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% Scope:
%   An explicit connected quantum response for a free massless scalar,
%   a complete normalization classification at first source order,
%   an exactly evaluated invariant test, and general descent criteria.
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% Important scope correction:
%   The affirmative example is a separately declared local free-field
%   benchmark. It is not the VII-D reflected triangle and does not
%   construct the unresolved gravitational characteristic coefficient.
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% Debugging:
%   - physical matching is specified before quotienting;
%   - all finite response differences define the ambiguity;
%   - continuous detection uses the weak closure determined by tests;
%   - finite test banks require coverage of the evaluated ambiguity;
%   - ordering changes are not changes of the prepared state;
%   - local corrections are propagated before being tested;
%   - the exact witness is also realized by delayed source/readout tests;
%   - no finite-part calculation is renamed as an interacting algebra;
%   - the supplied descriptions of VII-E are inconsistent: the displayed
%     version does contain a conditional distributional dictionary;
%     no quantum construction is attributed to that dictionary.
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  pdftitle={VII-H: Normalization-independent connected quantum responses},
  pdfsubject={Complete response ambiguity, observable descent, and a nonzero free-field benchmark}
}

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\title{\textbf{Landmark VII-H}\\[0.45em]
Normalization-independent connected quantum responses:\\
complete ambiguity, observable descent,\\
and a nonzero free-field benchmark}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We formulate prescription independence at the response level and
complete the three-stage construction for one explicitly declared
free-field benchmark. The source and readout are the vacuum-centred
Wick square of a massless scalar on four-dimensional Minkowski
spacetime. At first order in a compact external source, the complete
connected vacuum susceptibility has a punctured retarded kernel
\(\hbar\Theta(t)\delta'(t^2-|\mathbf x|^2)/(4\pi^3)\).

A locally integrable logarithm gives an explicit retarded extension.
Reality, translation covariance, agreement away from coincidence,
and scaling degree at most four imply that every admissible
normalization differs by exactly \(\hbar c\delta_0\).
Thus the complete response ambiguity is determined, rather than
inferred from two regulators. We construct an admissible annular test
whose pairing is exactly \(-\hbar t_0/(2\pi^2)\) for every prescription,
and provide a single delayed source/readout experiment with a
rigorously nonzero response.

We also prove the general observable-descent criterion, its
test-determined topological separation condition, reconstruction
descent, and a finite-bank certificate that remains valid when the
underlying ambiguity is infinite-dimensional. Calibration,
minimum-variance weights, regulator smoothing, and imperfect
annihilation are treated separately.

The benchmark is not the reflected triangle of VII-D and is not the
unresolved gravitational characteristic coefficient. The latter
construction remains open on the supplied data. Classical admission,
the nonzero auxiliary remainder of XII-F, and finite instruments in
VII-G are not substituted for that missing quantum response.
\end{abstract}

\scopebox{Exact release}{%
\textbf{Constructed:} one connected quantum response at first source
order; its complete ambiguity in a stated prescription class; an
exactly evaluated nonzero invariant test; a delayed source/readout
realization; and the general descent and stability results.

\smallskip
\textbf{Explicit restriction:} the affirmative result is for a local
free massless scalar benchmark with fixed vacuum, calibrated field
and composite normalization, and no physical spacetime boundary.

\smallskip
\textbf{Not constructed:} an embedding of VII-D's reflected triangle
in a complete interacting response; the inherited gravitational
characteristic quantum coefficient; an interacting gravitational
Ward identity; or a measurement instrument for that coefficient.

\smallskip
The benchmark demonstrates the proposed logical sequence without
claiming to close an unrelated analytic gate.}

\tableofcontents

%=====================================================================
\section{The comparison problem and the selected target}
\label{sec:introduction}
%=====================================================================

The question is not whether a preferred finite part can be assigned.
It is whether a response, after physical matching, has a nonzero
pairing independent of every permitted prescription change:
\begin{equation}
 \boxed{
 \text{defined response}
 \longrightarrow
 \text{complete response ambiguity}
 \longrightarrow
 \text{nonzero invariant observation}.}
 \label{eq:programme}
\end{equation}

Three distinctions govern the construction:
\begin{equation}
 \begin{gathered}
 \text{one graph coefficient}\neq\text{complete response},\\
 \text{a local numerical counterterm}
   \neq\text{its propagated response image},\\
 \text{a quotient criterion}\neq\text{a nonzero quotient class}.
 \label{eq:distinctions}
\end{gathered}
\end{equation}

Local renormalization gives extension and comparison theorems under
specified hypotheses \cite{BF,HW2001,HW2002}. Finite-renormalization
frameworks compare interactions and observables together
\cite{BDF,DF}. These results do not identify the ambiguity of an
unspecified nonlocal response.

\subsection{Why a benchmark is selected explicitly}

The supplied corpus does not contain a complete construction of the
proposed reflected-triangle or gravitational connected response.
In particular, the supplied XIV-I and XIV-J statements stop before
the required full characteristic composition
\cite{CheungXIVI,CheungXIVJ}.

We therefore do not announce a completion of that response.
Instead, we select a different model for which all three arrows in
\eqref{eq:programme} can be proved in one manuscript:
\[
 \text{free massless scalar}
 \quad+\quad
 \text{Wick-square source}
 \quad+\quad
 \text{Wick-square readout}.
\]
Its first source derivative is already quantum, singular, connected,
and normalization dependent at coincidence. Its full allowed
ambiguity can be classified and annihilated without deleting its
nonlocal signal.

This restriction is made before any calculation. It is not a
reinterpretation of VII-D's coefficient.

\subsection{The role of Cheung's observation and descent results}

VII-E provides an observation framework and, in the displayed version,
a conditional section entitled ``A dictionary for distributional
observations'' \cite{CheungVIIE}. The supplied descriptions that both
deny and attribute that dictionary cannot be used simultaneously.
We use only its limited interpretation: observations may be blind to
declared differences, but blindness does not construct the response.

Reconstruction descent is a separate condition, stated in Em01-A
and XIV-J \cite{CheungEm01A,CheungXIVJ}. The finite separation argument
also has a classical prototype in Landmark VI \cite{CheungVI}.
All mathematical criteria needed here are proved below.

VII-G's finite instruments distinguish selective maps, outcome
effects, and implementation comparison \cite{CheungVIIG}. That
distinction is retained: constructing a scalar response distribution
does not by itself construct an instrument measuring it.

%=====================================================================
\section{Physical matching and complete response ambiguity}
\label{sec:general}
%=====================================================================

All spaces in this section are real locally convex spaces. Complex
responses can be treated by adjoining their real and imaginary
components.

\begin{definition}[A matched response family]
\label{def:matched}
Fix:
\begin{enumerate}
\item a physical model, preparation, source/readout calibration, and
      physical boundary problem;
\item a perturbative order \(N\);
\item an admitted test space \(\Tests\);
\item a nonempty class \(\Pres_0\) of physically matched prescriptions;
\item a constructed response
      \(\Resp_N^{\mathsf P}\in\Tests'\) for every
      \(\mathsf P\in\Pres_0\).
\end{enumerate}
Every response must be expressed in the same calibrated test
coordinates. Its complete difference space is
\begin{equation}
 \Amb_N
 =
 \spanR\{
 \Resp_N^{\mathsf P}-\Resp_N^{\mathsf Q}:
           \mathsf P,\mathsf Q\in\Pres_0\}.
 \label{eq:complete-ambiguity}
\end{equation}
No finite-dimensionality is assumed.
\end{definition}

For formal truncated responses, the definition is applied
coefficientwise. Equivalently, collect the finitely many coefficients
into a finite direct sum of distribution spaces. An identity through
order \(N\) is an identity of those coefficients, not an uncontrolled
numerical estimate of the omitted orders.

\begin{definition}[Invariant tests and observable closure]
Set
\begin{equation}
 \Amb_N^\perp
 =
 \{F\in\Tests:\pair{a}{F}=0\text{ for every }a\in\Amb_N\}.
 \label{eq:annihilator}
\end{equation}
The closure used for continuous observations is
\[
 \overline{\Amb_N}^{\,\sigma(\Tests',\Tests)},
\]
where \(\sigma(\Tests',\Tests)\) is the weak topology generated by
the actual admitted test pairings.
\end{definition}

\begin{theorem}[Exact observable descent and nonzero detection]
\label{thm:descent-observation}
Fix a reference response \(\Resp_*\) in the matched family.
\begin{enumerate}[label=\textup{(\roman*)}]
\item The pairing \(\pair{\Resp}{F}\) is independent of every
      prescription in \(\Pres_0\) if and only if
      \(F\in\Amb_N^\perp\).
\item There exists \(F\in\Amb_N^\perp\) with
      \(\pair{\Resp_*}{F}\neq0\) if and only if
\begin{equation}
 \Resp_*\notin
 \overline{\Amb_N}^{\,\sigma(\Tests',\Tests)}.
 \label{eq:detectability}
\end{equation}
\item Every invariant test defines a continuous linear functional
      on the weak Hausdorff quotient
\[
 \Tests'/\overline{\Amb_N}^{\,\sigma(\Tests',\Tests)}.
\]
\end{enumerate}
\end{theorem}

\begin{proof}
For (i), prescription independence is exactly the vanishing of the
pairing on every generating difference in
\eqref{eq:complete-ambiguity}. Linearity extends it to their span.

