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\hypersetup{
  pdftitle={X-B-ST: One scalar retarded map on the X-A-ST host},
  pdfauthor={},
  pdfsubject={Fixed-ramp admission, first-order background covariance,
  and a local Grade S finite interface}
}

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\title{\textbf{X-B-ST: One scalar retarded map\\
on the X-A-ST host}\\[5pt]
\large Fixed-ramp admission, first-order background covariance,\\
and the one-contraction boundary}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We construct a local Grade S realization of Cheung's
fixed-regulator scalar-only X-B prescription on the
supported host presentation already fixed by X-A-ST.
The quantum field is the free Klein--Gordon scalar on
the declared Minkowski completion; the metric coefficients
remain classical and off shell. The host geometry is not
reconstructed or evolved.

At every fixed positive ramp width, the bare relational
dressing is admitted through second background order,
both by a joint polynomial wavefront test and, separately,
as a scalar-linear functional with a smooth compact test.
For every specified off-shell linear representative
\(F_c\), the normalized response is
\[
 R_1(V_{1,g}[h];F_c)
   =\boldsymbol\phi(-L_{1,g}[h]\mathsf E_{\mathrm A}c).
\]
We prove continuity, retarded support, independence from
the Hadamard contraction kernel, and the absence of
positive-\(\hbar\) terms in its output test. The completed
first coefficient is invariant in the unchanged free
algebra on the gauge algebra frozen before the calculation.
An explicit equation-ideal representative separates this
identity from the retained edge and switch defects.

The four X-B-ii scope closures are incorporated into the
same file. Classical normalization on the entire insertion
domain determines the degree-one output class; compatible
contact modifications are therefore algebraically zero.
Conversely, the one-contraction verification does not
construct a time-ordered family: the Feynman kernel's
square fails the ordinary wavefront product criterion
on the diagonal. This is a limitation of the present
construction, not a nonexistence theorem for renormalized
products.

The contribution is a supported transfer, normalization
audit, and explicit scope separation of the source
results, not a new priority claim for them. Its potential
value is a reproducible starting domain for later response
analysis without conflating free-field admission,
classical covariance, singular-product construction,
and measurement. A finite destructive-clock interface
encodes supplied evaluations; it does not acquire a
continuum field.
\end{abstract}

\statementbox{Two orders, one host, and one map}{%
Bare dressing admission is established through
\(\kappa^2\). The completed background identity is
established through \(\kappa^1\), namely modulo
\(\kappa^2\). These are different statements.

Only the single scalar-only map
\(R_1(V_{1,g}[h];\cdot)\) is released.
The displayed classical \(\rho_{2,g}\) is not promoted
to a constructed second-order quantum map.
X-B-ii supplies scope closures inside this file, not
another spacetime or another interaction programme.}

\tableofcontents

%=====================================================================
\section{Purpose, Enter background, and the inherited host}
\label{sec:host}
%=====================================================================

\subsection{Purpose and attribution}

The analytical source records are Cheung's X-B,
version 3, and X-B-ii, version 1
\cite{XB,XBii}. The first constructs one normalized
scalar response; the second clarifies its scope,
one-contraction boundary, and normalization.
The present manuscript transfers that package to the
supported presentation supplied in X-A-ST
\cite{XAST}. It does not enlarge the source's quantum
field content.

The distinction between a classical relational
coefficient and its quantum admission is essential.
A classical identity does not define a distributional
product, while an admitted free-field expression does
not automatically define an interacting response.
The standard perturbative algebraic framework makes
these domain distinctions explicit
\cite{FR,DFreference,BF}.

\subsection{Enter and its interpretation}

The Enter background distinguishes destructive readout
from quantum return. A measured quantum name is consumed;
the declared output is a classical record together with
the explicitly retained residual. Recovery from an
accessible subsystem is a different task
\cite{EnterI,EnterPhilosophy}.

We use this discipline only after constructing the
field-theoretic objects. In particular:
\begin{enumerate}
\item an encoded number is not the continuum field from
which that number was mathematically evaluated;
\item a channel identity is not a thermodynamic
implementation or a field detector;
\item a finite dependency schedule is not a derivation of
a light cone;
\item a prescribed spacetime supporting finite operations
is not spacetime emerging from those operations.
\end{enumerate}
The spacetime Enter model of I-B supplies the relevant
fixed-host interpretation \cite{IB}. The finite
instruments needed here are proved directly below.

\subsection{Host pointer: no second cone}

\begin{assumption}[Inherited supported presentation]
\label{ass:host}
Retain exactly the host instance fixed by X-A-ST:
\begin{enumerate}[label=\textup{(H\arabic*)}]
\item the time-oriented regular conical product
\((\mathcal M^\circ,G)\), its prescribed worldsheet
\(\mathcal W\), and its supplied future relation;
\item its supported injective developing chart
\[
 \iota:\mathcal L\longrightarrow\mathcal M^\circ,
\]
whose retained compact subsets stay off \(\mathcal W\);
\item the declared flat completion
\[
 \widehat M=\RR^{1+3},\qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 M_{T_*}=[0,T_*]\times\RR^3;
\]
\item the signature ledger
\[
 \iota^*G=-\eta,\qquad
 t=c_{\mathrm{host}}(T-T_\Sigma).
\]
The host has signature \((-,+,+,+)\), while all
analytical equations in this manuscript retain
\((+,-,-,-)\);
\item a retained observation region
\(\mathcal O\Subset (0,T_*)\times\RR^3\), with
\[
 D_*:=J_\eta^-(\overline{\mathcal O})\cap M_{T_*}
                         \Subset\mathcal L.
\]
Compact laboratory tests are transported to
\(\widehat M\) and extended by zero.
\end{enumerate}
The regular host is not assumed globally hyperbolic.
The global free field and all Green distributions below
live on \(\widehat M\), not on that regular product.
\end{assumption}

These are the X-A-ST host data, not new geometric
conclusions. The worldsheet, signature, speed, lapse
interpretation, chart, and future relation remain
inputs.

A local developing chart is not a global Minkowski
representation, an intrinsic conical Green operator,
or a global field-controlled unitary. We make none of
those identifications.

\begin{definition}[Supported meaning of this transfer]
\label{def:supported}
The algebraic construction is defined on the flat
completion. Its \emph{supported host presentation}
consists of the same pairings and test actions,
restricted to retained compact causal calculations
and transported through \(\iota\).

For \(c\in C_c^\infty(\mathcal O)\), set
\[
 D_c=J_\eta^-(\supp c)\cap M_{T_*}.
\]
The support contract is that every coordinate calculation
needed by the chosen dressing test lies in \(D_*\);
interaction contributions outside \(D_c\) do not enter
its response.
\end{definition}

The map will also be defined for arbitrary compact
completion tests. This larger analytical domain is not
a claim that every such test has a supported conical
presentation.

\begin{definition}[Local Grade S in this file]
\label{def:grade}
A local Grade S release consists of:
\begin{enumerate}[label=\textup{(S\arabic*)}]
\item the inherited supported host presentation;
\item explicitly admitted field-theoretic objects with
their domains and normalization;
\item a finite installed evaluation dictionary;
\item a complete destructive-clock instrument, including
untouched references;
\item scheduled locality and a nonempty regular,
reference-preserving route under declared finite
operational permissions.
\end{enumerate}
This definition contains no emergence or Einstein-dynamics
requirement. Grade E is not claimed.
\end{definition}

%=====================================================================
\section{The frozen classical handoff}
\label{sec:handoff}
%=====================================================================

This section states the chart data supplied by X-A-ST.
Its classical covariance proof is not repeated.
The new identities proved later are not inferred merely
by citing the parent's main theorem.

\subsection{Backgrounds, ramp, plateau, and gauge action}

Let
\[
 \Bg=C^\infty(\widehat M;\Sym^2T^*\widehat M)^2
\]
with the smooth compact-open topology, and let
\[
 q=\eta+\kappa h+\kappa^2k,\qquad (h,k)\in\Bg.
\]
The backgrounds are real, smooth, classical, and off
shell. No Einstein equation, graviton propagator, or
quantum metric variable is introduced.

Fix \(m\ge0\), \(0<\varepsilon<T_*\), and
\[
 g\in C_c^\infty((0,T_*)\times\RR^3;[0,1]).
\]
The ramp is
\[
 \sigma(t)=\chi(t/\varepsilon),\qquad
 \chi(s)=
 \begin{cases}
 0,&s\le0,\\[1mm]
 \displaystyle
 \frac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},&0<s<1,\\[3mm]
 1,&s\ge1.
 \end{cases}
 \label{eq:ramp}
\]
It is smooth and flat at both endpoints.

Retain the same nonempty relatively compact open
plateau \(U\) as in X-A-ST:
\begin{equation}
 \overline U\Subset
 \mathcal L\cap\operatorname{int}\{g=1\}
                     \cap\{\varepsilon<t<T_*\},
 \qquad
 \gp=C_c^\infty(U;T\widehat M).
 \label{eq:plateau}
\end{equation}
The formal group \(\Gp\) is generated through
\(\kappa^2\) by flows of
\(\kappa\xi+\kappa^2\zeta\), \(\xi,\zeta\in\gp\).
The vector fields are extended by zero.
They preserve \(g,\sigma\), are initial-trivial, and
do not meet the worldsheet.

Write \(X_\xi=\xi^\mu\partial_\mu\) on scalars and
\(D_\xi r=\partial_\mu(\xi^\mu r)\) on scalar densities.
The formal metric transformation is
\begin{align}
 h'&=h+K\xi,\qquad
 (K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu,
 \nonumber\\
 k'&=k+\Lie_\xi h+K\zeta+\tfrac12\Lie_\xi^2\eta.
 \label{eq:metriclaw}
\end{align}
The last term is not omitted in a second-order
diagnostic.

\subsection{Density and coordinate coefficients}

Indices are raised with \(\eta\). Set \(H=\tr_\eta h\) and
\begin{align}
 b_1[h]&=\tfrac12H,
 &
 b_2[h]&=\tfrac18H^2-\tfrac14h_{\alpha\beta}h^{\alpha\beta},
 \nonumber\\
 a_1^{\mu\nu}[h]&=\tfrac12H\eta^{\mu\nu}-h^{\mu\nu},
 &
 a_2^{\mu\nu}[h]
  &=h^\mu{}_\lambda h^{\lambda\nu}
      -\tfrac12Hh^{\mu\nu}+b_2[h]\eta^{\mu\nu}.
 \label{eq:densities}
\end{align}
Thus
\[
 \sqrt{-\det q}\,q^{\mu\nu}
  =\eta^{\mu\nu}+\kappa a_1^{\mu\nu}[h]
              +\kappa^2A_2^{\mu\nu}[h,k],
\]
where
\[
 A_2[h,k]=a_1[k]+a_2[h],\qquad
 B_2[h,k]=b_1[k]+b_2[h].
\]

The supplied coordinate coefficients are
\begin{align}
 w^\mu[h]
   &=-\ERzero\partial_\alpha
                    \bigl(\sigma a_1^{\alpha\mu}[h]\bigr),
 \nonumber\\
 v^\mu[h,k]
   &=-\ERzero\partial_\alpha\left[
       \sigma A_2^{\alpha\mu}[h,k]
       +\sigma a_1^{\alpha\beta}[h]\partial_\beta w^\mu[h]
                              \right].
 \label{eq:coordinates}
\end{align}
The retarded massless inverse on these smooth
past-compact sources is understood by finite propagation,
equivalently by the zero-data prescription at \(t=0\).
No spatial decay is assumed on \(h,k\).

These are the coefficients of the divergence-form
density prescription
\[
 A_\varepsilon[q]
   =\eta^{-1}+\sigma(A_q-\eta^{-1}).
\]
They are not coefficients of a convolution regulator
or of a wave equation obtained by multiplying a
different source equation by \(\sigma\).

The handoff includes smoothness, fixed-width continuity,
retained-past dependence, and the laws
\begin{align}
 w[h']&=w[h]+\xi,\nonumber\\
 v[h',k']&=v[h,k]+X_\xi w[h]
                       +\zeta+\tfrac12X_\xi\xi.
 \label{eq:coordinatelaw}
\end{align}
In particular,
\begin{equation}
 w[K\xi]=\xi,\qquad \xi\in\gp.
 \label{eq:preflight}
\end{equation}
These are supplied classical chart data
\cite{XA,XAST}, not a quantum multiplication theorem.

For later kernel arguments the linear coefficient has
the explicit distributional representation
\begin{align}
 w^\mu[h](x)
   &=\int\Lambda_\varepsilon^{\mu,\alpha\beta}(x,z)
                          h_{\alpha\beta}(z)\,\dd^4z,
 \nonumber\\
 \Lambda_\varepsilon^{\mu,\alpha\beta}(x,z)
   &=\sigma(z^0)\mathsf a^{\nu\mu,\alpha\beta}
                          \partial_{z^\nu}G_{0,\mathrm R}(x-z),
 \label{eq:Lambda}\\
 \mathsf a^{\nu\mu,\alpha\beta}
   &=\tfrac12\eta^{\nu\mu}\eta^{\alpha\beta}
       -\tfrac12(\eta^{\nu\alpha}\eta^{\mu\beta}
                         +\eta^{\nu\beta}\eta^{\mu\alpha}).
 \nonumber
\end{align}
The unsmeared kernel is distributional. Smoothness of
its observation-smeared first derivatives does not
mean that \(\Lambda_\varepsilon(x,z)\) is a smooth
two-point function.

