prop:productorder}, and
Theorem~\ref{thm:escaping}.

The contact family was not proved exhaustive
or quantum-admissible. The conditional repair
theorem assumes a constructed quantum defect
and operation. The ordinary symbol calculation
does not supply them. The attached XX-D
explicitly retains unverified native infrared
hypotheses. Thus none of the new mathematical
results discharges the absent application data,
and the stated gate ending follows.
\end{proof}

%=====================================================================
\section{Separation and negative-control ledger}
\label{sec:ledger}
%=====================================================================

\small
\begin{longtable}{@{}p{.23\linewidth}p{.68\linewidth}@{}}
\toprule
Record or correction & Separation retained; transfer refused\\
\midrule
\endfirsthead
\toprule
Record or correction & Separation retained; transfer refused\\
\midrule
\endhead
XX-A &
Independent original-rod variant, matter-scalar calibration,
fixed label density, positive classical branch, and coupled
two-jet identity. No exact identification with every
Em02-E convention.
\\[4pt]

XX-B &
Action-Hessian normalization and compact-test free input.
Rods are not propagating fields; retarded reachability is
not compact-group reachability; algebraic Hadamard data
are not a positive physical state.
\\[4pt]

XX-C &
Local subtraction is not state selection or global decay.
Same wavefront upper bounds do not prove subtraction.
Compact-source dipole tails do not prove covariance tails.
The two-rod inverse-distance component and smooth
subsidiary defect are retained.
\\[4pt]

XX-D &
Classical insertion, compact-cutoff symbols, and conditional
native one-contraction symbols remain separate.
Compact output tests do not compactify the Hessian.
A fixed classical pullback is not quantum substitution.
Central second jets are not automatically a gauge algebra.
\\[4pt]

XIX-A \cite{CheungXIXA} &
Harmonic scalar, original-rod lapse, incidence register,
and density ramp remain different prescriptions.
Original rods are not lapse-weighted rods.
A non-identification is not a claim that all numerical
values must differ.
\\[4pt]

XIX-B \cite{CheungXIXB} &
One direct coordinate mismatch is not a universal VI
obstruction or a quantum anomaly. Positive
\(\hbar\)-orders do not repair its fixed classical residual.
\\[4pt]

XIX-C \cite{CheungXIXC} &
Equal records do not generally determine residual
instruments or sources. Scheduled support is a finite
support bound, not a product theorem. Refinement
limits require their own comparison.
\\[4pt]

XIX-D \cite{CheungXIXD} &
Quantum slots are same-target obligations, not objects
supplied by an inventory. Q4 remainder and Q5 identity
are not closed by renaming the revised joint plan.
\\[4pt]

XIX-E \cite{CheungXIXE} &
No catalogue, witness, source change, or finite
comparison opens the inherited admission exit or
constructs external-time-free evolution.
\\[4pt]

XVIII-A \cite{CheungXVIIIA} &
Pure-trace values do not determine full kernels.
Exact repair uses the actual image. A classical
formal coefficient cannot be canceled by a
positive quantum order.
\\[4pt]

XVIII-B \cite{CheungXVIIIB} &
The free massive-scalar stress response on classical
Minkowski space is not this coupled cosmological
lapse, covariance, or counterterm image.
\\[4pt]

Em02-E \cite{CheungEm02E} &
Classical original-rod data and incomplete quantum
audit remain distinct. The present independent
variant does not rewrite that release.
\\[4pt]

XVI-Q \cite{CheungXVIQ} &
The reported narrow one-loop order-\(\kappa^2\) bulk
two-point closure is a different target and prescription.
It supplies no cosmological insertion counterterms,
QdS state, general bulk restoration, or enlarged
O.1 closure.
\\[4pt]

XVI-N; XV-B
\cite{CheungXVIN,CheungXVB} &
The specified conditional insertion obstruction and
technical lifting interfaces are not discharged by
a classical lapse calculation or a contact image.
The downstream admission exit remains unopened.
\\[4pt]

OpenAI catalogue; Oai069-A; Oai282-B
\cite{Catalogue,CheungOai069,CheungOai282} &
Catalogue records and finite/free Ward patterns are
documentary comparisons, not gravitational counterterm
values, positive-state certificates, dictionaries,
or independently verified family proofs.
\\[4pt]