If a test annihilates \(\Amb_N\), weak continuity makes it annihilate
its weak closure. A nonzero pairing with \(\Resp_*\) therefore implies
\eqref{eq:detectability}.

Conversely, a weak neighbourhood separating \(\Resp_*\) from the
closure is determined by finitely many tests \(F_1,\ldots,F_m\).
Define
\[
 T(u)=(\pair{u}{F_1},\ldots,\pair{u}{F_m}).
\]
Then \(T(\Resp_*)\notin T(\Amb_N)\); otherwise an ambiguity would
match all those pairings and belong to the separating neighbourhood.
The linear subspace \(T(\Amb_N)\subset\R^m\) is closed. Choose \(w\)
orthogonal to it with \(w^{\mathsf T}T(\Resp_*)\neq0\).
The test \(F=\sum_iw_iF_i\) has the required properties.

The pairing is continuous in the weak topology and vanishes on the
closed subspace. The universal property of the quotient proves (iii).
\end{proof}

\begin{remark}[Which topology is justified]
An algebraically nonzero class need not be detectable by the admitted
tests. Replacing the weak closure in
\eqref{eq:detectability} by a stronger closure requires a separate
duality theorem. No reflexivity or completeness assumption is hidden
in Theorem~\ref{thm:descent-observation}.
\end{remark}

\subsection{Finite changes, not merely tangent directions}

\begin{example}[An infinitesimal catalogue can miss a finite difference]
Let \(a,b\in\Tests'\) be linearly independent and consider
\[
 \Resp^\theta=\Resp^0+\theta a+\theta^2b.
\]
The derivative at \(\theta=0\) detects only \(a\), but the span of all
finite differences contains both \(a\) and \(b\).
A test annihilating \(a\) alone need not give a prescription-independent
value.
\end{example}

Thus the definition \eqref{eq:complete-ambiguity} is used even when
prescription changes have a nonlinear parametrization.

%=====================================================================
\section{Propagation and reconstruction before observation}
\label{sec:propagation}
%=====================================================================

\begin{proposition}[Propagated ambiguity]
\label{prop:propagation}
Let \(P:\mathscr U\to\Tests'\) be a defined continuous linear map
and let \(u^{\mathsf P}\in\mathscr U\).
If
\[
 u^{\mathsf P}-u^{\mathsf Q}=c_{\mathsf P\mathsf Q}r,
\]
then
\[
 P u^{\mathsf P}-P u^{\mathsf Q}
       =c_{\mathsf P\mathsf Q}P r.
\]
The observation condition is \(\pair{P r}{F}=0\), not
\(\pair{r}{F}=0\) unless the two objects have actually been identified.

More generally, if
\[
 \Resp^{\mathsf P}
 =\Resp_*+\sum_\nu P_\nu c_\nu^{\mathsf P},
\]
the ambiguity is contained in the span of the images of the allowed
joint changes \((c_\nu^{\mathsf P}-c_\nu^{\mathsf Q})_\nu\).
It equals a proposed image span only when those changes exhaust its
generators.
\end{proposition}

\begin{proof}
The first assertions are linearity. For the last assertion, subtract
the two response formulas. Independent variation of every catalogue
entry is not implied by the formula; correlations among allowed
changes must be retained.
\end{proof}

If the propagation map also changes, the exact comparison is
\begin{equation}
 P^{\mathsf P}u^{\mathsf P}-P^{\mathsf Q}u^{\mathsf Q}
 =
 P^{\mathsf P}(u^{\mathsf P}-u^{\mathsf Q})
 +(P^{\mathsf P}-P^{\mathsf Q})u^{\mathsf Q}.
 \label{eq:changing-map}
\end{equation}
The second term cannot be omitted.

Finite renormalizations of an interacting theory require precisely
such attention to interactions and observable identifications
\cite{BDF,DF}. The present proposition does not assert that the
needed maps exist for the inherited characteristic graph.

\begin{theorem}[Reconstruction descent]
\label{thm:reconstruction}
Let \(\Rec:\mathscr P\to\mathscr K\) be a surjective linear
presentation map. A linear map
\(\widetilde{\Mmap}:\mathscr P\to Y\) factors uniquely as
\[
 \widetilde{\Mmap}=\Mmap\Rec
\]
if and only if
\begin{equation}
 \ker\Rec\subseteq\ker\widetilde{\Mmap}.
 \label{eq:reconstruction-condition}
\end{equation}
If \(\mathscr K\) has the quotient topology and
\(\widetilde{\Mmap}\) is continuous, the induced map is continuous.
\end{theorem}

\begin{proof}
Factorization annihilates every null presentation.
Conversely, two presentations of the same kernel differ by an
element of \(\ker\Rec\), so
\eqref{eq:reconstruction-condition} makes their images equal.
Set \(\Mmap(\Rec p)=\widetilde{\Mmap}(p)\).
Surjectivity gives uniqueness, and the quotient topology gives
continuity.
\end{proof}

This is the criterion stated in Cheung's reconstruction manuscripts
\cite{CheungEm01A,CheungXIVJ}. It is not the free-shell equation
quotient and not prescription equivalence.

\begin{example}[A null presentation with nonzero summands]
For a one-dimensional coordinate \(q\),
\[
 q\delta'(q)+\delta(q)=0.
\]
Indeed,
\[
 \pair{q\delta'}{\psi}
       =-\partial_q(q\psi(q))|_{q=0}=-\psi(0).
\]
Tensoring with a nonzero distribution in other variables leaves the
identity unchanged. A rule extracting a highest polynomial degree
from the two summands need not descend to their zero reconstruction.
This is the elementary mechanism used in XIV-J's null-presentation
test \cite{CheungXIVJ}.
\end{example}

%=====================================================================
\section{Definition sheet for the quantum benchmark}
\label{sec:model}
%=====================================================================

\subsection{Field, state, source, and readout}

Set
\[
 M=\R^4,\qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 z=(t,\mathbf x),\qquad
 s(z)=t^2-|\mathbf x|^2,\qquad
 \Box=\partial_t^2-\Delta_{\mathbf x}.
\]
There is no physical reflecting or absorbing boundary.

Let \(\varphi\) be the real massless free scalar in its Minkowski
vacuum \(\omega_0\), with
\begin{equation}
 \omega_0(\varphi(x)\varphi(y))=\hbar W(x-y),
 \qquad
 W(t,\mathbf x)
 =
 \int_{\R^3}
 \frac{e^{-\ii|\mathbf p|t+\ii\mathbf p\cdot\mathbf x}}
      {(2\pi)^3\,2|\mathbf p|}\dd^3\mathbf p.
 \label{eq:W}
\end{equation}
Equivalently,
\begin{equation}
 W(t,\mathbf x)
 =
 -\frac1{4\pi^2}
 \lim_{\epsilon\downarrow0}
 \frac1{(t-\ii\epsilon)^2-|\mathbf x|^2}.
 \label{eq:W-boundary}
\end{equation}
The parameter \(\hbar>0\) is retained explicitly.

Define the calibrated vacuum-centred Wick square
\[
 \Qfield(x)=:\!\varphi(x)^2\!:\,_{\omega_0}.
\]
Its field-degree coefficient is one and
\(\omega_0(\Qfield(x))=0\).
For real compact tests,
\begin{equation}
 A(f)=\int f(x)\Qfield(x)\dd^4x,\qquad
 B(h)=\int h(y)\Qfield(y)\dd^4y.
 \label{eq:AB}
\end{equation}

The external source enters with Hamiltonian density
\[
 \mathcal H_{\mathrm{src}}(y)=-\lambda h(y)\Qfield(y).
\]
The initial state remains \(\omega_0\). At first source order,
the retarded readout is
\begin{equation}
 \omega_0(A_{\lambda,h}^{\mathrm{ret}}(f))
 =
 \lambda\chi(f,h)+O(\lambda^2),
 \label{eq:source-expansion}
\end{equation}
where, away from coincident source/readout points,
\begin{equation}
 \chi(f,h)
 =
 \frac{\ii}{\hbar}
 \int f(x)h(y)\Theta(x^0-y^0)
       \omega_0([\Qfield(x),\Qfield(y)])\dd^4x\dd^4y.
 \label{eq:kubo}
\end{equation}
The sign follows by expanding source evolution with
\(\mathcal H_{\mathrm{src}}=-\lambda h\Qfield\).

The response is connected: its two insertions are joined by vacuum
contractions, and the disconnected subtraction
\(\omega_0(\Qfield(x))\omega_0(\Qfield(y))\) is zero.
The selected perturbative order is \(N=1\) in \(\lambda\).
No bulk self-interaction is present.