\subsection{Bare insertion representatives}

Fix \(f\in C_c^\infty(\mathcal O;\RR)\), let
\(K_f=\supp f\), and put
\[
 B^\mu[h,k]=w^\nu[h]\partial_\nu w^\mu[h]-v^\mu[h,k].
\]
The bare coefficients, on a smooth scalar configuration
\(\psi\), are
\begin{align}
 O^{(0)}(\psi)&=\int f\psi\,\dd^4x,\nonumber\\
 O^{(1)}[h](\psi)&=-\int f w^\mu[h]\partial_\mu\psi\,\dd^4x,
 \nonumber\\
 O^{(2)}[h,k](\psi)
   &=\int f\left(B^\mu[h,k]\partial_\mu\psi
                +\tfrac12w^\mu[h]w^\nu[h]
                                  \partial_\mu\partial_\nu\psi\right)
                                  \,\dd^4x.
 \label{eq:barecoeff}
\end{align}
Integration by parts fixes the representatives
\begin{align}
 c_0&=f,\qquad c_1[h]=\partial_\mu(fw^\mu[h]),\nonumber\\
 c_2[h,k]&=-\partial_\mu(fB^\mu[h,k])
          +\tfrac12\partial_\mu\partial_\nu
                              (fw^\mu[h]w^\nu[h]).
 \label{eq:tests}
\end{align}
Write
\[
 F_c(\psi)=\int c\psi\,\dd^4x,\qquad
 O_\kappa=F_{c_0}+\kappa F_{c_1}+\kappa^2F_{c_2}.
\]
The test \(f\) is one fixed label-space density.
It is not transformed during the background comparison.

\subsection{The one inherited scalar vertex}

The only interaction vertex used in the response
construction is
\begin{align}
 V_{1,g}[h;\psi]
   &=\tfrac12\int g\left(
        a_1^{\mu\nu}[h]\partial_\mu\psi\partial_\nu\psi
                            -m^2b_1[h]\psi^2\right)\,\dd^4x
 \nonumber\\
   &=-\tfrac12\pair{\psi}{L_{1,g}[h]\psi},
 \label{eq:vertex}\\
 L_{1,g}[h]u
   &=\partial_\mu(g a_1^{\mu\nu}[h]\partial_\nu u)
                                  +gm^2b_1[h]u.
 \label{eq:Lone}
\end{align}
In particular,
\begin{equation}
 L_{1,g}[h]u
  =g\left(\partial_\mu(a_1^{\mu\nu}[h]\partial_\nu u)
                              +m^2b_1[h]u\right)
       +(\partial_\mu g)a_1^{\mu\nu}[h]\partial_\nu u.
 \label{eq:switchterms}
\end{equation}
The derivative of \(g\) is part of the operator.

Formal self-adjointness follows by compactly supported
integration by parts. The sign ledger is
\begin{equation}
 S_0''=-P_m,\qquad
 V_{1,g}'=-L_{1,g}[h]\psi,\qquad
 V_{1,g}''=-L_{1,g}[h].
 \label{eq:Hessians}
\end{equation}
The positive Euler operator \(P_m\) is not the
variational Hessian \(S_0''\).

%=====================================================================
\section{The completion free field and its linear sector}
\label{sec:free}
%=====================================================================

\subsection{Green operators and topology}

Set
\[
 P_m=\Box+m^2,\qquad
 \Box=\partial_t^2-\Delta_{\mathbf x},\qquad
 \DD=C_c^\infty(\widehat M;\CC).
\]
The test space has its standard LF topology; on a fixed
compact \(K\), \(\DD_K\) has all smooth seminorms.

Let
\[
 \ER,\EA:\DD\longrightarrow C^\infty(\widehat M)
\]
be the positive Klein--Gordon retarded and advanced
Green operators. They satisfy
\begin{align}
 P_m\ERc=\ER P_mc=c,\qquad
 P_m\EAc=\EA P_mc=c,\nonumber\\
 \supp\ERc\subset J_\eta^+(\supp c),\qquad
 \supp\EAc\subset J_\eta^-(\supp c),\qquad
 \ER^{\mathrm T}=\EA.
 \label{eq:Green}
\end{align}
Existence, smoothness, finite propagation, and continuity
are the standard Green-operator facts on the globally
hyperbolic \emph{completion} \cite{BGP}.

For example, after spatial Fourier transformation,
\[
 (\ERc)(t)=\int_{-\infty}^t
       \frac{\sin((t-s)\Omega_m)}{\Omega_m}c(s)\,\dd s,
 \qquad \Omega_m=(-\Delta_{\mathbf x}+m^2)^{1/2},
\]
and the advanced formula is the transposed one.
The quotient at zero frequency has its continuous
value. These equations prescribe Green operators,
not quantum Cauchy data.

Put
\begin{equation}
 \DS=\EA-\ER,\qquad
 \Delta_{\mathrm S}^{\mathrm R}=-\ER,\qquad
 \Delta_{\mathrm S}^{\mathrm A}=-\EA.
 \label{eq:signs}
\end{equation}
Then \(\DS^{\mathrm T}=-\DS\).

\subsection{The scalar Wick algebra}

Fix one scalar Hadamard bisolution \(\Dplus\) on
\(\widehat M\times\widehat M\), with
\begin{equation}
 P_{m,x}\Dplus=P_{m,y}\Dplus=0,\qquad
 \Dplus-\Dplus^{\mathrm T}=i\DS.
 \label{eq:Hadamard}
\end{equation}
Its wavefront set has the Hadamard null-relation form:
both slot covectors are nonzero, and the first lies
in one fixed half of the null cone
\cite{Radzikowski}. No state on the conical product is
constructed by this choice.

Include zero covectors in the closed Minkowski causal
cones \(\overline V_\pm\), and define
\[
 \Xi_n=T^*\widehat M^n
            \setminus(\overline V_+^{\,n}\cup\overline V_-^{\,n}).
\]
Use the polynomial microcausal free presentation:
a polynomial functional has compact distributional
derivative kernels satisfying
\[
 \WF(F^{(n)}[\psi])\subset\Xi_n
\]
for every positive derivative order.

Denote the corresponding off-shell scalar algebra by
\(\Aoff\). Its Wick product is
\begin{equation}
 F\star G
   =\sum_{n\ge0}\frac{\hbar^n}{n!}
          \pair{F^{(n)}}{\Dplus^{\otimes n}G^{(n)}}.
 \label{eq:Wick}
\end{equation}
Each polynomial pair has a finite sum. The usual
microcausal contraction theorem supplies this free
product and its associativity \cite{FR}.
No interacting product theorem is imported from it.

Let \(\Jzero\) be the pointwise ideal generated by
\[
 F_{P_ma},\qquad a\in\DD.
\]
Because \(\Dplus\) is a bisolution,
\[
 F_{P_ma}\star G=F_{P_ma}G=G\star F_{P_ma}.
\]
The same observation applied inside a pointwise
multiple shows that \(\Jzero\) is a star ideal. Define
\begin{equation}
 \Afree=\Aoff/\Jzero,\qquad
 \boldsymbol\phi(c)=[F_c].
 \label{eq:freequotient}
\end{equation}
In particular,
\begin{equation}
 \boldsymbol\phi(P_ma)=0,\qquad
 [\boldsymbol\phi(c),\boldsymbol\phi(d)]
       =i\hbar\pair{c}{\DS d}.
 \label{eq:CCR}
\end{equation}

The off-shell insertion domain and the output sector are
\[
 \Loff=\{F_c:c\in\DD\}\cong\DD,\qquad
 \Afree^{(1)}=\{\boldsymbol\phi(d):d\in\DD\}.
\]
Zero is included in this homogeneous degree-one sector.
Its topology is the quotient test topology on
\(\DD/P_m\DD\). Formal \(\hbar\)-series are treated
coefficientwise.

\begin{remark}[No curved split calculus]
All wavefront calculations, covector sums, diagonal
kernels, \(\DF\), \(\EA\), and \(\ER\) in this file
belong to the completion. A sum of covectors using
Minkowski's affine identifications is not, by itself,
a cone condition on \(\mathcal M^\circ\).
\end{remark}

%=====================================================================
\part*{Stage B: admission and the single retarded map}
\addcontentsline{toc}{part}{Stage B: admission and the single retarded map}
%=====================================================================

\section{Two distinct admission statements}
\label{sec:admission}
%=====================================================================

\subsection{Split-type kernels on the completion}

\begin{definition}[Split type]
\label{def:split}
A distribution \(u\) on \(\widehat M^n\) is of
\emph{split type} if
\[
 (x_1,p_1;\ldots;x_n,p_n)\in\WF(u)
       \quad\Longrightarrow\quad p_1+\cdots+p_n=0.
 \label{eq:split}
\]
The sum uses the fixed affine cotangent identification
on \(\widehat M\).
\end{definition}

\begin{lemma}[Split type implies the cone condition]
\label{lem:split-cone}
A split-type distribution satisfies
\(\WF(u)\subset\Xi_n\).
\end{lemma}

\begin{proof}
If all \(p_j\) are future causal, their time components
are nonnegative. Their sum can vanish only if every
time component vanishes, which forces every causal
\(p_j\) to vanish. The zero total covector is not in a
wavefront set. The past case is identical.
\end{proof}

\begin{lemma}[Split calculus]
\label{lem:split-calculus}
The following operations preserve split type:
\begin{enumerate}[label=\textup{(\roman*)}]
\item translation kernels \(u(x-z)\) and diagonal delta
kernels;
\item differentiation, smooth multiplication, and adjoining
unused variables;
\item multiplication of two split distributions whose
variable sets have exactly one common slot;
\item integration of internal variables when the projection
is proper on the relevant support.
\end{enumerate}
The product in \textup{(iii)} is defined by the ordinary
wavefront product criterion.
\end{lemma}

\begin{proof}
For (i), pullback by the difference map gives endpoint
covectors \((p,-p)\); the delta cases are analogous.
The operations in (ii) do not enlarge wavefront sets,
and unused variables have zero covectors.

For (iii), lift both distributions to the union of their
variables. An opposing pair would have zero covectors
in every unshared slot. Split type would then force
the covector in the single shared slot to be zero
for each factor as well, contradicting a nonzero
wavefront covector. The product criterion therefore
holds. Its wavefront bound consists of sums of factor
covectors, allowing a zero factor contribution.
Those sums remain split.

For (iv), the pushforward bound retains only covectors
with zero internal components. The surviving external
components must still sum to zero. These are the
standard pullback, product, and proper-pushforward
rules \cite[Chapter~8]{Hormander}.
\end{proof}

\subsection{Joint polynomial admission}

\begin{proposition}[Joint admission through \(\kappa^2\)]
\label{prop:joint}
Regarded as polynomial functionals of the auxiliary
multiplet \((h,k,\psi)\), the three bare coefficients
\[
 F_{c_0},\qquad F_{c_1[h]},\qquad F_{c_2[h,k]}
\]
have compact derivative kernels of split type.
They therefore satisfy the joint microcausal cone test
on the completion.

This is a mathematical admission test on classical
background slots. It does not quantize those slots.
\end{proposition}

\begin{proof}
The zeroth coefficient has a smooth first derivative.
The first mixed kernel is a derivative, in its scalar
slot, of
\[
 f(x)\Lambda_\varepsilon(x,z).
\]
It is split by \eqref{eq:Lambda} and
Lemma~\ref{lem:split-calculus}.

It remains to inspect the highest kernels at second
order. With fixed tensor contractions and finite
derivatives understood, they are generated by
\begin{align}
 &f(x)\Lambda_\varepsilon(x,z)
                           \Lambda_\varepsilon(x,z'),\nonumber\\
 &f(x)G_{0,\mathrm R}(x-w)
       \partial_w\bigl(\sigma(w^0)
                       \delta(w-z)\delta(w-z')\bigr),\nonumber\\
 &f(x)G_{0,\mathrm R}(x-w)
       \partial_w\bigl(\sigma(w^0)\delta(w-z)
                            \partial_w\Lambda_\varepsilon(w,z')\bigr).
 \label{eq:generatingkernels}
\end{align}
The \(k\)-linear term is the corresponding two-slot
coordinate kernel.

In the first line the factors share only \(x\).
In the second line the two deltas share only \(w\);
their product then meets the Green kernel only at
\(w\). In the third line attach the delta to the
inner coordinate kernel at \(w\), then attach the
outer Green kernel at that single slot. Every step
is covered by Lemma~\ref{lem:split-calculus}.
Derivatives do not change this conclusion.

Every retarded path ends in \(K_f\), and every
coordinate source vertex carries \(\sigma\).
Consequently all effective background and internal
vertices lie in
\[
 D_f=J_\eta^-(K_f)\cap M_{T_*}.
\]
This set is compact. A smooth compact localization equal
to one near it can be inserted before each internal
pushforward; it does not change the kernel. The
pushforwards are thus proper.

Lower derivative kernels are obtained by multiplying by
smooth configurations and integrating the unused slots
with that same localization. They remain split.
Compactness follows from the same causal support
argument. Finally, \(\sigma\) is a smooth multiplier,
including at \(t=0\); no sharp boundary distribution
has been inserted. Lemma~\ref{lem:split-cone} completes
the proof.
\end{proof}

This establishes the supported form of the joint
admission reported in X-B \cite[\loc{prop:joint}]{XB}.
Its proof does not contain the product of a propagator
with itself at both identified endpoints.