XIV-H--XIV-J &
An internal tree amplitude and an off-characteristic
sector reference are not the native cosmological
Q3 insertion. Their characteristic graph is not
substituted for the target of this manuscript.
\\
\bottomrule
\end{longtable}
\normalsize

%=====================================================================
\clearpage
\section*{One-page Track A release note}
\addcontentsline{toc}{section}{One-page Track A release note}
\label{sec:onepage}
%=====================================================================

\begingroup
\footnotesize

\paragraph{Exact frozen prescription.}
The target is the independently frozen original-rod lapse on
\(\barg=-\dd t^2+(t/t_*)^{2p}\dd\mathbf x^2\),
\(\barphi=\phi_*+\sqrt{2p}M\log(t/t_*)\), with \(p>1\)
and \(0<t_-<t<t_+<\infty\). The calibrated coordinate
clock is \(T=t_*e^{(\phi-\phi_*)/A}\); the lapse remains
\((-g^{-1}(\dd\phi,\dd\phi))^{-1/2}\).
Rods are harmonic on the actual clock slices with
decaying spatial corrections and no lapse weight.
Smearing is the XX-A fixed label density, with
\eqref{eq:density} used to compare XX-D.

\begin{tabularx}{\linewidth}{@{}p{.12\linewidth}Y@{}}
\toprule
Gate & Current scope and ending\\
\midrule
Q1 &
XX-B: closed only for its compact-test gauge-fixed free
multiplet, normalized causal inverses, and algebraic
Hadamard contraction class.\\
Q2 &
XX-C: not closed. Local subtraction and compact-source
tails are supplied; physical positivity, selected native
infrared estimates, and compatible subsidiary repair
are not supplied.\\
Q3 &
XX-D: open. Classical insertion and compact-cutoff symbols
are constructed; its native one-contraction theorem remains
conditional. Two-endpoint products require their stated domains.\\
Q4 &
XX-E, Theorem~\ref{thm:release}: native quantum breaking
calculation and complete remainder control are not entered.
Theorems~\ref{thm:cutoff} and~\ref{thm:contactimage}
construct only their named classical/contact comparisons.\\
Q5 &
Not closed. No native quantum Ward operation, complete
admissible quantum image, or evaluated cancellation.
Theorem~\ref{thm:repair} is conditional and post-calculation.\\
\bottomrule
\end{tabularx}

\paragraph{Retained negative controls.}
The ledger in Section~\ref{sec:ledger} retains every
XIX-A/B/C/D/E prescription, residual, record, refinement,
slot, and boundary separation; XVIII-A's pure-trace,
formal-order, and actual-image controls; XVIII-B's
different stress benchmark; Em02-E's separate native
audit; XVI-Q's narrow bulk scope; XVI-N/Route B and
XV-B boundaries; and the XX-B/C/D normalization,
positivity, tail, coordinate, second-jet, and product
corrections. OpenAI 069/282 and catalogue records
supply no gravitational counterterms or dictionary.
The subsidiary defect, conditional one-contraction
status, unverified infrared hypotheses, two-rod
inverse-distance component, and pure-trace blind sector
remain explicit.

\paragraph{Standing boundary (unchanged; reproduced verbatim).}
The target remains the independently frozen original-rod lapse
variant on the explicit on-shell power-law Einstein--scalar
background with external cosmic time. XX-B closed Q1 only for
the compact-test gauge-fixed free multiplet. XX-C supplies local
subtraction, compact-source dipole tails, a conditional one-sided
covariance-tail theorem, a same-class non-positive representative,
the native two-rod inverse-distance obstruction, and the
subsidiary-defect actual-image criterion; it does not select a
positive physical state or unconditional native tails. XX-D
constructs the complete classical second jet (with \(Y_2\)
absent because \(\partial_i N_0=0\)), the native unsmoothed
classical insertion on the spatial Schwartz domain,
compact-source-cutoff microcausal Wick symbols, and a conditional
native one-contraction Wick symbol whose infrared hypotheses
remain unverified for any selected XX-B/XX-C representative.
Products requiring two noncompact rod legs are undefined without
further control, and a universal inverse-square two-rod bound is
excluded. The native quantum Q3 gate therefore remains open.
Every link to prior manuscripts is a separation or checklist
(declared path null). Source records are never premises. No
quantum evolution without external time, no inherited
characteristic quantum coefficient, no compilation, no discharge
of XVI-N/Route B, no opening of XV-B, and O.1 remains
NOT\_CLOSED outside the narrow XVI-Q scope.