\subsection{Relative and joint test spaces}

Take
\[
 \Tests=\D(M;\R).
\]
The response kernel in relative coordinates is
\(\Resp\in\D'(M;\R)\). Its joint kernel is the pullback
\[
 \mathbf R(x,y)=\Resp(x-y).
\]
The difference map \((x,y)\mapsto x-y\) is a submersion, so this
pullback exists.

For \(H\in\D(M\times M;\R)\), define
\begin{equation}
 F_H(z)=\int H(y+z,y)\dd^4y.
 \label{eq:joint-relative}
\end{equation}
This is a compact smooth test, and
\[
 \pair{\mathbf R}{H}=\pair{\Resp}{F_H}.
\]
In particular, for \(H=f\otimes h\),
\begin{equation}
 F_{f,h}(z)=\int f(y+z)h(y)\dd^4y,
 \qquad
 \chi(f,h)=\pair{\Resp}{F_{f,h}}.
 \label{eq:correlation-test}
\end{equation}

\subsection{Physical matching and the normalization ledger}

The following data are fixed throughout the benchmark.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.30\linewidth}X@{}}
\toprule
Datum & Treatment\\
\midrule
Free equation and field normalization &
\(\Box\varphi=0\) and the two-point function
\eqref{eq:W}; no mass or kinetic counterterm is compared\\
Preparation &
The same Minkowski vacuum, not an arbitrary Hadamard state\\
Source and readout &
The same calibrated \(\Qfield\), \(h\), \(f\), and source sign\\
Ordering coordinates &
Permitted only with the corresponding algebra and state transport\\
Coincident response normalization &
A source-local contact convention, classified below\\
Regulator &
Auxiliary; it must yield a limit satisfying the declared response
axioms if it is used\\
Boundary conditions &
None are varied; introducing a reflecting wall changes the model\\
\bottomrule
\end{tabularx}
\end{center}

A multiplicative change of \(\Qfield\), a source rescaling, a new
vacuum, or a mass shift is not a normalization convention in this
comparison.

If a different normal-ordering representation writes
\(\Qfield\) as a Wick square plus a smooth c-number, that c-number
must be included so that the same calibrated operator and state
are represented. It is not legitimate to keep the changed symbol
and call the resulting preparation unchanged. The distinction
between product data and a physical state is also explicit in
XII-A \cite{CheungXIIA}.

The source-local contact freedom concerns the convention for
coincident differentiated composite insertions. Formally, an
addition
\[
 \frac{\hbar c}{2}\int j(y)^2\dd^4y
\]
to a connected source-functional normalization has mixed second
derivative \(\hbar c\int f h\). It does not change the free
field equation or the zero-source composite.

If an apparatus calibration also fixes this coincident response,
then \(c\) is fixed and the admissible class becomes smaller.
The invariant experiment constructed below remains valid in that
smaller class. No equivalence of unmatched overlapping
source/readout calibrations is claimed.

%=====================================================================
\section{The complete first-order contraction and its domain}
\label{sec:contraction}
%=====================================================================

\begin{lemma}[Connected two-insertion coefficient]
\label{lem:wick}
For the calibrated Wick square,
\begin{equation}
 \omega_0(\Qfield(x)\Qfield(y))
       =2\hbar^2 W(x-y)^2.
 \label{eq:wick-square}
\end{equation}
Consequently the punctured retarded response is
\begin{equation}
 \Resp^\circ(z)
 =
 2\ii\hbar\Theta(t)\bigl(W(z)^2-W(-z)^2\bigr),
 \qquad z\neq0.
 \label{eq:punctured-response}
\end{equation}
\end{lemma}

\begin{proof}
Vacuum normal ordering removes self-contractions at each insertion.
There are exactly two pairings connecting the two fields at \(x\)
to the two fields at \(y\). Each contributes \(\hbar^2W(x-y)^2\).
Terms with an uncontracted normal-ordered field have zero vacuum
expectation. This proves \eqref{eq:wick-square}; subtraction in
the commutator gives \eqref{eq:punctured-response}.
\end{proof}

Thus the first-order calculation contains the complete two-insertion
vacuum sum. There is no omitted tadpole, disconnected graph, action
insertion, or scalar-sourced gravitational slot in this model.

Products of positive-frequency two-point distributions and the
Wick-square construction belong to the usual microlocal Wick
framework \cite{BF,HW2001}. The next lemma checks the subsequent
retarded operation rather than assuming it.

\begin{lemma}[Punctured retarded kernel]
\label{lem:kernel}
The distribution \eqref{eq:punctured-response} is defined on
\(M\setminus\{0\}\), has scaling degree four at the omitted point,
and equals
\begin{equation}
 \boxed{
 \Resp^\circ(z)
 =
 \frac{\hbar}{4\pi^3}\Theta(t)\delta'(s(z)),
 \qquad z\neq0.}
 \label{eq:cone-kernel}
\end{equation}
Here \(\delta'\) denotes differentiation with respect to the
scalar argument \(s\).
\end{lemma}

\begin{proof}
The nonzero wavefront covectors of \(W\) have one time orientation.
There is no opposing pair within \(W\cdot W\), so this square is a
defined distributional product. The same holds for \(W(-z)^2\).

On the nonzero future null cone, \(t>0\) and \(\dd s\neq0\).
The one-dimensional boundary-value identity
\[
 (u-\ii0)^{-2}-(u+\ii0)^{-2}
       =-2\ii\pi\delta'(u)
\]
therefore pulls back to \(s\).
Using \eqref{eq:W-boundary} gives
\[
 W(z)^2-W(-z)^2
       =-\frac{\ii}{8\pi^3}\delta'(s(z))
       \quad(t>0).
\]
Substitution proves \eqref{eq:cone-kernel}.

Away from the origin, the commutator is supported on the nonzero
null cones, where the time cut is locally smooth. At \(t=0\) and
\(\mathbf x\neq0\) it is locally zero. Hence the time cut creates
no additional punctured product problem.

Under positive dilation, \(s\) has degree two and
\(\delta'(s)\) has degree minus four. The punctured kernel is a
nonzero homogeneous distribution of degree minus four, giving
scaling degree four.
\end{proof}

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.25\linewidth}X@{}}
\toprule
Operation & Domain and justification\\
\midrule
\(W^2\) &
Oriented wavefront product; no opposite covectors\\
Retarded restriction &
Defined away from the origin; no multiplication at the tip is
assumed\\
Tip extension &
Constructed explicitly in Section~\ref{sec:extension}\\
Joint pullback &
The difference map is a submersion\\
Compact observation &
Proper integration in \eqref{eq:joint-relative}\\
Differentiation &
Weak distributional differentiation, not a sharp boundary limit\\
\bottomrule
\end{tabularx}
\end{center}

The wavefront assertions use the product and pullback calculus
\cite{Hormander}. Normal-topology continuity theorems apply only
on compatible cones and support-proper stages \cite{BDH}; no
unverified characteristic product is introduced.

%=====================================================================
\section{An explicit extension and the full normalization class}
\label{sec:extension}
%=====================================================================

\subsection{A logarithmic reference prescription}

For \(\mu>0\), let
\begin{equation}
 L_\mu(z)=
 \begin{cases}
 \log(\mu^2s(z)),&t>0,\ s(z)>0,\\
 0,&\text{otherwise}.
 \end{cases}
 \label{eq:log-function}
\end{equation}
This is a locally integrable function. Its definition is not a
product of an undefined distribution with a characteristic
function.

Define
\begin{equation}
 \boxed{
 \Resp_\mu
 =
 \frac{\hbar}{64\pi^3}\Box^2 L_\mu.}
 \label{eq:reference-response}
\end{equation}
The square denotes two applications of \(\Box\), hence four
derivatives in total.

\begin{lemma}[Reference extension]
\label{lem:reference}
The distribution \eqref{eq:reference-response} is real, invariant
under proper orthochronous Lorentz transformations, retarded, and
extends \(\Resp^\circ\). Its scaling degree is four. For every
compact \(K\subset M\), there is a finite constant \(C_{K,\mu}\)
such that
\begin{equation}
 \abs{\pair{\Resp_\mu}{F}}
       \leq C_{K,\mu}\hbar\,p_{4,K}(F),
 \qquad \supp F\subset K,
 \label{eq:reference-estimate}
\end{equation}
where \(p_{j,K}\) is the maximum of the derivatives through order
\(j\) on \(K\).
\end{lemma}

\begin{proof}
Logarithmic singularities are locally integrable transverse to the
nonzero cone. At the tip, radial integration also gives a finite
integral of a logarithm. Reality, Lorentz invariance, and retarded
support follow from the defining function and the operator
\(\Box^2\).