\subsection{The scalar-only admission actually used}

\begin{proposition}[Smooth scalar-linear admission]
\label{prop:linear}
For fixed \((h,k)\), all tests in \eqref{eq:tests} lie
in \(\DD_{K_f}\). Their dependence on backgrounds is
continuous and polynomial at fixed \(\varepsilon\):
\(c_1\) is linear in \(h\), and \(c_2\) is quadratic
in \(h\) plus linear in \(k\).

Thus each \(F_{c_j}\) lies in the specified off-shell
linear insertion domain, and
\(\boldsymbol\phi(c_j)\) is a well-defined element of
\(\Afree^{(1)}\).
\end{proposition}

\begin{proof}
The supplied coordinate maps are smooth and continuous
in the background topology at fixed width.
Equations \eqref{eq:tests} use their finite products
and derivatives multiplied by \(f\).
This proves smoothness, support, polynomial dependence,
and continuity in every \(\DD_{K_f}\) seminorm.

As a functional of the only quantum configuration
\(\psi\), \(F_{c_j}\) has smooth first derivative \(c_j\)
and vanishing higher derivatives.
\end{proof}

The scalar test support lies in \(K_f\), while dependence
on classical backgrounds can extend through \(D_f\).
Neither inclusion is asserted to be an equality for
every configuration. Joint admission and
Proposition~\ref{prop:linear} are separate statements.

%=====================================================================
\section{The normalized scalar retarded map}
\label{sec:retarded}
%=====================================================================

\subsection{Definition, sign, normalization, and support}

\begin{definition}[The single map]
\label{def:retarded}
For \(h\) classical and the specified representative
\(F_c\in\Loff\), define
\begin{equation}
 \boxed{
 \Rone(V_{1,g}[h];F_c)
       =\boldsymbol\phi(d_c[h]),\qquad
 d_c[h]=-L_{1,g}[h]\EA c.}
 \label{eq:Rone}
\end{equation}
The notation
\[
 -\pair{c}{\ER L_{1,g}[h]\boldsymbol\phi}
\]
means \eqref{eq:Rone} by transposition. It is not an
application of a Green operator to an unspecified
unsmeared quantum field.
\end{definition}

For an arbitrary smooth scalar \(u\), define the smooth
linear response expression
\[
 \rho_{1,g}[h;u]=-\ER L_{1,g}[h]u.
\]
When \(P_mu=0\), this is the first retarded correction
with the inherited free input. For arbitrary \(u\),
it is simply its displayed smooth linear extension.

\begin{theorem}[Existence, support, normalization, and continuity]
\label{thm:retarded}
Equation \eqref{eq:Rone} defines a map
\[
 \Loff\longrightarrow\Afree^{(1)}
\]
linear in the insertion test and linear in the
quadratic vertex. Its output test is smooth and compact,
with
\begin{equation}
 \supp d_c[h]\subset
           \supp g\cap J_\eta^-(\supp c).
 \label{eq:retardedsupport}
\end{equation}
For every smooth \(u\),
\begin{equation}
 \pair{d_c[h]}{u}
   =-\pair{c}{\ER L_{1,g}[h]u}
   =\int c\,\rho_{1,g}[h;u]\,\dd^4x.
 \label{eq:normalization}
\end{equation}
The map is continuous in the smooth background and
compact-test topologies. It is additive in the vertex
and vanishes when
\[
 \supp g\cap J_\eta^-(\supp c)=\varnothing.
\]
No diagonal extension or subtraction is used.
\end{theorem}

\begin{proof}
The advanced Green operator sends \(c\) to a smooth
function supported in its causal past. The differential
operator \(L_{1,g}[h]\) has smooth coefficients supported
in \(\supp g\), giving \eqref{eq:retardedsupport}.

Since \(L_{1,g}[h]u\) is compactly supported,
transposition in \eqref{eq:Green} gives
\[
 -\pair{c}{\ER L_{1,g}[h]u}
       =-\pair{\EA c}{L_{1,g}[h]u}
       =-\pair{L_{1,g}[h]\EA c}{u}.
\]
The final integration by parts is legitimate because
the operator coefficients are compactly supported.

On each fixed-support test stage, Green-operator
continuity gives smooth seminorm estimates on the
compact interaction region. Multiplication by the
background coefficients and their finitely many
derivatives is continuous. The output lies in the fixed
space \(\DD_{\supp g}\). These estimates, and continuity
of the Green operator on the LF test space, give the
stated continuity and its passage to the quotient.
Linearity and the causal vanishing follow directly.
\end{proof}

This proves the principal X-B normalization and support
statement in the present presentation
\cite[\loc{def:retarded}, \loc{thm:retarded}]{XB}.
The signs agree with \eqref{eq:Hessians}:
the action inverse is \(-\ER\), not \(+\ER\).

\begin{proposition}[Supported patch agreement]
\label{prop:patch}
For retained laboratory tests, the response and dressing
tests agree for two auxiliary completion extensions
that coincide near the complete causal support used
by their calculation and have the same inherited
coordinate initial data.

The resulting compact pairings are unchanged by
transport through \(\iota\). This is exact patch
agreement, not a global conical representation.
\end{proposition}

\begin{proof}
Coordinate coefficients agree by the supplied
X-A-ST completion contract. For the response,
\(\EA c\) at an interaction point uses only the
causal region joining that point to \(\supp c\).
On a retained calculation this region lies in the
declared saturated compact support. The Green
equations there have the same source and terminal
prescription, so uniqueness gives the same test
action. Applying the same local \(L_{1,g}[h]\)
gives the same \(d_c\).

Transport of a compact distributional pairing through
a chart is defined by transposition on tests and
densities. It rewrites, rather than changes, that
pairing.
\end{proof}

For arbitrary completion tests outside the support
contract, no such conical transfer is asserted.

\subsection{The insertion domain is genuinely off shell}

\begin{proposition}[Failure of input descent to free classes]
\label{prop:input}
Although its output is in \(\Afree\), the map need not
factor through the free class of its input:
\begin{equation}
 \Rone(V_{1,g}[h];F_{P_ma})
        =\boldsymbol\phi(-L_{1,g}[h]a)
 \label{eq:inputwarning}
\end{equation}
need not vanish.
\end{proposition}

\begin{proof}
Use \(\EA P_ma=a\). For a concrete example let \(m=0\),
choose a real bump \(r\) supported where \(g=1\), and
take \(h=r\eta\). Then
\[
 L_{1,g}[h]u=\partial_\mu(r\partial^\mu u).
\]
Choose \(a=\partial_t r\), with \(\partial_t r\ne0\).
Against the smooth free solution \(u=t\),
\[
 \pair{-L_{1,g}[h]a}{t}
     =-\pair{a}{L_{1,g}[h]t}
     =-\int(\partial_t r)^2\,\dd^4x\ne0.
\]
Every element of the equation ideal vanishes on smooth
free solutions. Hence the output class is nonzero,
although \([F_{P_0a}]=0\).
\end{proof}

Thus the geometric prescription must supply the
representative before the map is applied.
This is not an ambiguity of \eqref{eq:Rone}.

\subsection{Tree-exactness on the named sector}

\begin{theorem}[No positive-\(\hbar\) correction]
\label{thm:hbar}
For the prescribed scalar-only pair, the output test
in \eqref{eq:Rone} has no \(\hbar\)-dependence:
\[
 [\Rone(V_{1,g}[h];F_c)]_{\hbar^r}=0,
                         \qquad r\ge1.
\]
It is also independent of the choice of Hadamard
contraction kernel with the stated commutator.
This is tree-exactness of this one map, not an
all-orders interacting identity.
\end{theorem}

\begin{proof}
For the verification, introduce only the auxiliary
kernel
\[
 \DF=\Dplus-i\EA
\]
and the expression
\[
 \CF(V,F_c)=VF_c+\hbar\pair{V'}{\DF c}.
\]
Here \(V=-\tfrac12\pair{\psi}{L\psi}\), with \(L\)
formally self-adjoint and compactly supported.
Both \(\EA c\) and \(\Dplus c\) are smooth: the
Hadamard kernel has no wavefront covector with a
zero second slot. Thus the expression is well defined.

Since \(F_c^{(n)}=0\) for \(n\ge2\),
\[
 V\star F_c=VF_c+\hbar\pair{V'}{\Dplus c}.
\]
Consequently
\[
 \frac{i}{\hbar}\bigl(\CF(V,F_c)-V\star F_c\bigr)
       =\pair{V'}{\EA c}
       =-\pair{\psi}{L\EA c}.
 \label{eq:onecheck}
\]
The numerator is exactly divisible by \(\hbar\).
The zero-contraction term and the Hadamard kernel
cancel; a second cross-contraction is absent.
Taking \(L=L_{1,g}[h]\) proves the assertion.

Changing a quadratic vertex's free Wick ordering adds
only a scalar-independent term. Its first functional
derivative is zero and does not change this calculation.
\end{proof}

The auxiliary expression has not been named a
time-ordered family. Its exact boundary is proved in
Section~\ref{sec:oneboundary}.

%=====================================================================
\section{First-order background identity and retained defects}
\label{sec:ward}
%=====================================================================

\subsection{An off-shell operator calculation}

Define the unswitched positive Euler coefficient
\[
 Q_h=L_{1,1}[h].
\]

\begin{lemma}[Pure-gauge Euler identity]
\label{lem:Q}
For every smooth vector field \(\xi\),
\begin{equation}
 Q_{K\xi}=D_\xi P_m-P_mX_\xi.
 \label{eq:Qidentity}
\end{equation}
This is an identity on arbitrary smooth scalars.
\end{lemma}

\begin{proof}
The density coefficients are
\[
 a_1^{\mu\nu}[K\xi]
    =(\partial\cdot\xi)\eta^{\mu\nu}
                       -\partial^\mu\xi^\nu-\partial^\nu\xi^\mu,
 \qquad
 b_1[K\xi]=\partial\cdot\xi.
\]
In particular,
\[
 \partial_\mu a_1^{\mu\nu}[K\xi]=-\Box\xi^\nu.
\]
Expansion of \(Q_{K\xi}u\) gives
\[
 (\partial\cdot\xi)P_mu
   -(\Box\xi^\nu)\partial_\nu u
   -2(\partial^\mu\xi^\nu)\partial_\mu\partial_\nu u.
\]
Expanding \(D_\xi P_mu-P_m(\xi^\nu\partial_\nu u)\)
gives the same expression.
\end{proof}

Define the off-shell completed first test and its class
by
\begin{equation}
 e_1[h]=c_1[h]-L_{1,g}[h]\EA f,\qquad
 \widehat A_1[h]=\boldsymbol\phi(e_1[h]).
 \label{eq:Aone}
\end{equation}
The free quantum field is held fixed in this comparison.

\begin{proposition}[Equation-ideal, coordinate, and switch terms]
\label{prop:variation}
For an arbitrary compact interior parameter \(\xi\),
not necessarily in \(\gp\),
\begin{align}
 e_1[h+K\xi]-e_1[h]
 ={}&P_m(X_\xi\EA f)
       +\partial_\mu\bigl(f(w[K\xi]-\xi)^\mu\bigr)
       -\Sigma_{g,\xi}\EA f,
 \label{eq:variation}\\
 \Sigma_{g,\xi}u
 ={}&(g-1)Q_{K\xi}u
       +(\partial_\mu g)a_1^{\mu\nu}[K\xi]\partial_\nu u.
 \label{eq:Sigma}
\end{align}
All three displayed tests are smooth and compact.
The first is explicitly in \(P_m\DD\).
\end{proposition}

\begin{proof}
The changed test is
\[
 \partial_\mu(fw^\mu[K\xi])
                          -L_{1,g}[K\xi]\EA f.
\]
Equation \eqref{eq:switchterms} gives
\(L_{1,g}[K\xi]=Q_{K\xi}+\Sigma_{g,\xi}\).
By Lemma~\ref{lem:Q} and \(P_m\EA f=f\),
\[
 Q_{K\xi}\EA f=D_\xi f-P_m(X_\xi\EA f).
\]
Substitute this identity and separate
\[
 \partial_\mu(fw^\mu[K\xi])
   =D_\xi f+
          \partial_\mu\bigl(f(w[K\xi]-\xi)^\mu\bigr).
\]
Compactness of the gauge support makes
\(X_\xi\EA f\) compact. The other supports are compact
because \(f\), or the coefficients of \(\Sigma\), are
compact.
\end{proof}

The coefficients of \(\Sigma_{g,\xi}\) are supported in
\[
 \supp\xi\cap
      \bigl(\supp(1-g)\cup\supp\dd g\bigr).
\]
This uses closed supports; it does not replace them by
the open set \(\{g\ne1\}\).

\begin{theorem}[First-order background identity]
\label{thm:ward}
For every smooth classical \(h\) and every
\(\xi\in\gp\),
\begin{equation}
 \widehat A_1[h+K\xi]=\widehat A_1[h]
                                  \quad\text{in }\Afree.
 \label{eq:ward}
\end{equation}
Equivalently,
\[
 \boldsymbol\phi(f)+\kappa\widehat A_1[h]
\]
is background invariant modulo \(\kappa^2\).
\end{theorem}

\begin{proof}
The coordinate term in \eqref{eq:variation} vanishes by
the supplied preflight identity \eqref{eq:preflight}.
The switch operator vanishes because \(g=1\) and
\(\dd g=0\) on a neighbourhood of \(\supp\xi\).
The remaining difference is
\[
 P_m(X_\xi\EA f)\in P_m\DD,
\]
whose field class is zero.
\end{proof}

This is the X-B first-order background identity
\cite[\loc{prop:variation}, \loc{thm:ward}]{XB}.
Off shell the difference is generally the nonzero
equation-ideal representative just displayed.
A general scalar pullback is not asserted to be an
automorphism of the unchanged free Klein--Gordon shell.
No interacting BRST differential or quantum master
equation has been defined.