\paragraph{Milestone decision.}
This note records the actual release; it is not a substitute
for closing Q2--Q5. Track A's quantum relational observable
remains unresolved.

\endgroup

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

\paragraph{Complete supplied texts.}
XX-A, XX-B, XX-C, and XX-D have been supplied in full in the
present exchange. Earlier statements that the full XX-A or
XX-B text was unavailable are historical statements of
those drafts, not statements about the materials available
for XX-E.

\paragraph{What ``proof'' means in this release.}
The classical residuals, contact images, domain tests, and
linear-algebraic implications are the printed mathematical
results. The quantum application hypotheses are expressly
not announced as verified. A formatted theorem environment
does not make an absent covariance, quantum operation, or
counterterm classification exist.

\paragraph{Source attribution.}
Cheung's registered manuscripts are cited as Cheung, N.,
by supplied identifier and version. XX-A--XX-D are cited
as supplied drafts without inferred authorship. Catalogue
records retain their separate supplied OpenAI attribution.
No source record is independently authenticated here.

\paragraph{Verification.}
This source uses standard pdfLaTeX packages, no external
inputs, and one internal bibliography. No successful
pdfLaTeX build, proof-assistant verification, independent
symbolic or numerical execution, referee decision, or
journal acceptance is asserted.

\paragraph{Submission metadata.}
The manuscript is anonymous. A signed submission must
supply authorized authorship, affiliation, and stable
copies or archival locations for unpublished records.
Citation of N.~Cheung does not establish authorship
of this draft.

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

\bibitem{XXA}
\emph{XX-A: Original-Rod Lapse: Classical Freeze and Quantum
Admission Contract},
supplied draft, \sid{XX-A.tex}, n.d.
Complete text supplied in the present manuscript exchange.
Relevant locators:
\sid{def:rel-lapse}, \sid{cor:coupled},
\sid{lem:regular-wick}, and \sid{thm:repair}.

\bibitem{XXB}
\emph{XX-B: Coupled Gauge-Fixed Free Input for the Original-Rod
Lapse},
supplied draft, \sid{XX-B.tex}, n.d.
Complete text supplied in the present manuscript exchange.
Relevant locators:
\sid{thm:NH}, \sid{prop:residual},
\sid{prop:intertwining}, \sid{thm:contractions},
and \sid{def:closure}.

\bibitem{XXC}
\emph{XX-C: State, Subtraction, and Spatial-Tail Audit for
the Original-Rod Lapse},
supplied draft, \sid{XX-C.tex}, n.d.
Complete text supplied in the present manuscript exchange.
Relevant locators:
\sid{thm:subtraction}, \sid{thm:conditional},
\sid{thm:IRfailure}, \sid{thm:tworod},
and \sid{prop:repair}.

\bibitem{XXD}
\emph{XX-D: The Original-Rod Relational Lapse:
Composite Formula and Product Domains},
supplied draft, \sid{XX-D.tex}, n.d.
Complete text supplied in the present manuscript exchange.
Relevant locators:
\sid{eq:qloc}, \sid{thm:classicaldomain},
\sid{thm:compactWick}, \sid{thm:native},
and \sid{thm:release}.

\bibitem{CheungEm02E}
N.~Cheung,
\emph{Em02-E},
version 1, \sid{Em02-E-v1.tex}, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Supplied registered record:
original inflaton-slice rods, classical lapse,
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, \sid{XVIII-A-v2.tex}, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Relevant supplied locators:
\sid{thm:witness}, \sid{cor:smoothwitness},
\sid{thm:ward}, and \sid{thm:release}.

\bibitem{CheungXVIIIB}
N.~Cheung,
\emph{Landmark XVIII-B},
version 2, \sid{XVIII-B-v2.tex}, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Separately frozen free massive-scalar stress-response
benchmark; not the present lapse.

\bibitem{CheungXIXA}
N.~Cheung,
\emph{XIX-A},
version 1, \sid{XIX-A.tex}, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Supplied prescription and clock-separation record,
including original-versus-modified rods.