We verify the constant using the retarded Riesz family. In dimension
four its normalization is
\[
 \Riesz_\alpha(z)
 =
 \frac{2^{1-\alpha}}
 {\pi\Gamma(\alpha/2)\Gamma((\alpha-2)/2)}
 \Theta(t)\Theta(s)s^{(\alpha-4)/2}
\]
on its locally integrable domain, continued holomorphically as
a distribution. The identities
\[
 \Box\Riesz_{\alpha+2}=\Riesz_\alpha,\qquad
 \Riesz_0=\delta_0
\]
are the standard Riesz identities \cite[Section 1.2]{BGP}.

On the punctured future cone, put
\(Y_\beta(u)=u_+^{\beta-1}/\Gamma(\beta)\), continued in \(\beta\).
Since \(Y_{-1}=\delta'\) and
\(1/\Gamma(\alpha/2)=\alpha/2+O(\alpha^2)\),
\[
 \left.\partial_\alpha\Riesz_\alpha\right|_{\alpha=0}
       =\frac1\pi\Theta(t)\delta'(s)
       \quad\text{away from }0.
\]
On the other hand,
\[
 \Riesz_{\alpha+4}
 =
 C(\alpha+4)\Theta(t)\Theta(s)s^{\alpha/2},
 \qquad C(4)=\frac1{8\pi}.
\]
Differentiate \(\Box^2\Riesz_{\alpha+4}=\Riesz_\alpha\)
at zero. The derivative of \(C\) contributes only a multiple of
\(\delta_0\), and the logarithmic term contributes
\[
 \frac1{16\pi}\Box^2
       \bigl(\Theta(t)\Theta(s)\log s\bigr).
\]
It follows that \eqref{eq:reference-response} restricts to
\(\hbar\Theta(t)\delta'(s)/(4\pi^3)\), as required.

For the estimate, weak differentiation gives
\[
 \pair{\Resp_\mu}{F}
 =
 \frac{\hbar}{64\pi^3}
 \int_{t>0,\ s>0}\log(\mu^2s)\,\Box^2F(z)\dd^4z.
\]
The absolute logarithmic integral on \(K\) is finite, and
\(\abs{\Box^2F}\leq16p_{4,K}(F)\).

Finally, positive dilation of this formula gives the exact
logarithmic scaling identity proved below. It implies scaling
degree at most four. The nonzero punctured homogeneous restriction
makes the degree exactly four.
\end{proof}

\subsection{The allowed prescriptions}

\begin{definition}[Admissible first-order prescription]
\label{def:prescriptions}
An admissible scalar response prescription is a real distribution
\(\Resp^{\mathsf P}=\hbar K^{\mathsf P}\) on \(M\) satisfying:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(K^{\mathsf P}|_{M\setminus\{0\}}
      =\Resp^\circ/\hbar\);
\item \(\supp K^{\mathsf P}\subset J^+(0)\);
\item proper orthochronous Lorentz invariance; its associated
      joint kernel depends only on \(x-y\);
\item \(\sd_0(K^{\mathsf P})\leq4\).
\end{enumerate}
The physical data and calibrations of Section~\ref{sec:model} are
fixed. If a regulator is used, its limit must exist and satisfy
these conditions. An unsuccessful regulator does not become a
prescription merely by being named.
\end{definition}

These are axioms for the selected scalar vacuum response at this
order, not a classification of all higher time-ordered products.
The allowance for a contact term is deliberate; no extra
coincident-response calibration has been imposed.

\begin{theorem}[Complete first-order ambiguity]
\label{thm:ambiguity}
The class in Definition~\ref{def:prescriptions} is nonempty.
For every fixed reference scale \(\mu_0>0\), all its members are
exactly
\begin{equation}
 \boxed{
 \Resp^{\mathsf P}
       =\Resp_{\mu_0}+\hbar c_{\mathsf P}\delta_0,
       \qquad c_{\mathsf P}\in\R.}
 \label{eq:all-prescriptions}
\end{equation}
Every real \(c_{\mathsf P}\) is allowed. Consequently,
\begin{equation}
 \Amb_{\mathrm{resp}}=\R\,\hbar\delta_0,\qquad
 \Amb_{\mathrm{resp}}^\perp
       =\{F\in\D(M;\R):F(0)=0\}.
 \label{eq:actual-annihilator}
\end{equation}
For the joint kernel, the difference is
\[
 \hbar(c_{\mathsf P}-c_{\mathsf Q})\delta^{(4)}(x-y).
\]
\end{theorem}

\begin{proof}
Lemma~\ref{lem:reference} supplies a member.
The difference between any member and the reference is supported
at the origin. A point-supported distribution is a finite sum
of derivatives of \(\delta_0\).
The scaling degree of \(\partial^\alpha\delta_0\) in dimension
four is \(4+|\alpha|\). The order bound therefore permits only
\(c\delta_0\). Reality makes \(c\) real.

Conversely, adding a real multiple of \(\hbar\delta_0\) preserves
every condition in Definition~\ref{def:prescriptions}.
Thus the catalogue is exhaustive, not merely an upper bound.
Evaluation of \(\delta_0\) proves the annihilator formula, and
the joint statement follows from the difference-map pullback.
\end{proof}

This is the point-extension mechanism of microlocal
renormalization \cite[Theorem 5.3]{BF}. Cheung's XVI-J has the
same type of one-dimensional extension freedom in its stated
narrowed numerical sector \cite{CheungXVIJ}. No identification of
those two sectors is needed or asserted here.

\subsection{Scale conversion and its exact meaning}

\begin{proposition}[Scale and dilation identities]
\label{prop:scale}
For positive scales \(\mu,\nu\),
\begin{equation}
 \Resp_\mu-\Resp_\nu
       =\frac{\hbar}{4\pi^2}
          \log\frac{\mu}{\nu}\,\delta_0.
 \label{eq:scale-conversion}
\end{equation}
For \(F_\ell(z)=\ell^{-4}F(z/\ell)\), \(\ell>0\),
\begin{equation}
 \pair{\Resp_\mu}{F_\ell}
 =
 \ell^{-4}
 \left[
  \pair{\Resp_\mu}{F}
  +\frac{\hbar}{4\pi^2}\log\ell\,F(0)
 \right].
 \label{eq:dilation}
\end{equation}
Scale conversion is additive and every test in
\eqref{eq:actual-annihilator} is insensitive to it.
\end{proposition}

\begin{proof}
The Riesz identity at \(\alpha=4\) gives
\[
 \Box^2\bigl(\Theta(t)\Theta(s)\bigr)=8\pi\delta_0.
\]
Subtract the logarithms in \eqref{eq:reference-response} to obtain
\eqref{eq:scale-conversion}.
For \eqref{eq:dilation}, change variables \(z=\ell w\) in the
weak integral. Four differentiations give the factor
\(\ell^{-4}\), while
\(\log(\mu^2s(\ell w))=\log(\mu^2s(w))+2\log\ell\).
The same delta identity completes the calculation.
\end{proof}

The logarithm in \eqref{eq:scale-conversion} is a normalization
conversion for this response. It is not a physical beta function
or a gravitational Wilson coefficient.

%=====================================================================
\section{An exactly evaluated nonzero invariant}
\label{sec:invariant}
%=====================================================================

Choose \(t_0>a>0\), and let
\[
 \rho(u)=
 \begin{cases}
 e^{-1/(1-u^2)},&|u|<1,\\
 0,&|u|\geq1.
 \end{cases}
\]
Set
\begin{equation}
 b(t)=
 \frac{\rho((t-t_0)/a)}
      {\sqrt{a\int_{-1}^1\rho(u)^2\dd u}}.
 \label{eq:bump}
\end{equation}
Then
\begin{equation}
 \int b(t)^2\dd t=1,\qquad
 \int t\,b(t)^2\dd t=t_0.
 \label{eq:bump-moments}
\end{equation}
The second equality uses the evenness of \(\rho\).

Define the real compact test
\begin{equation}
 \boxed{
 F_*(t,\mathbf x)
   =(t^2-|\mathbf x|^2)b(t)b(|\mathbf x|).}
 \label{eq:Fstar}
\end{equation}
It is smooth because its spatial support avoids \(\mathbf x=0\).
Its support avoids \(z=0\), although it crosses the nonzero future
null cone.

\begin{theorem}[Explicit invariant quantum response]
\label{thm:main}
For every admissible prescription in
Definition~\ref{def:prescriptions},
\begin{equation}
 \boxed{
 \pair{\Resp^{\mathsf P}}{F_*}
       =-\frac{\hbar t_0}{2\pi^2}\neq0.}
 \label{eq:main-value}
\end{equation}
This value is unchanged by every finite normalization change in
the complete class, not only by a selected regulator conversion.
\end{theorem}

\begin{proof}
Since \(F_*(0)=0\), Theorem~\ref{thm:ambiguity} removes every
allowed difference. The common value can be evaluated with the
punctured cone kernel.