\subsection{The retained edge and switch mechanisms}

Passing \eqref{eq:variation} to the free algebra gives
\begin{equation}
 \delta_\xi\widehat A_1
  =\int f(\xi-w[K\xi])^\mu\partial_\mu
                   \boldsymbol\phi\,\dd^4x
                    -\boldsymbol\phi(\Sigma_{g,\xi}\EA f),
 \label{eq:defectclass}
\end{equation}
where the integral denotes its compact test action.

For an initial-trivial compact parameter,
\begin{equation}
 \xi-w[K\xi]
   =\ERzero\bigl((1-\sigma)\Box\xi\bigr)
      +\ERzero\bigl(\sigma'a_1^{0\,\cdot}[K\xi]\bigr).
 \label{eq:edgegap}
\end{equation}
Indeed,
\[
 \Box w^\mu[K\xi]
      =\sigma\Box\xi^\mu-\sigma'a_1^{0\mu}[K\xi],
 \qquad \xi=\ERzero\Box\xi.
\]

The inherited edge witness is
\[
 \xi=\alpha(t)\beta(\mathbf x)\partial_1,\qquad
 0\ne\alpha\ge0,\quad
 \supp\alpha\subset(0,\varepsilon),\quad
 \int\beta\,\dd^3x=1,
\]
for which
\begin{equation}
 \int_{\RR^3}(w^1[K\xi]-\xi^1)(t,\mathbf x)\,\dd^3x
       =-\int_0^\varepsilon\sigma'(s)\alpha(s)\,\dd s<0,
       \qquad t\ge\varepsilon.
 \label{eq:edgewitness}
\end{equation}
It is outside \(\gp\).

\begin{remark}[What the witness does not prove]
A coordinate gap does not by itself prove a nonzero total
smeared on-shell response for every fixed \(f,g\).
The test can miss it, a resulting compact test can be
an equation zero, and the switch contribution in
\eqref{eq:defectclass} can also be present.
If an excluded edge parameter happens to lie in an
already chosen interaction plateau, the switch term
vanishes and the coordinate term is isolated. No edge
plateau is assumed to exist for every \(g\).
\end{remark}

Any established nonzero order-\(\kappa\),
order-\(\hbar^0\) defect cannot be canceled by terms
starting at \(\kappa^2\), or at positive \(\hbar\)-order,
while its lower-order prescription is fixed.
This coefficientwise observation does not identify
the present defect with the different VI-B defect
\cite{VIB}. The original algebra is neither shrunk
nor enlarged after the calculation.

%=====================================================================
\section{The mixed term, second-order boundary, and diagnostics}
\label{sec:second}
%=====================================================================

\subsection{Action on the first dressing}

Define
\[
 M[h]=\Rone(V_{1,g}[h];F_{c_1[h]})
        =\boldsymbol\phi(-L_{1,g}[h]\EA c_1[h]).
\]

\begin{proposition}[Mixed-term identification]
\label{prop:mixed}
For every smooth \(u\),
\begin{equation}
 \pair{-L_{1,g}[h]\EA c_1[h]}{u}
      =-\int f w^\mu[h]\partial_\mu
                         \rho_{1,g}[h;u]\,\dd^4x.
 \label{eq:mixed}
\end{equation}
Thus the map on the first bare insertion is exactly the
mixed term of the classical relational expansion,
with its minus sign.
\end{proposition}

\begin{proof}
Theorem~\ref{thm:retarded} identifies the left side with
\(\pair{c_1[h]}{\rho_{1,g}[h;u]}\).
Use \(c_1[h]=\partial_\mu(fw^\mu[h])\) and integrate
by parts.
\end{proof}

\begin{proposition}[Transformation of the mixed term]
\label{prop:mixed-transformation}
For \(\xi\in\gp\),
\begin{align}
 M[h+K\xi]-M[h]
  ={}&\boldsymbol\phi(-L_{1,g}[h]\EA D_\xi f)
       -\boldsymbol\phi(D_\xi c_1[h])
       -\boldsymbol\phi(D_\xi^2f).
 \label{eq:mixedchange}
\end{align}
This is a transformation law, not an invariance claim.
\end{proposition}

\begin{proof}
The two maps \(h\mapsto L_{1,g}[h]\) and
\(h\mapsto c_1[h]\) are linear, while
\(c_1[K\xi]=D_\xi f\). Expand
\[
 -(L_{1,g}[h]+L_{1,g}[K\xi])
                  \EA(c_1[h]+D_\xi f).
\]
On the plateau \(L_{1,g}[K\xi]=Q_{K\xi}\). For every
compact test \(a\),
\[
 -Q_{K\xi}\EA a
       =-D_\xi a+P_m(X_\xi\EA a).
\]
The second term vanishes in \(\Afree\).
Apply this with \(a=c_1[h]\) and \(a=D_\xi f\).
\end{proof}

For comparison, the supplied coordinate laws imply the
bare-test identities
\begin{align}
 c_1[h']-c_1[h]&=D_\xi f,\nonumber\\
 c_2[h',k']-c_2[h,k]
       &=D_\xi c_1[h]+\tfrac12D_\xi^2f+D_\zeta f.
 \label{eq:c2change}
\end{align}
To verify the second line algebraically, substitute
\eqref{eq:coordinatelaw} into \(B=w\cdot\partial w-v\).
It gives
\[
 B'=B+w\cdot\partial\xi+
                     \tfrac12\xi\cdot\partial\xi-\zeta.
\]
The change of \eqref{eq:barecoeff} is then
\[
 \int f\left(
    X_wX_\xi\psi+\tfrac12X_\xi^2\psi-X_\zeta\psi\right)\,\dd^4x.
\]
Transposition yields \eqref{eq:c2change}.

\subsection{What is missing at second dynamical order}

The physical first-vertex correction has an additional
factor \(\kappa\). Its retained expansion is
\begin{equation}
 O_\kappa+\kappa\Rone(V_{1,g};O_\kappa)
  =O^{(0)}+\kappa\widehat A_1
             +\kappa^2\bigl(O^{(2)}+M[h]\bigr)
                     \formal.
 \label{eq:truncation}
\end{equation}
Only \(c_0,c_1\) enter the response part through this
physical order.

For the classical second Euler coefficient write
\[
 L_{2,g}[h,k]u
   =\partial_\mu(gA_2^{\mu\nu}[h,k]\partial_\nu u)
                                  +gm^2B_2[h,k]u.
\]
The supplied classical response is
\begin{equation}
 \rho_{2,g}[h,k;u]
    =\ER L_{1,g}[h]\ER L_{1,g}[h]u
                             -\ER L_{2,g}[h,k]u.
 \label{eq:rhotwo}
\end{equation}
For \(P_mu=0\), the handoff includes
\begin{align}
 \rho_{1,g}[h';u]
   &=\rho_{1,g}[h;u]+X_\xi u,\nonumber\\
 \rho_{2,g}[h',k';u]
   &=\rho_{2,g}[h,k;u]+X_\xi\rho_{1,g}[h;u]
                         +X_\zeta u+\tfrac12X_\xi^2u.
 \label{eq:rhogaugelaw}
\end{align}
These are classical statements.

The full classical second relational coefficient is
\begin{equation}
 A_{2,\mathrm{cl}}[h,k;u]
    =O^{(2)}[h,k](u)
       -\int f w[h]\cdot\partial\rho_{1,g}[h;u]\,\dd^4x
       +\int f\rho_{2,g}[h,k;u]\,\dd^4x.
 \label{eq:classicalsecond}
\end{equation}
The last term is not supplied by the constructed
single first-vertex map. Writing its classical formula
does not define a normalized second-vertex map, an
iterated causal product, or a second-order quantum
identity.

\begin{proposition}[Pure-gauge diagnostic and the order boundary]
\label{prop:puregauge}
Let \(\xi\in\gp\), and take the complete two-jet
\[
 h=K\xi,\qquad k=\tfrac12\Lie_\xi^2\eta.
\]
Then in the degree-one free sector,
\[
 \widehat A_1[K\xi]=0,\qquad
 O^{(2)}=\tfrac12\int fX_\xi^2\boldsymbol\phi\,\dd^4x,
 \qquad
 M[K\xi]=-\int fX_\xi^2\boldsymbol\phi\,\dd^4x.
\]
Consequently the incomplete second coefficient in
\eqref{eq:truncation} is
\begin{equation}
 -\tfrac12\int fX_\xi^2\boldsymbol\phi\,\dd^4x,
 \label{eq:puregaugeremainder}
\end{equation}
which need not vanish.
\end{proposition}

\begin{proof}
The inherited laws give \(w=\xi\) and
\(v=\tfrac12X_\xi\xi\).
Hence the bare second coefficient is
\(\tfrac12\int fX_\xi^2\psi\).
Theorem~\ref{thm:ward}, compared with \(h=0\), gives
the first identity. Applying
\[
 -Q_{K\xi}\EA a=-D_\xi a
                  \quad\text{modulo }P_m\DD
\]
to \(a=D_\xi f\) gives \(M=-\boldsymbol\phi(D_\xi^2f)\),
which is the stated integral.

For nonvanishing, in the massless case choose
\(\xi=a(t,\mathbf x)\partial_t\) compactly supported
in \(U\), with \(a\partial_ta\ne0\) somewhere.
Choose an admissible observation test supported in a
small region where it has fixed sign.
Against the smooth free solution \(u=t\),
\(X_\xi^2t=a\partial_ta\), so the pairing is nonzero.
This diagnostic applies when those observation choices
are made; it is not a nonvanishing assertion for every
previously fixed test.
\end{proof}

Classically the missing term is
\(\rho_{2,g}=\tfrac12X_\xi^2u\) on this jet and cancels
\eqref{eq:puregaugeremainder}. This is a check of the
boundary, not a constructed quantum completion.

\subsection{Zero switch and a genuine nonzero class}

\begin{proposition}[Zero-switch diagnostic]
\label{prop:zero}
In the separate specialization \(g=0\),
\[
 \Rone(0;F_c)=0,
\]
but
\[
 O_\kappa
   =\boldsymbol\phi(f+\kappa c_1[h]+\kappa^2c_2[h,k])
\]
retains its coordinate dressing and differs from
\(\boldsymbol\phi(f)\) in general.
The zero-switch plateau algebra is trivial.
\end{proposition}

\begin{proof}
The response vanishes because \(L_{1,0}=0\), while
\eqref{eq:coordinates} does not contain \(g\).

For a nonzero class, take \(m=0\) and real
\(f\ge0\), \(f\ne0\). Let \(t_-=\min t(K_f)>0\),
choose \(0<a<b<t_-\), and let
\(0\ne\psi\in C_c^\infty((a,b))\) be nonnegative.
Choose a compact spatial cutoff equal to one near
the spatial projection of \(D_f\), and use it to
define \(h=\psi(t)\eta\) near the retained past.
Finite propagation gives on the observation region
\[
 w^0=-\Psi(t),\qquad w^i=0,\qquad
 \Psi(t)=\int_0^t\sigma(s)\psi(s)\,\dd s>0.
\]
Thus \(c_1=-\partial_t(f\Psi)\), and
\[
 \pair{c_1}{t}=\int f\Psi\,\dd^4x>0.
\]
An equation-ideal element would vanish on the smooth
free solution \(t\), so \(\boldsymbol\phi(c_1)\ne0\).

Finally \(\operatorname{int}\{g=1\}\) is empty at
\(g=0\). This specialization creates no new nonempty
gauge algebra.
\end{proof}

\subsection{A sourced sign check}

For \(m=0\), \(h=\psi\eta\), one has
\[
 a_1[h]=\psi\eta^{-1},\qquad a_2[h]=0,\qquad
 L_{1,g}[h]u=\partial_\mu(g\psi\partial^\mu u).
\]
On a time-dependent region of the retained causal
compact, let \(g=g_0(t)\) and \(\psi=\psi(t)\), with
spatial cutoffs outside that compact. Define
\[
 \Psi(t)=\int_0^t\sigma\psi\,\dd s,\qquad
 \Gamma(t)=\int_0^t g_0\psi\,\dd s.
\]
For the free solution \(u=t\),
\[
 w^0=-\Psi,\qquad \rho_{1,g}[h;t]=-\Gamma.
\]
Therefore
\begin{equation}
 \ev_t\widehat A_1[h]
      =\int f(\Psi-\Gamma)\,\dd^4x,\qquad
 \ev_t M[h]=-\int f\Psi\,g_0\psi\,\dd^4x.
 \label{eq:sourcedsign}
\end{equation}
Here \(\ev_t\) means classical evaluation of a linear
class, justified precisely in
Section~\ref{sec:normalization}. It is not a quantum
expectation value or a state on the Wick algebra.