\bibitem{CheungXIXB}
N.~Cheung,
\emph{XIX-B},
version 1, \sid{XIX-B.tex}, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Supplied direct-coordinate mismatch and formal-order
nontransfer record.

\bibitem{CheungXIXC}
N.~Cheung,
\emph{XIX-C},
version 1, \sid{XIX-C.tex}, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Relevant supplied locator: \sid{thm:main}.
Finite incidence, records, residuals, scheduled support,
and refinement non-determination.

\bibitem{CheungXIXD}
N.~Cheung,
\emph{XIX-D},
version 1, \sid{XIX-D.tex}, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Relevant supplied sections:
``CE: the original-rod relational lapse''
and ``Exact affine cancellation criterion.''
Q1--Q5 same-target audit.

\bibitem{CheungXIXE}
N.~Cheung,
\emph{XIX-E},
version 1, \sid{XIX-E.tex}, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Supplied boundary inventory and unchanged obligations.

\bibitem{CheungXVIQ}
N.~Cheung,
\emph{XVI-Q},
release Q11, \sid{XVI-Q.tex}, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Relevant supplied locators:
\sid{rem:o1}, \sid{rem:qds},
\sid{rem:insertion}, and \sid{rem:package}.
Narrow bulk two-point report; no lapse identity or
series-wide verification is imported.

\bibitem{CheungXVIN}
N.~Cheung,
\emph{XVI-N},
version r12, \sid{XVI-N.tex}, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Relevant supplied locator:
\sid{thm:massless-obstruction}.
Conditional special-family insertion obstruction;
spatially compact Route B remains separate.

\bibitem{CheungXVB}
N.~Cheung,
\emph{XV-B},
version 2, \sid{XV-B.tex}, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Supplied relative-lifting and extension interfaces.
No inherited full-map or downstream quantum-admission
conclusion is imported.

\bibitem{CheungOai069}
N.~Cheung,
\emph{Oai069-A},
version 1, \sid{Oai069-A.tex}, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Finite-monodromy and free Ward comparison record;
not a gravitational counterterm premise.

\bibitem{CheungOai282}
N.~Cheung,
\emph{Oai282-B},
version 1, \sid{Oai282-B.tex}, n.d.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Declared free-scalar local Ward action and finite
coverage class; not the present cosmological identity.

\bibitem{Catalogue}
OpenAI,
\emph{OpenAI mathematics catalogue and companion scope records},
supplied record dated 7 October 2026,
\sid{OpenAI.tex}.
Enter Calculus Companion,
\url{https://enter.grok.me}.
Documentary family comparisons only.
No proof or authorship authentication is inferred.

\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, 2007.
DOI: \sid{10.4171/037}.
arXiv:0806.1036.
Normally hyperbolic Cauchy uniqueness and causal
Green operators.

\bibitem{BF}
R.~Brunetti and K.~Fredenhagen,
``Microlocal analysis and interacting quantum field theories:
Renormalization on physical backgrounds,''
\emph{Communications in Mathematical Physics}
\textbf{208} (2000), 623--661.
DOI: \sid{10.1007/s002200050004}.
arXiv:math-ph/9903028.
Extension and normalization framework for already
specified distributional constructions.

\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 normalization and covariance conditions.
No automatic nonlocal Newton insertion theorem is
imported.

\bibitem{FRBV}
K.~Fredenhagen and K.~Rejzner,
``Batalin--Vilkovisky formalism in perturbative algebraic
quantum field theory,''
\emph{Communications in Mathematical Physics}
\textbf{317} (2013), 697--725.
arXiv:1110.5232.
Renormalized BV operations and their relation to
the anomalous Master Ward Identity.
No native lapse application 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.
Locally covariant effective perturbative gravity
and renormalized BV framework.

\bibitem{BDFrR}
R.~Brunetti, M.~D\"utsch, K.~Fredenhagen, and K.~Rejzner,
``Unitary, anomalous Master Ward Identity and its connections
to the Wess--Zumino condition, BV formalism and
\(L_\infty\)-algebras,''
arXiv:2210.05908.
DOI: \sid{10.1007/s00023-023-01388-w}.
Primary anomaly-consistency framework;
not a calculation of the present nonlocal lapse breaking.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory},
arXiv:1208.1428.
Functional products, formal perturbative orders, and
the distinction between algebraic constructions and
state interpretation.

\end{thebibliography}

\end{document}