For fixed \(t>0\), integrate spatially in polar coordinates and
put \(u=t^2-r^2\). Near \(u=0\),
\[
 4\pi r^2\dd r=2\pi r\,|\dd u|.
\]
The \(\delta'\) pairing therefore gives
\[
 \begin{split}
 \int_{\R^3}\delta'(t^2-r^2)
             (t^2-r^2)b(t)b(r)\dd^3\mathbf x
 &=-\left.
   \frac{\dd}{\dd u}
     \bigl(2\pi\sqrt{t^2-u}\,u\,b(t)
                  b(\sqrt{t^2-u})\bigr)
   \right|_{u=0}\\
 &=-2\pi t\,b(t)^2.
 \end{split}
\]
Multiply by \(\hbar/(4\pi^3)\), integrate in \(t\), and apply
\eqref{eq:bump-moments}.
\end{proof}

\begin{corollary}[A nonzero detectable quotient class]
The class of \(\Resp_{\mu_0}\) in
\[
 \D'(M;\R)/\R\,\hbar\delta_0
\]
is nonzero and is detected by the continuous test \(F_*\).
\end{corollary}

\begin{proof}
A response in the ambiguity subspace would pair to zero with
\(F_*\), contrary to \eqref{eq:main-value}.
\end{proof}

The result is quantum: it is the connected response of a quantum
composite in a declared vacuum and has coefficient \(\hbar\).
It is not an auxiliary classical metric variation.

\subsection{A compact joint test with the same exact value}

Choose any \(v\in\D(M;\R)\) with \(\int v=1\), and set
\[
 H_*(x,y)=v(y)F_*(x-y).
\]
This is compact and smooth; \eqref{eq:joint-relative} gives
\(F_{H_*}=F_*\). Hence
\[
 \pair{\mathbf R^{\mathsf P}}{H_*}
       =-\frac{\hbar t_0}{2\pi^2}.
\]
This is an exact joint-test certificate. A general joint weight is
not automatically one product \(f\otimes h\). The next subsection
provides a single product experiment instead.

%=====================================================================
\section{A delayed source/readout realization}
\label{sec:experiment}
%=====================================================================

The invariant can be observed without coincident source/readout
support and without treating an arbitrary joint test as a single
experiment.

\subsection{A useful cone estimate}

\begin{lemma}[Annular cone pairing]
\label{lem:annular}
Let \(G\) be a smooth compact test supported at positive times
away from the origin. On the cone,
\begin{equation}
 \pair{\Resp^\circ}{G}
 =
 \frac{\hbar}{16\pi^3}
 \int_0^\infty\int_{S^2}
 \left[
   \frac{G(t,t\mathbf n)}{t}
       +\partial_rG(t,t\mathbf n)
 \right]\dd\mathbf n\dd t.
 \label{eq:annular-pairing}
\end{equation}
If its cone sampling lies in \(t\in[d,D]\), \(d>0\), then
\begin{equation}
 \abs{\pair{\Resp^\circ}{G}}
 \leq
 \frac{\hbar(D-d)}{4\pi^2}
 \left[
   d^{-1}\norm G_\infty+
   \norm{\partial_rG}_\infty
 \right].
 \label{eq:annular-bound}
\end{equation}
\end{lemma}

\begin{proof}
For each \(t,\mathbf n\), the substitution \(u=t^2-r^2\)
changes \(r^2\dd r\) into \(r\,|\dd u|/2\).
The negative derivative at zero of
\(rG(t,r\mathbf n)/2\) is
\[
 \frac{G(t,t\mathbf n)}{4t}
       +\frac{\partial_rG(t,t\mathbf n)}4.
\]
This proves \eqref{eq:annular-pairing}.
Use \(\operatorname{area}(S^2)=4\pi\) to obtain the bound.
\end{proof}

\subsection{Explicit product tests}

Put \(q=t_0-a>0\) and
\[
 p_2(F_*)=\max_{|\alpha|\leq2}\sup_z|\partial^\alpha F_*(z)|.
\]
Choose a positive number \(\delta\) satisfying
\begin{equation}
 0<\delta<
 \min\left\{
 a,\frac q4,\,
 \frac{t_0}
 {32a\,p_2(F_*)\,(2/q+\sqrt3)}
 \right\}.
 \label{eq:delta-choice}
\end{equation}
Let
\[
 \eta(u)=\frac{\rho(u)}{\int_{-1}^1\rho(v)\dd v},\qquad
 y_0=(2\delta,0,0,0),
\]
and define
\begin{equation}
 h_\delta(y)
 =
 \delta^{-4}
 \prod_{\alpha=0}^3
 \eta\!\left(\frac{y^\alpha-y_0^\alpha}{\delta}\right),
 \qquad
 f_\delta(x)=F_*(x-y_0).
 \label{eq:experimental-tests}
\end{equation}
The source test is nonnegative and has integral one.
The source lies in \(\delta<y^0<3\delta\), while the readout lies
at \(x^0>2\delta+q>3\delta\). They are strictly time ordered and
disjoint.

\begin{theorem}[One delayed experiment detects the invariant]
\label{thm:experiment}
For every admissible prescription,
\[
 \chi^{\mathsf P}(f_\delta,h_\delta)
       =\pair{\Resp^{\mathsf P}}{F_{f_\delta,h_\delta}}
\]
has the same value and satisfies
\begin{equation}
 \boxed{
 \chi^{\mathsf P}(f_\delta,h_\delta)
       \leq-\frac{3\hbar t_0}{8\pi^2}<0.}
 \label{eq:experiment-negative}
\end{equation}
The parameter \(\delta\) is an experimental source width, not an
ultraviolet regulator.
\end{theorem}

\begin{proof}
Write \(y=y_0+\xi\). Formula \eqref{eq:correlation-test} becomes
\[
 F_{f_\delta,h_\delta}(z)
       =\int h_\delta(y_0+\xi)F_*(z+\xi)\dd^4\xi,
 \qquad |\xi^\alpha|\leq\delta.
\]
It vanishes at zero because source and readout supports are
disjoint. Every contact difference therefore pairs to zero.

Set \(G=F_{f_\delta,h_\delta}-F_*\). The mean-value theorem gives
\[
 \norm G_\infty\leq4\delta p_2(F_*),\qquad
 \norm{\partial_rG}_\infty
       \leq4\sqrt3\,\delta p_2(F_*).
\]
The cone is sampled only in
\[
 q-\delta\leq t\leq t_0+a+\delta.
\]
Its lower endpoint is at least \(q/2\), and the interval length
is at most \(4a\). Lemma~\ref{lem:annular} yields
\[
 \abs{\pair{\Resp^\circ}{G}}
 \leq
 \frac{4\hbar a\delta}{\pi^2}
       p_2(F_*)\,(2/q+\sqrt3)
 <\frac{\hbar t_0}{8\pi^2}.
\]
Combine this estimate with
\(\pair{\Resp^\circ}{F_*}=-\hbar t_0/(2\pi^2)\).
\end{proof}

The operational interpretation is the first derivative of the
vacuum expectation of the smeared Wick-square readout under a
compact quadratic source pulse. The readout smearing is real and
signed, as is permissible for a linear observable.

This interpretation does not construct a bounded finite-dimensional
instrument for the unbounded Wick-square observable. Such an
implementation would require its own measurement specification,
as emphasized by the selective-map comparison in VII-G
\cite{CheungVIIG}.

%=====================================================================
\section{Finite banks, calibration, and error control}
\label{sec:bank}
%=====================================================================

\subsection{A complete finite-bank criterion}

Let \(\Amb\subset\Tests'\) be any linear ambiguity space, possibly
infinite-dimensional. Fix tests \(F_1,\ldots,F_m\), and define
\[
 T(u)=(\pair{u}{F_i})_{i=1}^m,\qquad s=T(\Resp_*).
\]
The finite-dimensional evaluated ambiguity is \(T(\Amb)\).

\begin{theorem}[Finite-bank separation with coverage]
\label{thm:bank}
Choose \(a_1,\ldots,a_k\in\Amb\) such that their evaluated vectors
span \(T(\Amb)\), and set
\[
 M_{ij}=\pair{a_j}{F_i}.
\]
There is a weight vector \(w\in\R^m\) with
\[
 M^{\mathsf T}w=0,\qquad w^{\mathsf T}s\neq0
\]
if and only if \(s\notin\col(M)\).
Every such \(w\) gives a test
\[
 F_w=\sum_iw_iF_i
\]
annihilating the entire ambiguity \(\Amb\).