%=====================================================================
\part*{Stage B-ii: four scope closures inside the same file}
\addcontentsline{toc}{part}{Stage B-ii: four scope closures}
%=====================================================================

\section{C1: the response-scope proposition}
\label{sec:scope}
%=====================================================================

\begin{definition}[The retained response catalogue]
\label{def:catalogue}
The retained field-theoretic outputs are
\[
 \boldsymbol\phi(c_0),\quad
 \boldsymbol\phi(c_1[h]),\quad
 \boldsymbol\phi(c_2[h,k]),\quad
 \Rone(V_{1,g}[h];F_c),\quad
 \widehat A_1[h],\quad M[h],
\]
and their stated background differences.
In the fourth entry \(c\) is an independently specified
test. The permitted background-dependent insertion
needed through physical order \(\kappa^2\) is \(c_1[h]\),
which gives \(M[h]\).
\end{definition}

\begin{proposition}[Scope of the constructed responses]
\label{prop:scope}
Every output in Definition~\ref{def:catalogue}:
\begin{enumerate}[label=\textup{(\roman*)}]
\item belongs to the homogeneous degree-one sector,
with zero allowed;
\item has a smooth compact representative test with no
\(\hbar\) in it;
\item depends continuously on the classical backgrounds
at fixed \(\varepsilon\);
\item is polynomial in those backgrounds, of degree at
most two in \(h\) and one in \(k\);
\item contains no contraction between two background
slots, no quantum metric line, and no gravitational
action.
\end{enumerate}
\end{proposition}

\begin{proof}
For the bare coefficients this is
Proposition~\ref{prop:linear}. For the response it is
Theorems~\ref{thm:retarded} and \ref{thm:hbar}.
The map is linear in \(h\) for independent \(c\);
substituting \(c_1[h]\) gives quadratic \(M[h]\).
The completed coefficient and its differences are sums
of the same expressions.

Every quantum output is of the form
\(\boldsymbol\phi(d)\). Background slots are evaluated
on \(h,k\), not contracted through a background
two-point kernel. The response uses a scalar Green
kernel against a smooth test.
\end{proof}

\begin{remark}[Two necessary scope qualifications]
The proposition concerns the retained \emph{responses},
not every auxiliary object written in this file.
The vertex is quadratic, the one-contraction verification
contains a cubic pointwise term, the commutator is a
scalar multiple of the identity, and the finite
instruments are channels, not degree-one field elements.

Also, applying the unweighted formula to the
background-dependent \(c_2[h,k]\) would produce
degree-three \(h\)-terms and mixed \(hk\)-terms.
Such a substitution is not in the degree bound of
Proposition~\ref{prop:scope} and contributes only at
physical order \(\kappa^3\) after the extra vertex factor.
No universal degree-two bound for arbitrary compositions
is claimed.
\end{remark}

These qualifications retain the intended X-B-ii scope
without confusing its list of outputs with the larger
algebra containing the auxiliary vertex
\cite[\loc{prop:scope}]{XBii}.

%=====================================================================
\section{C2: the one-contraction boundary}
\label{sec:oneboundary}
%=====================================================================

\subsection{The auxiliary Feynman kernel}

\begin{lemma}[Properties of \(\DF\)]
\label{lem:DF}
The completion distribution
\[
 \DF=\Dplus-i\EA
\]
has the following properties:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(\DF^{\mathrm T}=\DF\);
\item \(P_{m,x}\DF=-i\delta_{\mathrm{diag}}\);
\item \(\DF c\) is smooth for \(c\in\DD\);
\item
\[
 \WF(\DF)\supset
 \{(x,x;\xi,-\xi):x\in\widehat M,\ \xi\ne0\};
 \label{eq:fullconormal}
\]
\item
\[
 \DF-\Dplus=-i\EA,\qquad
 \DF-\Dplus^{\mathrm T}=-i\ER.
\]
\end{enumerate}
The transpose symmetry in \textup{(i)} is not a
Hermiticity assertion.
\end{lemma}

\begin{proof}
Use \(\EA^{\mathrm T}=\ER\) and
\(\Dplus^{\mathrm T}=\Dplus-i(\EA-\ER)\):
\[
 \DF^{\mathrm T}
   =\Dplus^{\mathrm T}-i\ER
   =\Dplus-i\EA=\DF.
\]
The bisolution equation and \(P_m\EA=I\) give (ii).
The no-zero-slot wavefront condition implies that
\(\Dplus\) maps compact smooth tests to smooth
functions; adding \(-i\EA c\) gives (iii).

The wavefront set of the diagonal delta is
\[
 \WF(\delta_{\mathrm{diag}})
   =\{(x,x;\xi,-\xi):\xi\ne0\}.
\]
A differential operator with smooth coefficients
does not enlarge wavefront sets. Hence (ii) implies
\[
 \WF(\delta_{\mathrm{diag}})
           \subset\WF(\DF),
\]
which proves (iv). These are standard kernel and
wavefront facts \cite[Chapter~8]{Hormander}.
The last identities follow directly from the definition
and \eqref{eq:Hadamard}.
\end{proof}

\subsection{The one pair and its causal requirements}

For this subsection the admissible pair has the exact
form
\begin{align}
 \textup{(A1)}\quad&
 V(\psi)=-\tfrac12\pair{\psi}{L\psi},
 \quad L\text{ formally self-adjoint with smooth compact
 differential coefficients},\nonumber\\
 \textup{(A2)}\quad&
 F_c,\qquad c\in\DD.
 \label{eq:pair}
\end{align}
The physical vertex is the specialization
\(L=L_{1,g}[h]\). The general pair is used to describe
the scope of the same one-contraction calculation,
not to install another physical interaction.

Define
\begin{align}
 \CF(V,F_c)&=VF_c+\hbar\pair{V'}{\DF c},\nonumber\\
 \CF(F_c,V)&=F_cV+\hbar\pair{c}{\DF V'}.
 \label{eq:CF}
\end{align}
These are polynomial functionals on smooth
configurations. The compact function \(V'=-L\psi\)
can be used as a smooth test for each configuration.

\begin{proposition}[Symmetry and the one-contraction identity]
\label{prop:check}
For \textup{(A1)--(A2)},
\[
 \CF(V,F_c)=\CF(F_c,V),
\]
and
\begin{equation}
 \frac{i}{\hbar}\bigl(\CF(V,F_c)-V\star F_c\bigr)
       =\pair{V'}{\EA c}
       =-F_{L\EA c}.
 \label{eq:check}
\end{equation}
For \(L=L_{1,g}[h]\), its free class is exactly
\eqref{eq:Rone}.
\end{proposition}

\begin{proof}
Transpose symmetry of \(\DF\) proves the first claim.
The Wick expansion terminates at one contraction
because \(F_c^{(2)}=0\). Thus the difference is
\[
 \hbar\pair{V'}{(\DF-\Dplus)c}
        =-i\hbar\pair{V'}{\EA c}.
\]
Multiply by \(i/\hbar\), then use \(V'=-L\psi\)
and formal self-adjointness of \(L\).
\end{proof}

\begin{proposition}[Restricted causal factorization]
\label{prop:causal}
Let \(S\) contain the compact coefficient support of \(L\).
Then:
\begin{enumerate}[label=\textup{(\roman*)}]
\item if \(S\cap J_\eta^-(\supp c)=\varnothing\),
\[
 \CF(V,F_c)=V\star F_c;
\]
\item if \(S\cap J_\eta^+(\supp c)=\varnothing\),
\[
 \CF(V,F_c)=F_c\star V.
\]
\end{enumerate}
For the physical vertex, the retarded map vanishes
under condition \textup{(i)}.
\end{proposition}

\begin{proof}
The first difference is
\(-i\hbar\pair{V'}{\EA c}\), and its supports are
disjoint under (i).
For (ii), use
\[
 F_c\star V=VF_c+\hbar\pair{V'}{\Dplus^{\mathrm T}c}
\]
and \(\DF-\Dplus^{\mathrm T}=-i\ER\).
The supports are again disjoint.
\end{proof}

These are the restrictions of symmetry, contraction,
and causal factorization to the pair
\eqref{eq:pair}. They are not a causal hierarchy.

\subsection{Why the same tools do not supply a family}

A time-ordered family would require multilinear maps
\(T_n\) on declared domains, including symmetry,
causal factorization, compatible Wick expansions,
and extension normalizations
\cite{DFreference,BF,FR}. Its quadratic--quadratic
contractions would require more than
\eqref{eq:CF}.

\begin{theorem}[Non-extension by the present product tools]
\label{thm:nonextension}
\begin{enumerate}[label=\textup{(\roman*)}]
\item The square \(\DF\cdot\DF\) fails the ordinary
wavefront product criterion at every point
\((x,x)\) of the diagonal.
\item The corresponding Wick square
\(\Dplus\cdot\Dplus\), and its finite differentiated
products, pass that criterion.
\item A general quadratic--quadratic time-ordered
contraction requires an identified-endpoint product
of two Feynman lines, with the vertex differential
operators applied. The undifferentiated case is
exactly \(\DF\cdot\DF\).
The present file selects neither a diagonal extension
nor its normalization.
\item Consequently no value of \(T_2(V,V)\) for the
required general quadratic sector, no \(R(V;V)\), and
no general time-ordered family is released by the
one-contraction check.
\end{enumerate}
This is not a theorem that renormalized extensions or
time-ordered families do not exist.
\end{theorem}

\begin{proof}
At \(z=(x,x)\), Lemma~\ref{lem:DF} gives both
\[
 (z;\xi,-\xi),\qquad(z;-\xi,\xi)
\]
in \(\WF(\DF)\), for every \(\xi\ne0\).
They sum to zero. This is precisely the failure of the
sufficient product criterion
\cite[Theorem~8.2.10]{Hormander}.

For \(\Dplus\), opposite wavefront covectors would
require a nonzero first-slot covector and its negative
both in the same fixed half null cone. This is
impossible. Differentiation cannot enlarge the
wavefront set, so the same argument handles its
finite differentiated products.

For the general quadratic pair the formal
two-contraction term is
\[
 \frac{\hbar^2}{2}
        \pair{V''}{\DF^{\otimes2}V''}.
\]
The kernels of the local differential Hessians have
the form \(-L_x\delta(x-x')\). Contracting them identifies
the two first endpoints and the two second endpoints
of the tensor product of Feynman lines.
Before differentiation, that operation is the
pullback of
\[
 \DF(x_1,y_1)\otimes\DF(x_2,y_2)
\]
to \(x_1=x_2,\ y_1=y_2\).

For an order-zero compact coefficient \(L=\lambda(x)\),
which is a test case for a proposed general
quadratic rule, the expression is literally the
weighted square
\(\lambda(x)\lambda(y)\DF(x,y)^2\).
Thus a general prescription cannot be obtained from
the present unextended product criterion.
Differential vertices require separately justified
differentiated versions of the same endpoint operation;
their formal notation supplies no such theorem.

Away from the diagonal the Hadamard Feynman wavefront
orientation permits the usual square: for positive
time separation the kernel agrees locally with
\(\Dplus\), for negative separation with
\(\Dplus^{\mathrm T}\), and at equal-time separated
points the relevant distributions are smooth.
What is missing is the diagonal extension and a
normalization for it. Those are additional
renormalization data \cite{Radzikowski,BF}.
None is chosen here.

The original quadratic--linear pair avoids the
two-line operation because \(F_c^{(2)}=0\).
Its successful check therefore supplies no value
for the missing quadratic sector and no family.
\end{proof}

\begin{remark}[The exact quantifier]
Failure of the sufficient criterion is not a proof
of nonexistence of a distributional extension.
Nor does it prove that every specially chosen
differential numerator has a nonzero obstruction
after all cancellations. No such universal
noncancellation claim is needed.

The phrase ``no second pair'' means no second
quadratic pair completing the asserted time-ordered
programme by these tools. It does not mean that
every other elementary pair, such as a pair of
smooth linear fields, is undefined.
\end{remark}

This is the scope boundary of X-B-ii, read as a
non-extension statement for the sheet's tools
\cite[\loc{thm:nonextension}]{XBii}, not as a new
universal no-go theorem.

%=====================================================================
\section{C3: normalization and degree-one uniqueness}
\label{sec:normalization}
%=====================================================================

\subsection{The equation ideal and separation by solutions}

\begin{lemma}[Degree-one part of the ideal]
\label{lem:graded}
The pointwise ideal \(\Jzero\) is graded by field degree,
and
\[
 \Jzero\cap\Aoff^{(1)}
        =\{F_{P_ma}:a\in\DD\}.
\]
Hence
\[
 \boldsymbol\phi(d)=0
          \quad\Longleftrightarrow\quad d\in P_m\DD.
 \label{eq:linearzero}
\]
\end{lemma}

\begin{proof}
An ideal element is a finite sum
\(\sum_iG_iF_{P_ma_i}\). Decompose each polynomial
\(G_i\) into its homogeneous parts. Each homogeneous
component of the sum is still in the ideal.
Its degree-one part consists exactly of constant
multiples of the linear generators, hence is
\(F_{P_ma}\) for some compact \(a\).

The equality \(F_d=F_{P_ma}\) implies
\(d=P_ma\), since the functionals can be tested on
arbitrary compact smooth configurations.
\end{proof}

Let \(\mathrm{Sol}_{\mathrm{sc}}(P_m)\) denote smooth
solutions with spatially compact support, equivalently
compactly supported Cauchy data.

\begin{lemma}[Detection by spatially compact solutions]
\label{lem:detect}
For \(d\in\DD\), the following are equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(d\in P_m\DD\);
\item \(\DS d=0\);
\item \(\pair{d}{u}=0\) for every
\(u\in\mathrm{Sol}_{\mathrm{sc}}(P_m)\);
\item \(\pair{d}{u}=0\) for every smooth free solution.
\end{enumerate}
Also
\[
 \DS\DD=\mathrm{Sol}_{\mathrm{sc}}(P_m).
\]
\end{lemma}

\begin{proof}
Green identities immediately give (i)\(\Rightarrow\)(ii)
and integration by parts gives (i)\(\Rightarrow\)(iv).
The implication (iv)\(\Rightarrow\)(iii) is immediate.