If the evaluated coverage
\(\col(M)=T(\Amb)\) has not been proved, the same matrix calculation
certifies blindness only to the listed generators.
\end{theorem}

\begin{proof}
The orthogonal complement of \(\col(M)\) is
\(\ker M^{\mathsf T}\). A vector in this complement has nonzero
pairing with \(s\) exactly when \(s\) is not in \(\col(M)\).
For any \(a\in\Amb\), coverage gives \(T(a)\in\col(M)\), so
\[
 \pair{a}{F_w}=w^{\mathsf T}T(a)=0.
\]
Without coverage, this reasoning applies only to the known span.
\end{proof}

This is the finite-certificate logic used in Cheung's earlier
linear quotient analysis \cite{CheungVI}. It neither constructs
an absent response nor proves coverage of an infinite catalogue.

\begin{example}[An exact two-test certificate for the benchmark]
Choose \(p\in\D(M;\R)\) with \(p(0)=1\), and set
\[
 F_1=F_*+p,\qquad F_2=p.
\]
Using the complete generator \(a_1=\hbar\delta_0\),
\[
 M=\hbar\begin{pmatrix}1\\1\end{pmatrix},
 \qquad w=\begin{pmatrix}1\\-1\end{pmatrix}.
\]
The exact contrast is
\[
 w^{\mathsf T}s=-\frac{\hbar t_0}{2\pi^2}.
\]
This is a certificate for the complete benchmark ambiguity because
Theorem~\ref{thm:ambiguity} already proved its coverage.
\end{example}

\subsection{Calibration is not automatic annihilation}

Suppose finite parameters \(\theta\in\R^k\) produce response changes
\(L\theta\), while calibration data change by \(C\theta\).
After exact calibration, the residual allowed parameters are
\(\ker C\).

\begin{proposition}[Residual calibrated ambiguity]
The response ambiguity remaining after this linear calibration is
\[
 L(\ker C).
\]
An invariant test need annihilate that residual space, not physical
parameter directions that have already been fixed.
\end{proposition}

\begin{proof}
Matched parameter changes have \(C\theta=0\), and their response
images are precisely \(L\theta\).
\end{proof}

For nonlinear finite matching, replace this formula by the actual
finite differences satisfying the calibration conditions.
Interaction and observable matching under finite renormalization
must be performed together \cite{BDF}.

\subsection{Minimum-variance invariant weights}

\begin{proposition}[Optimal weights for a covered bank]
\label{prop:optimal}
Assume the complete evaluated ambiguity is \(\col(M)\) and that
the measurement covariance \(\Sigma\) is positive definite.
Set
\[
 C=\Sigma^{-1/2}M,\qquad
 q=\Sigma^{-1/2}s,
\]
and let \(\Pi\) be the Euclidean orthogonal projection onto
\(\ker C^{\mathsf T}\).
If \(\Pi q\neq0\), the unique weights minimizing
\(w^{\mathsf T}\Sigma w\) subject to
\[
 M^{\mathsf T}w=0,\qquad s^{\mathsf T}w=1
\]
are
\begin{equation}
 w_*=
 \frac{\Sigma^{-1/2}\Pi q}{\norm{\Pi q}^2},
 \qquad
 w_*^{\mathsf T}\Sigma w_*=\frac1{\norm{\Pi q}^2}.
 \label{eq:optimal}
\end{equation}
They are unchanged when \(s\) is replaced by \(s+M\theta\).
\end{proposition}

\begin{proof}
Put \(y=\Sigma^{1/2}w\). The constraints become
\(C^{\mathsf T}y=0\) and \(q^{\mathsf T}y=1\), and the variance
is \(\norm y^2\).
On the permitted subspace, \(q^{\mathsf T}y=(\Pi q)^{\mathsf T}y\).
Cauchy--Schwarz gives the minimum at
\(y=\Pi q/\norm{\Pi q}^2\).
Finally,
\(\Sigma^{-1/2}M\theta\in\col(C)\), which \(\Pi\) annihilates.
\end{proof}

This is a mathematical measurement-design result. It assumes a
specified covariance model and does not derive that model from
the field theory.

\subsection{Imperfect annihilation and numerical errors}

\begin{proposition}[Uniform sensitivity bound]
\label{prop:error}
Suppose an actual evaluated prescription difference is \(M\theta\)
with \(\norm\theta\leq C_0\). Then
\begin{equation}
 \abs{\Delta I_w}
       \leq C_0\norm{M^{\mathsf T}w}.
 \label{eq:annihilation-error}
\end{equation}
If a numerical matrix \(\widehat M\) satisfies
\(\norm{M-\widehat M}_{\mathrm{op}}\leq\epsilon_M\) and
\(\widehat M^{\mathsf T}w=0\), then
\[
 \abs{\Delta I_w}\leq C_0\epsilon_M\norm w.
\]
If each estimated response pairing has deterministic error at most
\(\epsilon_i\), the contrast error is at most
\(\sum_i|w_i|\epsilon_i\).
\end{proposition}

\begin{proof}
Use
\[
 \Delta I_w=w^{\mathsf T}M\theta
       =(M^{\mathsf T}w)^{\mathsf T}\theta
\]
and Cauchy--Schwarz. The matrix and pairing estimates follow from
the operator norm and triangle inequality.
\end{proof}

Without a bound on the allowed coefficients, approximate
annihilation does not give a uniform prescription-independence
estimate.

\subsection{Smoothing a constructed response}

\begin{proposition}[Regulator smoothing after construction]
\label{prop:smoothing}
Let \(\rho_\epsilon\) be a real compact smooth approximate identity
with integral one, support in a Euclidean ball of radius
\(\epsilon\), and uniformly bounded \(L^1\)-norm.
For a fixed compactly supported test \(F\) whose support avoids
the origin,
\[
 \pair{\Resp^{\mathsf P}*\rho_\epsilon}{F}
       \longrightarrow\pair{\Resp^{\mathsf P}}{F}.
\]
For sufficiently small \(\epsilon\), the contact contribution
vanishes exactly, uniformly in \(c_{\mathsf P}\). On a fixed support
stage the remaining error is bounded by
\begin{equation}
 C\,\hbar\epsilon\,p_5(F),
 \label{eq:smoothing-rate}
\end{equation}
where \(C\) depends on a reference prescription and the uniform
mollifier bound, not on \(c_{\mathsf P}\).
\end{proposition}

\begin{proof}
Pair convolution with the reflected test convolution.
The mean-value theorem bounds its difference from \(F\), through
four derivatives, by a constant times \(\epsilon p_5(F)\).
Apply \eqref{eq:reference-estimate} on a common enlarged compact.

The difference between any prescription and the reference is
\(\hbar c_{\mathsf P}\delta_0\).
After convolution it is supported in the radius-\(\epsilon\) ball,
which is disjoint from \(\supp F\) for sufficiently small
\(\epsilon\). Its pairing is therefore exactly zero.
\end{proof}

This proposition smooths an already defined distribution.
It is not a construction of an otherwise undefined graph product,
and finite-\(\epsilon\) smoothing need not preserve exact causal
support.

The benchmark is an exact first derivative. No nonperturbative
bound on the \(O(\lambda^2)\) term in
\eqref{eq:source-expansion} is claimed. Such a bound, calibration
uncertainty, and apparatus noise are separate from
\eqref{eq:smoothing-rate} and \eqref{eq:annihilation-error}.

%=====================================================================
\section{Why the same shortcuts fail in a more general response}
\label{sec:cautions}
%=====================================================================

The benchmark has only one contact direction because its locality,
fixed physical data, punctured kernel, and scaling-degree condition
have been proved sufficient. That conclusion must not be transferred
to another graph.

\subsection{Finite normal order need not mean finite dimension}

\begin{proposition}[An infinite ambiguity at fixed normal order]
\label{prop:infinite}
On \(\R^d_y\times\R^k_n\), the distributions
\[
 a(y)\delta(n),\qquad a\in\D(\R^d),
\]
have normal order zero along \(\{n=0\}\), yet form an
infinite-dimensional vector space.
Even if their tangential support is restricted to one point,
allowing arbitrary tangential distribution order remains
infinite-dimensional.
\end{proposition}

\begin{proof}
Independent smooth functions \(a\) give independent distributions,
as follows by pairing with a test equal to one near \(n=0\).
For the point-supported claim, the distributions
\[
 \partial_y^\alpha\delta(y-y_0)\delta(n),
 \qquad \alpha\in\N^d,
\]
are independent, as monomial tests with a cutoff isolate their
coefficients.
\end{proof}

This is the distinction emphasized in Cheung's punctured
normal-jet analysis \cite{CheungXVF}. It is compatible with the
finite-dimensional point-extension result used in the benchmark:
the latter also bounds the total scaling degree.

\subsection{A local action correction can have a nonlocal image}

\begin{example}[Separated source and readout do not kill every contact]
Let \(E^{\mathrm R}\) be the wave retarded operator and \(c\) a
smooth compact function. A first-order change of the wave operator
by multiplication with \(c\) changes its inverse by
\[
 \delta E^{\mathrm R}
       =-E^{\mathrm R}cE^{\mathrm R}.
\]
For compact tests \(f,h\), the measured change is
\begin{equation}
 -\int c(y)(E^{\mathrm A}f)(y)
                 (E^{\mathrm R}h)(y)\dd^4y.
 \label{eq:propagated-action}
\end{equation}
The pairing is defined on smooth propagated tests and may be
nonzero with separated \(f,h\).
Choose \(h\) and \(f\) so their retarded and advanced responses are
nonzero at one intermediate point; continuity permits a compact
\(c\) of fixed sign in a sufficiently small neighbourhood.
\end{example}

Thus external separation eliminates the benchmark's completely
classified coincidence ambiguity, not an arbitrary action
counterterm. Cheung's Em02-E makes the same local-to-nonlocal
distinction for its spatial Green-kernel model
\cite{CheungEm02E}.