For completeness, the image statement follows from
Cauchy uniqueness. If \(u\) has compact Cauchy data,
choose a temporal cutoff \(\chi\) which is zero early
and one late. The source
\[
 a=P_m(\chi u)=[P_m,\chi]u
\]
is compact, because its time support is bounded and
\(u\) is spatially compact. Retarded and advanced
uniqueness give
\[
 \ERa=\chi u,\qquad \EAa=(\chi-1)u.
\]
Thus \(\DS(-a)=u\). The converse image inclusion is
finite propagation.

If (iii) holds, then for every \(a\in\DD\),
\[
 0=\pair{d}{\DS a}=-\pair{\DS d}{a},
\]
so (ii) follows. If (ii) holds, let
\(v=\EAd=\ERd\). Its support is contained in
\[
 J_\eta^-(\supp d)\cap J_\eta^+(\supp d),
\]
a compact causal diamond on Minkowski space.
Hence \(v\in\DD\) and \(P_mv=d\), proving (i).
This is the usual Green exactness mechanism
\cite[Section~3.4]{BGP}.
\end{proof}

For a smooth free solution define
\begin{equation}
 \ev_u\boldsymbol\phi(d)=\pair{d}{u}.
 \label{eq:evaluation}
\end{equation}
This is a well-defined linear evaluation on
\(\Afree^{(1)}\). The spatially compact solution
evaluations separate that sector.
It is not asserted to be a state on the full quantum
algebra.

\subsection{The whole-domain normalization sheet}

\begin{definition}[Normalization sheet]
\label{def:sheet}
For fixed \(h\), a map
\[
 \mathcal R:\Loff\longrightarrow\Afree^{(1)}
\]
satisfies the normalization sheet when
\begin{equation}
 \ev_u\mathcal R(F_c)
       =-\pair{c}{\ER L_{1,g}[h]u}
 \quad
 \text{for every }c\in\DD,\
 u\in\mathrm{Sol}_{\mathrm{sc}}(P_m).
 \label{eq:sheet}
\end{equation}
No linearity or continuity of the candidate is assumed.

If formal \(\hbar\)-series are allowed in the codomain,
\eqref{eq:sheet} is required coefficientwise as an
equality of the complete series.
\end{definition}

\begin{theorem}[Normalization determines the map]
\label{thm:unique}
The map \(\Rone(V_{1,g}[h];\cdot)\) is the unique map
satisfying Definition~\ref{def:sheet}.
\end{theorem}

\begin{proof}
Existence is \eqref{eq:normalization}.
For another candidate and each \(c\), write
\(\mathcal R(F_c)=\boldsymbol\phi(d'_c)\).
The difference of the two tests annihilates every
spatially compact free solution. By
Lemma~\ref{lem:detect},
\[
 d'_c-d_c[h]\in P_m\DD.
\]
Lemma~\ref{lem:graded} gives equality of the output
classes. No condition on the dependence of \(d'_c\)
on \(c\) is needed.

For formal series apply the same argument to every
\(\hbar\)-coefficient.
\end{proof}

The normalization is on all off-shell insertion tests,
not only on \(f,c_1,c_2\), a finite catalogue, or a
finite set of classical evaluations. This is the
whole-domain uniqueness recorded in X-B-ii
\cite[\loc{thm:unique}]{XBii}.

\begin{remark}[Exact normalization is stronger than a classical limit]
If one imposed only equality of the
\(\hbar^0\)-coefficient, positive-\(\hbar\) changes
would not be excluded by that weaker requirement.
Theorem~\ref{thm:unique} uses either the
\(\hbar\)-independent degree-one sheet, or the full
coefficientwise equality explicitly stated above.
It is not uniqueness among all possible interacting
normalizations.
\end{remark}

\subsection{Contact modifications}

\begin{corollary}[Sheet-compatible contacts are zero classes]
\label{cor:contact}
Let \(N_h:\DD\to\DD\) be any proposed modification,
and set
\[
 \mathcal R_N(F_c)
      =\boldsymbol\phi(d_c[h]+N_hc).
\]
Then the following are equivalent:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(\mathcal R_N\) satisfies the same normalization sheet;
\item \(N_hc\in P_m\DD\) for every \(c\);
\item \(\boldsymbol\phi(N_hc)=0\) for every \(c\).
\end{enumerate}
In that case \(\mathcal R_N=\Rone(V_{1,g}[h];\cdot)\).
The same statement applies coefficientwise to formal
modifications.
\end{corollary}

\begin{proof}
Subtract \eqref{eq:sheet} for the two maps and apply
Lemmas~\ref{lem:detect} and \ref{lem:graded}.
\end{proof}

For example \(N_hc=P_m(\chi c)\), with smooth \(\chi\),
is compatible and invisible in the free algebra.
A nonzero multiplication \(N_hc=wc\) is not compatible
on the whole domain: choose a spatially compact
solution nonzero at a point where \(w\ne0\), then
choose \(c\) in a small neighbourhood with
\(\pair{wc}{u}\ne0\). Compact Cauchy data can realize
such a local nonzero solution.

\begin{corollary}[The first-order identity is independent of compatible contacts]
\label{cor:ward}
No sheet-compatible degree-one contact changes
Theorem~\ref{thm:ward}. Such a contact is neither
needed to repair that identity nor able to break it
without leaving the sheet.
\end{corollary}

\begin{proof}
The modified and original response maps have equal
classes for each background and insertion.
Their completed first coefficients therefore coincide.
\end{proof}

This does not classify contact ambiguities of another
response. In particular, VII-H concerns a different
free massless susceptibility and a different
normalization class \cite{VIIH}; its contact theorem
is not a contact repair for \(\widehat A_1\).

%=====================================================================
\section{C4: Branch B is recorded and not opened}
\label{sec:branchB}
%=====================================================================

An independent quantum-metric branch would have to
supply at least:
\begin{enumerate}[label=\textup{(B\arabic*)}]
\item a quadratic graviton action, gauge fixing, and ghost
sector, with the action-Hessian normalization specified;
\item graviton retarded and advanced Green operators,
their transposition conventions, and support theorems;
\item a graviton Hadamard two-point matrix satisfying the
chosen equations, commutator, and metric-slot
wavefront condition;
\item a contraction domain in the scalar, metric, and
ghost variables, including the status of \(k\) and of
the coordinate kernels after \(h\) is quantized;
\item the mixed vertex with quantum \(h\), its Wick
ordering, field degrees, and coupling and
\(\hbar\)-gradings;
\item an actual wavefront and extension analysis for the
required mixed graphs and any sourced graviton tree;
\item a Ward or BRST formulation on that new domain;
\item a comparison rule specifying any intended agreement
with the present scalar-only map.
\end{enumerate}

None of those data is supplied here.
The classical coordinate kernel is not a graviton
propagator, \(\Dplus\) is not a metric two-point matrix,
and joint admission with classical background slots
is not a quantum-metric contraction theorem.
The scalar map has no latent graviton contribution.

XII-B is a different track and is not an input.
Retrieval summaries do not constitute analytical
premises. This section asserts neither existence,
failure, nor correction of Branch B
\cite[Section~C4]{XBii}.

%=====================================================================
\section{Fixed host and the retained comparison boundaries}
\label{sec:fixedhost}
%=====================================================================

\begin{proposition}[Host stability]
\label{prop:fixedhost}
The response map, a regulator-label replacement, a
change of installed evaluation table, or a proposed
subtraction-scale label does not evolve
\(G,\mathcal W,T\), or the supplied future relation.
A regulator replacement may change the auxiliary
classical prescription and its values.
\end{proposition}

\begin{proof}
The inherited host data are fixed before the field
construction. Every map constructed here returns a
free-algebra class or a finite operational output.
No map with a new host metric as its output has been
defined.
\end{proof}

The comparison boundaries are:
\begin{itemize}
\item VII-I's finite-width free algebra is not the
present response map, and its sharp-limit transmission
data are not a fixed-\(\varepsilon\) theorem for this
prescription \cite{VIII}.
\item V-B's admissible currents and Newton--wave cubic
response concern another representation and response
problem \cite{VB}.
\item IX-C-ST's partial product and IX-D-ST's support
interfaces are not the map \eqref{eq:Rone}.
Wordwise coherence is not a causal lift
\cite{IXcontrols}.
\item The separated-support logarithmic obstruction
of VII-C is not repaired by a scalar-linear,
tree-exact map \cite{VIIC}.
\item X-A-conv and X-Enter retain their different
convolution prescription and conditional readout
contract \cite{ConvControls}.
\end{itemize}
None of these comparisons is a premise of
Theorem~\ref{thm:retarded}.

%=====================================================================
\part*{The finite Enter interface on the inherited regular route}
\addcontentsline{toc}{part}{The finite Enter interface}
%=====================================================================

\section{Installed evaluations and a complete clock instrument}
\label{sec:instrument}
%=====================================================================

\subsection{What the finite table contains}

Choose a finite catalogue of already supplied data:
background descriptions, completion-test descriptions,
smooth free-solution descriptions, and numerical reports
or certified intervals for expressions such as
\[
 \ev_{u_i}\Rone(V_{1,g}[h_i];F_{c_i}),
 \qquad
 \ev_{u_i}\widehat A_1[h_i],
 \qquad
 \int f_i\rho_{2,g}[h_i,k_i;u_i]\,\dd^4x.
 \label{eq:tableentries}
\]
The last entry is explicitly classical.
It is not an evaluation of an unconstructed
second-order quantum map.

By \eqref{eq:normalization}, the first entry is a
well-defined classical number. Providing its finite
description or an enclosure is an input to the
finite encoding. No general algorithm for arbitrary
smooth data and no universal finite encoding of
arbitrary real numbers is asserted.

The catalogue can always be nonempty, for instance by
using a zero background. Nontrivial scalar response
values are possible: in the massless diagnostic of
Proposition~\ref{prop:input}, let
\[
 C=\int(\partial_t r)^2\,\dd^4x>0,\qquad
 a=C^{-1}\partial_t r,\qquad c=P_0a.
\]
Then
\[
 \ev_t\Rone(V_{1,g}[r\eta];F_c)=-1,
\]
whereas the same specified \(c\) at \(h=0\) gives zero.
This is a mathematical evaluation of representatives,
not a field preparation or measurement.
Its supported host use requires the stated laboratory
support contract.

\subsection{Clock extraction}

Retain one finite live clock
\(C\cong\CC^{2^q}\), \(q\ge1\), with calibration basis
\(\{\ket{\tau_k}\}\). Let \(S\) contain every surviving
finite register and define
\[
 D_k=\bra{\tau_k}_C\otimes I_S,\qquad
 P_k=\proj{\tau_k}.
\]

\begin{theorem}[Complete destructive-clock instrument]
\label{thm:instrument}
The channel
\begin{equation}
 \mathcal M_C(X)
   =\sum_k\proj k_R\otimes D_kXD_k^\dagger
   =\sum_k\proj k_R\otimes
      \Tr_C[(P_k\otimes I_S)X(P_k\otimes I_S)]
 \label{eq:clock}
\end{equation}
is CPTP and satisfies
\[
 \Tr_R\mathcal M_C(X)=\Tr_CX.
\]
These equalities hold after adjoining every untouched
reference. The quantum clock names are consumed;
there is no quantum \(C\)-output.
\end{theorem}

\begin{proof}
The recorded Kraus operators are
\(\ket k_R\otimes D_k\). Their squared moduli sum to
\[
 \sum_kP_k\otimes I_S=I_{CS}.
\]
Thus the map is completely positive and trace preserving.

Expand
\(X=\sum_{k,\ell}\kb{\tau_k}{\tau_\ell}\otimes X_{k\ell}\).
Each output block is \(X_{kk}\), which proves both
formulae and the discard identity. The same calculation
with \(S\) enlarged by any reference proves the complete
statement.
\end{proof}

This is the relevant I-B/En09-A instrument, here
proved on the declared finite interface
\cite{IB,En09}. Conditioning can steer an entangled
reference's conditional state; the identity concerns
its unconditional marginal.

\begin{proposition}[Calibration and recovery limitations]
\label{prop:clocklimits}
\begin{enumerate}[label=\textup{(\roman*)}]
\item If distinct calibrated labels \(t_k\) must all equal
the time of one fixed readout event whenever their
outcomes occur, at most one code has positive
probability.
\item The channel has no left inverse on all clock inputs.
\item Its record marginal does not determine the residual.
\item No finite clock matrices satisfy
\([T_C,H_C]=iI_C\).
\end{enumerate}
\end{proposition}

\begin{proof}
Distinctness gives (i).
The clock inputs
\((\ket{\tau_0}\pm\ket{\tau_1})/\sqrt2\), with the
same surviving state, have identical outputs, proving
(ii). Inputs \(P_0\otimes\rho\) and
\(P_0\otimes\widetilde\rho\) have the same record and
different residuals, proving (iii).
The trace of a commutator is zero, while
\(\Tr(iI_C)\ne0\), proving (iv).
\end{proof}

Thus a nontrivial multi-outcome table readout is a
programme/history calibration unless a separate
event-time contract is supplied. It does not consume
geometric time.