\subsection{Late source agreement does not erase preparation memory}

\begin{example}[An elementary zero-data memory term]
Choose \(F\in C_c^\infty((0,\varepsilon))\), \(F\geq0\),
\(F\not\equiv0\). The sources \(F\) and zero agree for
\(t\geq\varepsilon\), but their zero-data response difference is
\[
 z(t)=\int_0^t(t-s)F(s)\dd s.
\]
For \(t\geq\varepsilon\),
\[
 z(t)=t\int F-\int sF>0.
\]
\end{example}

This is the memory test recorded in XVI-B
\cite{CheungXVIB}. Source agreement, response agreement, and
observable agreement are separate assertions.

\subsection{A finite bank is not an injective reconstruction}

\begin{example}[An invisible nonzero distribution]
Given finitely many compact tests \(F_1,\ldots,F_m\) on an open
set, choose a nonzero smooth compact function \(u\) orthogonal
to their finite-dimensional span. Then
\[
 \pair{u}{F_i}=0\quad\text{for every }i,
 \qquad u\neq0.
\]
\end{example}

A vanishing bank therefore does not imply a zero response
distribution. Theorem~\ref{thm:bank} tests a declared quotient on
that bank; it is not a general inverse theorem.

%=====================================================================
\section{The VII-D conversion and the inherited analytic boundary}
\label{sec:inherited}
%=====================================================================

\subsection{What can be transported from a numerical conversion}

Suppose the supplied VII-D family satisfies
\begin{equation}
 u_\mu^a-u_\mu^b=c_{ab}r
 \label{eq:VII-D-conversion}
\end{equation}
as distributions on its joint test space. This is a comparison
input about that numerical construction
\cite{CheungVIID}, not the benchmark kernel of this paper.

\begin{proposition}[Conversion consistency and derivative transfer]
\label{prop:conversion}
If \(r\neq0\), the exact relation
\eqref{eq:VII-D-conversion} implies
\[
 c_{ab}+c_{bc}=c_{ac}.
\]
If a fixed linear response map \(P\) has been constructed, its
conversion is \(c_{ab}P(r)\).

Let \(T\) be a differential operator of order \(d\) with fixed
smooth coefficients on a compact test stage. An estimate
\[
 \abs{\pair{v_R}{F}}\leq C R^{-1}p_{K,64}(F)
\]
implies
\[
 \abs{\pair{T^{\mathsf T}v_R}{F}}
       \leq C'R^{-1}p_{K,64+d}(F).
\]
\end{proposition}

\begin{proof}
Subtract the three distributions to obtain
\((c_{ab}+c_{bc}-c_{ac})r=0\), and use \(r\neq0\).
The response statement is Proposition~\ref{prop:propagation}.
For the last estimate, pair \(v_R\) with \(TF\) and use the
compact-stage bound
\(p_{K,64}(TF)\leq C_Tp_{K,64+d}(F)\).
\end{proof}

This explains the test-order loss under the numerical transfer
discussed in IX-B \cite{CheungIXB}. It does not factor the numerical
finite part through a missing sharp binary multiplication.

If the response includes independent action, composite, preparation,
or boundary-normalization changes, their actual response images
must be added. Relation \eqref{eq:VII-D-conversion} alone does not
prove that the complete ambiguity is \(\R P(r)\).

\subsection{The sharp-domain obstruction is not removed by notation}

The supplied IX-A statement gives a continuous coordinate separating
the unchanged regular approximants from the proposed sharp
insertion. The supplied IX-B statement distinguishes a ternary
numerical finite part from a second sharp binary product
\cite{CheungIXA,CheungIXB}. The VII-C obstruction concerns its stated
extension and continuity hypotheses \cite{CheungVIIC}.

None of those problems is used in the benchmark. Conversely, the
benchmark's explicit distribution does not construct the missing
sharp insertion, change its admission topology, or contradict
those results.

\subsection{The gravitational branch remains separate}

The supplied XII-A--F chain has the following scope:
\begin{enumerate}
\item XII-A constructs a free-shell carrier and ordering transport.
\item XII-B distinguishes density weighting from source weighting.
\item XII-C selects the cubic-only density modification and proves
      its actual Euler source.
\item XII-D constructs a smooth classical retarded response and an
      admitted completed insertion.
\item XII-E assembles the second metric slot with the sign
      \(\Phi_{\mathrm D}^{(2)}
       =\mathcal A_2^{\mathrm{sc}}-\mathcal C_{3,\mathrm D}\).
\item XII-F evaluates a nonzero auxiliary remainder even when
      the propagated companion vanishes.
\end{enumerate}
These are distinct claims \cite{CheungXIIA,CheungXIIB,CheungXIIC,
CheungXIID,CheungXIIE,CheungXIIF}.

In particular, the supplied XII-F witness has the signed value
\[
 -8\int f(t,\mathbf x)\int_0^t(t-s)^2a(s)\dd s\,\dd^4x<0.
\]
It is a classical variation of a named off-shell lift tested on an
initial free-shell configuration. It is not the response
\eqref{eq:kubo}, not a normalization difference, and not a quantum
Ward identity.

The affine second-slot identity already determines the minus sign.
An admission theorem alone does not prove a kernel nonzero, and a
second co-transformation row must not be added to a slot whose
variation already contains that transformation.

\subsection{The chronological characteristic-map stop}

The supplied XIV-H statement leaves an internal amplitude obligation
open. XIV-I subsequently supplies its named internal amplitude
construction at I2, but leaves the full joint characteristic map
open at I3. XIV-J supplies reconstruction and an off-characteristic
reference for the supported sector, but leaves the full map open
at J2 \cite{CheungXIVH,CheungXIVI,CheungXIVJ}.

Thus it would be inaccurate to treat the earlier H2 statement as
if XIV-I had made no progress. It would also be inaccurate to
replace the later missing joint map by the already defined internal
tree kernels.

The remaining obligations for that inherited target are:
\begin{enumerate}
\item the complete joint characteristic map, with reconstruction
      descent;
\item a joint section and controlled complete remainder;
\item the required transverse estimates;
\item contact compatibility;
\item the output wavefront theorem;
\item admission of the actual quantum coefficient.
\end{enumerate}
No step is supplied by the scalar benchmark. Abstract extension or
descent interfaces are not that graph operation.

\scopebox{Two different final statuses}{%
\textbf{The local benchmark of this paper: Constructed.}
Its full first-order ambiguity and a nonzero invariant quantum
pairing are proved.

\smallskip
\textbf{The inherited reflected-triangle/gravitational target:
Unresolved on the supplied data.}
No embedding of VII-D's coefficient into a complete quantum
response is claimed, and no full characteristic coefficient is
admitted by this paper.}

%=====================================================================
\section{Final theorem and interpretation}
\label{sec:conclusion}
%=====================================================================

\begin{theorem}[VII-H release]
\label{thm:release}
For the free massless scalar, vacuum, calibrated Wick square, source
sign, test space, and prescription class declared in
Sections~\ref{sec:model}--\ref{sec:extension}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the complete first source-order connected vacuum response
      exists as a real retarded distribution;
\item every permitted finite normalization difference is a real
      multiple of \(\hbar\delta_0\);
\item the test \(F_*\) of \eqref{eq:Fstar} annihilates all such
      differences and has the exact nonzero value
      \(-\hbar t_0/(2\pi^2)\);
\item the delayed product experiment
      \eqref{eq:experimental-tests} is prescription independent
      and strictly nonzero;
\item the finite-bank, calibration, optimization, and error
      statements apply with the complete ambiguity already
      classified.
\end{enumerate}
This is a normalization-independent connected quantum response
within the declared model and order. It is not a completion of the
inherited gravitational quantum-response programme.
\end{theorem}

\begin{proof}
Lemmas~\ref{lem:wick}, \ref{lem:kernel}, and
\ref{lem:reference} construct the response.
Theorem~\ref{thm:ambiguity} gives the complete finite comparison.
Theorems~\ref{thm:main} and \ref{thm:experiment} give the invariant
and its delayed realization. Section~\ref{sec:bank} proves the
remaining assertions.
\end{proof}

For this benchmark the programme is therefore realized as
\[
 \boxed{
 \begin{gathered}
 \Resp^\circ
 =\frac{\hbar}{4\pi^3}\Theta(t)\delta'(t^2-|\mathbf x|^2),\\
 \Resp_\mu
 =\frac{\hbar}{64\pi^3}\Box^2
   \bigl[\mathbf1_{\{t>0,\ s>0\}}\log(\mu^2s)\bigr],\\
 \Resp^{\mathsf P}-\Resp^{\mathsf Q}
 =\hbar(c_{\mathsf P}-c_{\mathsf Q})\delta_0,\\
 F_*(0)=0,\qquad
 \pair{\Resp^{\mathsf P}}{F_*}
       =-\frac{\hbar t_0}{2\pi^2}\neq0.
 \end{gathered}}
\]

The surviving signal is not obtained by choosing a preferred
normalization. It is obtained by classifying the entire permitted
difference space and testing outside it.

For a later interacting or gravitational model, the same quotient
logic becomes available only after its response and its complete
physically matched ambiguity have been constructed. Neither a
finite graph coefficient, a selected extension, a classical shell
identity, nor a finite readout model substitutes for those tasks.

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

\paragraph{Model selection.}
The affirmative result concerns a separately declared local
free-field benchmark. It is not attributed to VII-D or to the
XII gravitational carrier.

\paragraph{Quantum scope.}
The constructed quantity is a connected scalar vacuum susceptibility
at first order in an external source. No higher source-order
renormalization catalogue, interacting gravitational differential,
or gravitational Ward identity is asserted.

\paragraph{Matching scope.}
The free equation, state, field normalization, source, and zero-source
composite calibration are fixed. The remaining contact convention
is explicitly declared. Changes of physical state, mass, boundary
conditions, or unmatched detector normalization are excluded.

\paragraph{Operational scope.}
The delayed experiment specifies a source pulse and smeared readout
whose linear response is nonzero. No finite measurement instrument
for the Wick-square observable is constructed.

\paragraph{Dependency scope.}
The primary literature supplies the Wick and distributional
framework and the stated Riesz identities. The response coefficient,
normalization classification, invariant evaluation, delayed
experiment, and linear separation arguments are proved here.
Cheung's manuscripts are cited with their stated domains; their
unresolved constructions are not treated as theorems of existence.

\paragraph{Verification.}
No successful \texttt{pdflatex} execution, proof-assistant
verification, numerical execution, independent referee review,
or journal acceptance is asserted.

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

\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.
arXiv:math-ph/9903028.
doi:10.1007/s002200050004.

\bibitem{HW2001}
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.
arXiv:gr-qc/0103074.
doi:10.1007/s002200100540.

\bibitem{HW2002}
S.~Hollands and R.~M.~Wald,
``Existence of local covariant time ordered products of quantum
fields in curved spacetime,''
\emph{Communications in Mathematical Physics}
\textbf{231} (2002), 309--345.
arXiv:gr-qc/0111108.
doi:10.1007/s00220-002-0719-y.

\bibitem{BDF}
R.~Brunetti, M.~D\"utsch, and K.~Fredenhagen,
\emph{Perturbative Algebraic Quantum Field Theory and the
Renormalization Groups}.
arXiv:0901.2038.
Finite-renormalization and interacting-observable comparison
framework.

\bibitem{DF}
M.~D\"utsch and K.~Fredenhagen,
\emph{Action Ward Identity and the Stueckelberg--Petermann
Renormalization Group}.
arXiv:hep-th/0501228.

\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.
arXiv:0806.1036.
Section 1.2: Riesz distributions on Minkowski space.

\bibitem{BDH}
C.~Brouder, N.~V.~Dang, and F.~H\'elein,
``Continuity of the fundamental operations on distributions
having a specified wave front set
(with a counter example by Semyon Alesker),''
\emph{Studia Mathematica}
\textbf{232} (2016), 201--226.
arXiv:1409.7662.

\bibitem{Hormander}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential Operators I},
second edition, Springer, 1990.
Distribution structure, wavefront products, pullbacks, and
pushforwards.

\bibitem{CheungVI}
N.~Cheung,
\emph{Landmark VI}.
Manuscript \texttt{VI.tex}, version 1, n.d.
The supplied discussion identifies the finite-certificate and
linear-quotient results. The linear algebra used here is proved
independently.

\bibitem{CheungVIIC}
N.~Cheung,
\emph{Landmark VII-C: A mixed-Wick obstruction to a normal-topology
boundary extension for sharp Cauchy harmonic insertions}.
Manuscript \texttt{VII-C.tex}, version 1, n.d.
Cited only with its declared extension hypotheses.

\bibitem{CheungVIID}
N.~Cheung,
\emph{Landmark VII-D: Finite part of the regulated
Cauchy-reflection triangle}.
Manuscript \texttt{VII-D.tex}, version 1, n.d.
Numerical finite part and regulator-conversion comparison.

\bibitem{CheungVIIE}
N.~Cheung,
\emph{Landmark VII-E: Observation boundaries for destructive
quantum readout and external geometry}.
Manuscript \texttt{VII-E.tex}, version 1, n.d.
Includes the conditional section
``A dictionary for distributional observations.''
No quantum graph construction is attributed to that dictionary.

\bibitem{CheungVIIG}
N.~Cheung,
\emph{Landmark VII-G: Completely positive observation boundaries
for finite matter--geometry feedback}.
Supplied manuscript \texttt{VII-G.tex}, n.d.
Selective-map comparison and finite operational scope.

\bibitem{CheungIXA}
N.~Cheung,
\emph{Landmark IX-A}.
Manuscript \texttt{IX-A.tex}, version 1, n.d.
Supplied sharp-admission nonconvergence statement
\texttt{thm:IX1}.

\bibitem{CheungIXB}
N.~Cheung,
\emph{Landmark IX-B}.
Manuscript \texttt{IX-B-v2.tex}, version 2, n.d.
Numerical transfer, test-order loss, and the missing-second-
multiplication statement \texttt{prop:missing}.

\bibitem{CheungXIIA}
N.~Cheung,
\emph{Landmark XII-A: One graded polynomial carrier for the
finite-ramp multiplet, with ordering-stable composites and a
robust observation exclusion}.
Supplied manuscript \texttt{XII-A-v2.tex}, n.d.

\bibitem{CheungXIIB}
N.~Cheung,
\emph{Landmark XII-B: Admission and one retarded response of the
ramped cubic Einstein source, at fixed regulator width:
handoff audit, variational consistency, and the unresolved release}.
Supplied manuscript \texttt{XII-B.tex}, n.d.

\bibitem{CheungXIIC}
N.~Cheung,
\emph{Landmark XII-C: One declared cubic weighting:
actual Euler source, changed Noether identity, and the
coupled-jet question}.
Supplied manuscript \texttt{XII-C.tex}, n.d.

\bibitem{CheungXIID}
N.~Cheung,
\emph{Landmark XII-D: Retarded response and carrier admission for
the cubic-only density-weighted Einstein source}.
Supplied manuscript \texttt{XII-D.tex}, n.d.

\bibitem{CheungXIIE}
N.~Cheung,
\emph{Landmark XII-E: Assembly of the second-order relational
coefficient after admission of the density-weighted cubic insertion}.
Supplied manuscript \texttt{XII-E.tex}, n.d.

\bibitem{CheungXIIF}
N.~Cheung,
\emph{Landmark XII-F: The gauge-fixed remainder of
\(\Phi_{\mathrm D}^{(2)}\): companion support, the auxiliary
pairing, and what is not a Ward identity}.
Supplied manuscript \texttt{XII-F.tex}, n.d.

\bibitem{CheungXIVH}
N.~Cheung,
\emph{Landmark XIV-H}.
Manuscript \texttt{XIV-H.tex}, version 1, n.d.
Supplied excerpts concerning the internal kernel construction
and the H2 boundary.

\bibitem{CheungXIVI}
N.~Cheung,
\emph{Landmark XIV-I}.
Manuscript \texttt{XIV-I.tex}, version 1, n.d.
Supplied excerpts: internal amplitude construction at I2 and
the unresolved joint characteristic map at I3.

\bibitem{CheungXIVJ}
N.~Cheung,
\emph{Landmark XIV-J}.
Manuscript \texttt{XIV-J.tex}, version 1, n.d.
Supplied excerpts: reconstruction descent, the null presentation,
the supported-sector reference, and the J2 verdict.

\bibitem{CheungXVF}
N.~Cheung,
\emph{Landmark XV-F}.
Manuscript \texttt{XV-F-v3.tex}, version 3, n.d.
Supplied statement on the structure and dimension of
puncture-supported normal-jet ambiguities.

\bibitem{CheungXVIB}
N.~Cheung,
\emph{Landmark XVI-B}.
Manuscript \texttt{XVI-B.tex}, version 1, n.d.
Supplied preparation-memory lemma \texttt{lem:memory}.

\bibitem{CheungXVIJ}
N.~Cheung,
\emph{Landmark XVI-J}.
Manuscript \texttt{XVI-J.tex}, version 1, n.d.
Supplied moment-normalized extension and contact-ambiguity
statements for its narrowed numerical sector.

\bibitem{CheungEm01A}
N.~Cheung,
\emph{Em01-A}.
Manuscript \texttt{Em01-A.tex}, version 1, n.d.
Supplied exact reconstruction-descent criterion
\texttt{thm:descent}.

\bibitem{CheungEm02E}
N.~Cheung,
\emph{Em02-E}.
Manuscript \texttt{Em02-E-v1.tex}, version 1, n.d.
Supplied spatial Green-kernel response images and noncompact
field-slot support discussion.

\end{thebibliography}

\end{document}