\subsection{A fixed-interface approximation theorem}

To state an actual channel error, choose scaled installed
values \(v_k\in[-1,1]\) and encode them by a binary
report with probabilities
\[
 p_y(v_k)=\frac{1+yv_k}{2},\qquad y=\pm1.
\]
Define
\[
 \mathcal I_v(X)
    =\sum_{k,y}\proj{k,y}_R\otimes
                     p_y(v_k)D_kXD_k^\dagger.
\]

\begin{proposition}[Finite-table channel and error]
\label{prop:table}
The map \(\mathcal I_v\) is CPTP, and on the same
complete interfaces
\begin{equation}
 \ddia(\mathcal I_v,\mathcal I_{\widetilde v})
     =\tfrac12\max_k|v_k-\widetilde v_k|,
 \qquad
 \ddia(\Lambda,\Gamma)=\tfrac12\|\Lambda-\Gamma\|_\diamond.
 \label{eq:tableerror}
\end{equation}
The identity holds with arbitrary retained references.
\end{proposition}

\begin{proof}
Use Kraus operators
\[
 \sqrt{p_y(v_k)}\,\ket{k,y}_R\otimes D_k.
\]
Their completeness is immediate.
For an input operator \(X\), with a reference included,
the difference has orthogonal output blocks
\[
 \frac{y(v_k-\widetilde v_k)}2X_{kk}.
\]
Its trace norm is at most
\[
 \max_k|v_k-\widetilde v_k|\sum_k\|X_{kk}\|_1
       \le\max_k|v_k-\widetilde v_k|\,\|X\|_1.
\]
The last inequality is trace-norm contraction of
pinching, which is an average of unitary conjugations.
A basis input at an index attaining the maximum
saturates the bound. Divide by two.
\end{proof}

This is a fixed binary-probability encoding.
Small numerical changes of deterministic, mutually
orthogonal output words do not in general have small
diamond distance. No such false continuity claim is
used.

The abstract instrument is implementable under the
inherited finite unitary and destructive-readout
permissions, using a calibrated basis change and,
for the binary report, a finite probe rotation.
An exact abstract gate permission is not a hardware
synthesis or a field-to-qubit preparation certificate.

%=====================================================================
\section{Scheduled locality, regular transport, and resources}
\label{sec:transport}
%=====================================================================

\subsection{The complete finite dependency graph}

Let a finite site set carry finite local matrix algebras.
At each tick the complete channel is a product of
channels on a prescribed partition into blocks,
including every retained classical record register.

For a final set \(Y\), define the backwards sets by
\[
 N_{t,t}(Y)=Y,\qquad
 N_{s,t}(Y)=
 \bigcup\{B:\ B\text{ is a block at tick }s+1,\
                         B\cap N_{s+1,t}(Y)\ne\varnothing\}.
\]

\begin{theorem}[Scheduled locality]
\label{thm:locality}
For the complete record-retaining network,
\[
 (\Phi_{s+1}^*\circ\cdots\circ\Phi_t^*)(\mathcal A(Y))
                   \subseteq\mathcal A(N_{s,t}(Y)).
 \label{eq:scheduled}
\]
Equal input reductions on the complete backwards
region give equal corresponding output reductions.
Interventions outside that region cannot change its
joint output law.

If every dependency edge is separately assigned a
future-causal regular host route, scheduled ancestry
is contained in the supplied host causal relation.
\end{theorem}

\begin{proof}
A channel adjoint is unital. A block disjoint from the
current observable support therefore sees only an
identity factor and cannot enlarge that support.
A meeting block can enlarge it only inside that block.
Backward induction proves \eqref{eq:scheduled}.
Testing all output observables proves the reduction
and intervention claims. Reference factors are retained
unchanged in the same argument.

Concatenating the separately certified dependency
routes proves the final containment.
\end{proof}

This is not a reconstruction of the host causal cone
from a schedule. Scheduler consistency alone does not
supply that identification \cite{En08A}.

\subsection{The inherited target-adapted route}

Choose a straight developed spatial segment of length
\(\ell>0\) inside a regular convex tube of the inherited
laboratory. After local clock readout, lookup, and a
positive processing buffer, depart at time \(t_{\mathrm d}\).

Install \(n+1\) stations,
\[
 \mathbf x_j=\mathbf x_0+
                \frac jn(\mathbf x_n-\mathbf x_0),
       \qquad j=0,\ldots,n,
\]
with payload registers \(A_j\cong\CC^{2^p}\), \(p\ge1\).
Initialize \(A_1,\ldots,A_n\) in independent blank states
\(\omega\). At hop \(j\), exchange \(A_{j-1}\) and \(A_j\).
The developed tick duration is \(\ell/n\), so the
cross-port dependencies are null; same-station
dependencies are timelike.

The operational permission is explicit: a finite
two-register SWAP is available on those declared
tick-boundary ports, with the certified regular
causal dependencies. This is the X-A-ST/I-B route
permission, not a derived continuous-time interaction
Hamiltonian. Classical report wires are separately
declared causal.

\begin{theorem}[Reference-preserving regular transport]
\label{thm:transport}
For every payload--reference input \(\rho_{A_0E}\),
after hop \(j\) the state, under the fixed register
identifications, is
\[
 \rho_{A_jE}\otimes
      \bigotimes_{\substack{0\le i\le n\\i\ne j}}\omega_{A_i}.
\]
The logical payload channel is exactly the identity,
including arbitrary references. The finite classical
report is forwarded with its complete joint law.
Arrival occurs at
\[
 t_{\mathrm d}+\ell,\qquad
 T_{\mathrm d}+\ell/c_{\mathrm{host}}.
\]
Every dependency route remains in the regular part.
\end{theorem}

\begin{proof}
At the first hop a SWAP exchanges the unknown payload
factor, including all correlations with \(E\), with
the next independent blank. The previous station is
left blank. Repeating gives the formula by induction.

Classical forwarding copies only diagonal record
data; it is not cloning an arbitrary quantum state.
The chosen tick durations make the spatial dependency
edges null in the developing chart, hence causal
in the supplied host geometry.
\end{proof}

The reference \(E\) may include the clock record and
any inaccessible spectators after the instrument.
Consequently the route transports the complete
conditional finite payload, not just its marginal.
This is finite register transport relative to a
prescribed cone \cite{IB,En08B}, not a field propagation
or gravity-generated-entanglement theorem.

\subsection{Resource and complete-output error ledger}

A conservative preallocated reserve is
\[
 q+(n+1)p+(n+1)b+w_{\mathrm{lookup}}+1
\]
qubits for a \(q\)-qubit clock, \(p\)-qubit payload
at each station, \(b\)-bit report reserves,
lookup workspace, and one optional probe.
Classical table storage, including its finitely many
reports, is separately charged. External references
are declared rather than counted as laboratory memory.

The route uses \(n\) register SWAPs and \(n\) classical
report transmissions. Clock basis change, destructive
readout, lookup, and their durations are also charged.
The clock names are absent after readout.
All returned or discarded blank factors and retained
report copies are included in the complete output
declaration.

If compatible complete implemented channels have
encoder, decoder, table, and per-stage allowances,
then
\begin{equation}
 \ddia(\widetilde\Gamma,\Gamma)
  \le\min\left\{1,\epsilon_{\mathrm{enc}}
        +\epsilon_{\mathrm{dec}}+\epsilon_{\mathrm{table}}
                                    +\sum_j\epsilon_j\right\}.
 \label{eq:telescoping}
\end{equation}
To prove this, subtract the two compositions, replace
one factor at a time, and use diamond-norm contraction
under channels and the triangle inequality
\cite{Watrous}. An arrival or failure probability is
controlled only if that record is part of the compared
complete output.

No statistical estimate, external oracle duration,
inaccessible storage, or decoder cost is silently
absorbed into a logical-only error allowance.

%=====================================================================
\section{Release, significance, and successor boundary}
\label{sec:release}
%=====================================================================

\begin{theorem}[X-B-ST local Grade S release]
\label{thm:release}
For the inherited X-A-ST host presentation and the
frozen scalar-only sheet:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the bare ramp dressing is admitted through
\(\kappa^2\), jointly on the completion and separately
on the scalar-linear domain actually used;
\item the single normalized map
\(\Rone(V_{1,g}[h];F_c)\) is constructed on all
specified off-shell compact linear representatives;
its output test is smooth, compact, continuous,
retarded, and \(\hbar\)-independent;
\item the completed coefficient \(\widehat A_1\) obeys
the background identity on the originally frozen
\(\gp\);
\item the mixed term and its transformation are constructed;
the classical \(\rho_{2,g}\) and its role are displayed,
without a constructed second-order quantum completion;
\item the one-contraction pair passes its restricted
symmetry and causal-factorization checks, but the
present tools do not define the required second
quadratic pair or a time-ordered family;
\item the whole-domain normalization uniquely determines
the degree-one response class; every sheet-compatible
contact modification vanishes in the free algebra;
\item Branch B is recorded and not opened;
\item the finite installed encoding has a complete
destructive-clock instrument, scheduled locality,
reference-preserving regular transport, and the
stated resource and error ledger.
\end{enumerate}
These statements constitute local Grade S only.
\end{theorem}

\begin{proof}
Propositions~\ref{prop:joint} and \ref{prop:linear}
prove (i). Theorems~\ref{thm:retarded} and
\ref{thm:hbar}, with Proposition~\ref{prop:patch},
prove (ii). Proposition~\ref{prop:variation} and
Theorem~\ref{thm:ward} prove (iii).
Section~\ref{sec:second} proves (iv).
Lemma~\ref{lem:DF}, Propositions~\ref{prop:check}
and \ref{prop:causal}, and
Theorem~\ref{thm:nonextension} prove (v).
Theorem~\ref{thm:unique} and
Corollaries~\ref{cor:contact}--\ref{cor:ward} prove
(vi). Section~\ref{sec:branchB} records exactly (vii).
Theorems~\ref{thm:instrument},
\ref{thm:locality}, and \ref{thm:transport}, together
with the finite-table and resource statements,
prove (viii). These provide the clauses of
Definition~\ref{def:grade}.
\end{proof}

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.34\linewidth}p{.20\linewidth}X@{}}
\toprule
Object & Status & Exact boundary\\
\midrule
Bare coefficients through \(\kappa^2\) &
Admitted &
No metric quantization\\
Single scalar retarded map &
Constructed &
Off-shell input representatives; degree-one outputs\\
Completed first background identity &
Proved &
Modulo \(\kappa^2\), not \(\kappa^3\)\\
Retained mixed coefficient &
Constructed &
Generally not invariant alone\\
Classical second scalar response &
Displayed &
Not a constructed second-order quantum map\\
Whole-domain normalization &
Determines the map &
Exact sheet, not classical-limit data alone\\
Compatible degree-one contacts &
Zero classes &
No classification for another response\\
General quadratic time ordering &
Not constructed &
Diagonal extension and normalization absent\\
Branch B &
Recorded only &
No quantum metric data supplied\\
Finite Enter interface &
Constructed under permissions &
No continuum-field acquisition\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Significance without a priority inflation}

The useful mathematical feature is the combination of
an explicit compact output test, a precise normalization
domain, and an equally explicit stopping point.
The construction shows how a first background identity
can hold exactly in the free quotient while edge and
switch defects remain visible outside its algebra.

The source results are credited to Cheung's X-B and
X-B-ii \cite{XB,XBii}. The present contribution is their
supported host transfer and audit. No claim of a new
physical prediction or a solution of interacting
quantum gravity follows from this packaging.

\subsection{Handoff toward X-C-ST}

A successor receives:
\begin{enumerate}
\item the single map \eqref{eq:Rone} on its off-shell
insertion domain;
\item its normalization, support, continuity, and
tree-exactness statements;
\item the first-order identity and its explicit off-shell
equation-ideal difference;
\item the bare second dressing and retained mixed term;
\item the displayed classical \(\rho_{2,g}\);
\item the one-contraction boundary and normalization
uniqueness theorem.
\end{enumerate}
It does not receive a \(T_2\) family, a second-order
quantum dynamical map, a graviton tree, a loop triangle,
a two-point response, or a sharp-width limit.

\subsection{Standing non-claims}

The inherited characteristic gravitational quantum
coefficient is not admitted here. This qualification
is essential: the scalar-linear dressings and the
single scalar response \emph{are} admitted and
constructed in the senses proved above.

XV-B is not opened as a downstream quantum-admission
exit. XVI-N is not discharged. O.1 remains
\loc{NOT_CLOSED} in the inherited scope.
The H2, I3, and J2 requirements are not passed by this
file. In the version-specific source record,
XIV-I's constructed internal-amplitude result does not
supply the full characteristic composition left open
at I3/J2 \cite{BoundaryRecords}.

No conclusion here repairs VII-C or X-A-conv.
No Einstein equation, metric emergence, gravity-generated
entanglement, Page curve, or ultraviolet completion is
derived.

%=====================================================================
\appendix
\section{Debugging and quantifier audit}
\label{app:debug}
%=====================================================================

\begin{longtable}{@{}p{.34\linewidth}p{.61\linewidth}@{}}
\toprule
Potential error & Correction\\
\midrule
\endfirsthead
\toprule
Potential error & Correction\\
\midrule
\endhead

A second conical construction &
The host is the already supplied X-A-ST instance.
No cone, signature, or worldsheet is rederived.\\[4pt]

Using global conical hyperbolicity &
All global Green operators and free fields live on
the declared Minkowski completion.\\[4pt]

Treating split type as coordinate-free &
The covector sum uses the fixed Minkowski affine
identification only.\\[4pt]

Equating joint admission and scalar-linear admission &
The former tests polynomial background-slot kernels;
the latter freezes backgrounds and admits smooth
scalar tests. Neither quantizes \(h,k\).\\[4pt]

Using the positive Euler inverse as an action inverse &
\(S_0''=-P_m\), so the action retarded inverse is
\(-\ER\). The response test is \(-L_{1,g}\EA c\).\\[4pt]

Dropping \(\partial g\) &
The divergence-form operator explicitly contains
\((\partial_\mu g)a_1^{\mu\nu}\partial_\nu\).\\[4pt]

Applying \(R_1\) to an unspecified on-shell class &
The input is the supplied representative \(F_c\).
Equation-zero input representatives can have nonzero
responses.\\[4pt]

Putting \(\hbar\) into the response test because of the CCR &
The CCR belongs to the free product.
The single contraction's \(\hbar\) cancels exactly
against the normalization factor.\\[4pt]

Calling every object degree one &
Only the retained response catalogue is degree one.
The vertex and auxiliary verification have other
field degrees, and instruments are channels.\\[4pt]

A universal background degree-two claim &
The degree bound applies to the retained list.
Substituting background-dependent \(c_2\) into the
unweighted map can raise background degree.\\[4pt]

Calling the mixed term positive &
Its sign is
\(-\int f w\cdot\partial\rho_{1,g}\).
The pure-gauge test verifies that sign.\\[4pt]

Claiming a second-order identity from bare admission &
The missing \(\rho_{2,g}\) contribution is explicitly
separated from the constructed truncation.\\[4pt]

Inferring a nonzero total defect for every test from
the coordinate edge integral &
The edge and switch terms are separate.
Smearing, equation-zero tests, and cancellations
must be checked for each proposed total witness.\\[4pt]

Using only \(h=K\xi\) in a quadratic pure-gauge test &
The complete \(k=\tfrac12\Lie_\xi^2\eta\) is retained.\\[4pt]

Turning failure of a sufficient wavefront criterion
into universal nonexistence &
The theorem is non-extension by the present tools.
No renormalized extension is ruled out universally.\\[4pt]

Claiming every nonzero differential vertex necessarily
has a nonzero uncancelled square obstruction &
The universal endpoint operation fails its criterion;
special numerators require separate analysis.
No complete cancellation or noncancellation theorem
is asserted.\\[4pt]

Interpreting ``no second pair'' literally for every pair &
The missing pair is the quadratic sector needed for
a time-ordered family. Elementary linear products
already exist in the free algebra.\\[4pt]

Deducing uniqueness from a finite table or from one \(f\) &
Normalization is imposed for all \(c\in\DD\) and a
separating family of free solutions.\\[4pt]

Deducing quantum uniqueness from a leading classical
limit alone &
The formal-series version requires full
coefficientwise normalization.\\[4pt]

Using a classical solution evaluation as a quantum state &
\(\ev_u\) is a linear evaluation on the degree-one
sector, not a state on the Wick algebra.\\[4pt]

Calling an installed table a field measurement &
Only supplied evaluations or enclosures are encoded.
No continuum-field preparation or detector is built.\\[4pt]

Calling history calibration exact event-time readout &
At one fixed event, exact agreement of distinct
labels is deterministic.\\[4pt]

Equating a schedule with the physical causal past &
Physical containment uses separately certified
regular routes.\\[4pt]

Treating a tick SWAP as a derived relativistic Hamiltonian &
The finite port-to-port permission is an explicit
host-model input.\\[4pt]

Claiming no quantum coefficient of any kind is admitted &
Only the separate inherited characteristic coefficient
remains unadmitted. The scalar classes constructed
in this file retain their proved status.\\

\bottomrule
\end{longtable}

%=====================================================================
\section{Comparison-only operational sources}
\label{app:comparisons}
%=====================================================================

The following source records are not premises of the
retarded-map theorem.

En08-A distinguishes scheduler consistency from a
supplied geometric causal representation.
En08-B treats finite transport relative to a prescribed
null boundary. En09-A treats the destructive relational
clock instrument. En10-A concerns estimation of a
source weight under a separate finite access model
\cite{En08A,En08B,En09,En10}.
None derives \(\EA\), acquires the present continuum
field, or evolves the host metric.

The OpenAI catalogue components mentioned in the plan
are retained only as possible finite-interface
comparisons:
\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.23\linewidth}XX@{}}
\toprule
Record & Possible comparison & Not supplied by that comparison\\
\midrule
Oai283-A &
Complete-channel error accounting &
\(R_1\) or admission of the inherited graph\\
Oai272-A &
A declared finite instrument support witness &
An all-orders completion\\
Oai276-A &
Finite clock-outcome information bounds &
Gravity-generated entanglement\\
Oai119-A &
A declared finite readout bound &
Exclusion of classical gravity\\
Oai279-A &
Finite gate-list comparison &
A field-to-qubit preparation\\
Oai212-A, Oai365-A &
Passage and boundary certificates &
A derived metric\\
Families 265, 271, 362, 376 &
No use here &
A Page curve, lapse, or Einstein dynamics\\
\bottomrule
\end{tabularx}
\end{center}

This table asserts no independent verification of those
catalogue proofs. The channel and transport identities
actually used in X-B-ST were proved in this file.

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

\paragraph{Authorship and submission metadata.}
The author field is intentionally empty.
Attribution of source results to N.~Cheung does not
assign authorship of this adaptation.
Authorized author, affiliation, and submission metadata
must be supplied for a signed submission.

\paragraph{Source status.}
X-B and X-B-ii are cited by their rendered companion
records and explicit versions. X-A-ST is the supplied
preceding unsigned manuscript. Comparison-only sheets
are identified by manuscript identifier and supplied
record, without invented publication dates or DOIs.
Source citation is not an assertion of catalogue-wide
proof verification.

\paragraph{Analytical scope.}
The background metric coefficients are classical.
The quantum construction is the scalar free algebra
and the single normalized degree-one response map.
The joint background-slot wavefront test does not
install a quantum metric.

\paragraph{Operational scope.}
Finite readout consumes a finite clock encoding.
It does not consume spacetime time, measure an
unrepresented field, or supply a thermodynamic
implementation. Transport uses declared finite
regular-route permissions.

\paragraph{Verification.}
No successful \loc{pdflatex} execution, proof-assistant
verification, numerical computation, symbolic-program
execution, hardware synthesis, experiment, independent
referee review, or journal acceptance is asserted.
The mathematical claims are the definitions and written
proofs in this file.

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

\bibitem{XB}
N.~Cheung,
\emph{Landmark X-B: Fixed-regulator admission and one
retarded correction for the ramp relational scalar,
with the metric kept classical}.
\loc{X-B-v3.tex}, version 3.
Enter Calculus Companion, rendered manuscript record.
\url{https://enter.grok.me/browse/X-B}.
Especially \loc{lem:split-cone},
\loc{lem:split-calculus}, \loc{prop:joint},
\loc{prop:linear}, \loc{def:retarded},
\loc{thm:retarded}, \loc{thm:hbar},
\loc{prop:variation}, \loc{thm:ward},
\loc{prop:mixed-transformation},
\loc{prop:zero}, and \loc{thm:release}.

\bibitem{XBii}
N.~Cheung,
\emph{Landmark X-B-ii: Scope closure for the scalar-only
retarded map: the one-contraction boundary, the
normalization sheet, and the Branch B input list}.
\loc{X-B-ii-v1.tex}, version 1.
Enter Calculus Companion, rendered manuscript record.
\url{https://enter.grok.me/browse/X-B-ii}.
Especially \loc{prop:scope}, \loc{lem:DF},
\loc{prop:check}, \loc{prop:causal},
\loc{thm:nonextension}, \loc{thm:unique},
and \loc{cor:contact}.

\bibitem{XAST}
\emph{X-A-ST: Supported classical covariance of a
fixed-ramp relational scalar}.
Supplied preceding unsigned manuscript,
\loc{X-A-ST.tex}.
Supported chart, declared flat completion, signature
ledger, plateau, classical handoff, and finite-interface
contract. This is the supplied adaptation, not an
independently attributed Cheung publication.

\bibitem{XA}
N.~Cheung,
\emph{Landmark X-A: Classical covariance of one
fixed-regulator switched relational scalar through
order \(\kappa^2\)}.
\loc{X-A.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-A}.

\bibitem{IB}
N.~Cheung,
\emph{I-B: A Finite Enter Spacetime}.
\loc{I-B-v1.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/I-B}.
Prescribed conical product, signature, finite clock,
calibration, locality, and transport.

\bibitem{EnterI}
N.~Cheung,
\emph{Manuscript I}.
\loc{I.tex}, version 2.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/I}.
Operational characterization of destructive readout
within its stated strategy class; background only.

\bibitem{EnterPhilosophy}
N.~Cheung,
\emph{The Philosophy of Enter: Destructive Readout,
Quantum Recovery, and the Refusal of Identification}.
\loc{EntPhil-v2.tex}, version 2.
Introductory essay.
\url{https://enter.grok.me/enter}.
Interpretive background, not a theorem premise.

\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.
\url{https://arxiv.org/abs/0806.1036}.
Green operators and the compact-source/free-solution
exact sequence are used on the Minkowski completion.

\bibitem{Hormander}
L.~H\"ormander,
\emph{The Analysis of Linear Partial Differential
Operators I: Distribution Theory and Fourier Analysis},
second edition, Springer, 1990;
Classics in Mathematics reprint, 2003.
\url{https://doi.org/10.1007/978-3-642-61497-2}.
Chapter~8: wavefront sets, products, pullbacks,
and pushforwards.

\bibitem{Radzikowski}
M.~J.~Radzikowski,
``Micro-local approach to the Hadamard condition in
quantum field theory on curved space-time,''
\emph{Communications in Mathematical Physics}
\textbf{179} (1996), 529--553.
\url{https://doi.org/10.1007/BF02100096}.
The Hadamard wavefront condition is applied here
only on the declared completion.

\bibitem{FR}
K.~Fredenhagen and K.~Rejzner,
\emph{Perturbative algebraic quantum field theory}.
arXiv:1208.1428, version 2, 2013.
\url{https://arxiv.org/abs/1208.1428}.
Free microcausal functional products and the distinction
from renormalized interacting maps.

\bibitem{DFreference}
M.~D\"utsch and K.~Fredenhagen,
``Causal perturbation theory in terms of retarded
products, and a proof of the Action Ward Identity,''
\emph{Reviews in Mathematical Physics}
\textbf{16} (2004), 1291--1348.
\url{https://arxiv.org/abs/hep-th/0403213}.
Off-shell insertion and retarded-product framework;
no higher family is imported into this construction.

\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.
\url{https://arxiv.org/abs/math-ph/9903028}.
Distribution extension and renormalization framework;
not a selected extension for the omitted pair.

\bibitem{VIIH}
N.~Cheung,
\emph{VII-H}.
\loc{VII-H.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-H}.
A different massless free-scalar susceptibility and its
declared contact ambiguity; comparison only.

\bibitem{VIII}
N.~Cheung,
\emph{VII-I}.
\loc{VII-I.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-I}.
Finite-width free algebra and sharp-transmission
boundaries; comparison only.

\bibitem{VB}
N.~Cheung,
\emph{V-B}.
\loc{V-B-v1.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/V-B}.
Admissible-current, Newton--wave cubic, and instrument
comparisons; not the present retarded map.

\bibitem{VIB}
N.~Cheung,
\emph{VI-B}.
\loc{VI-B-v1.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VI-B}.
Different switched Einstein--scalar defect and
coefficientwise order obstruction.

\bibitem{IXcontrols}
N.~Cheung,
\emph{IX-C-ST} and \emph{IX-D-ST}.
Version 1 supplied manuscript records,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/IX-C-ST};
\url{https://enter.grok.me/browse/IX-D-ST}.
Partial-product and causal-interface negative controls.

\bibitem{VIIC}
N.~Cheung,
\emph{VII-C}.
\loc{VII-C.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-C}.
Separated-support logarithmic obstruction under its
own declared extension requirements.

\bibitem{ConvControls}
N.~Cheung,
\emph{X-A-conv} and \emph{X-Enter}.
Version 1 supplied manuscript records,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-A-conv};
\url{https://enter.grok.me/browse/X-Enter}.
Different convolution prescription and readout contract.

\bibitem{En08A}
N.~Cheung,
\emph{En08-A}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En08-A}.
Scheduler consistency and geometric-representation
distinctions.

\bibitem{En08B}
N.~Cheung,
\emph{En08-B}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En08-B}.
Resource-controlled finite transport relative to
a prescribed null boundary.

\bibitem{En09}
N.~Cheung,
\emph{En09-A: Relational Enter}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En09-A}.
Complete destructive-clock instrument and conditioning.

\bibitem{En10}
N.~Cheung,
\emph{En10-A}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En10-A}.
Finite source-weight estimation under a separately
specified access contract.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Author's manuscript and errata:
\url{https://jhwatrous.github.io/books-and-courses.html}.
Finite channels, instruments, and diamond-norm
comparisons.

\bibitem{BoundaryRecords}
N.~Cheung,
\emph{XIV-H}, \emph{XIV-I}, and \emph{XIV-J},
version 1; technical \emph{XV-B}, version 2;
and \emph{XVI-N}, revision r12.
Version-specific supplied manuscript records,
Enter Calculus Companion.
\url{https://enter.grok.me/browse}.
Documentary analytical boundaries only; no
characteristic graph theorem is imported.

\end{thebibliography}

\end{document}