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\hypersetup{
  pdftitle={X-C-ST: Second-order scalar completion on the inherited host},
  pdfauthor={Ngo Cheung},
  pdfsubject={Direct linear response maps, completed background covariance,
  and a local Grade S finite interface}
}

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\title{\textbf{X-C-ST: Second-order scalar completion\\
on the inherited X-A-ST/X-B-ST host}\\[5pt]
\large Direct linear response maps, completed background covariance,\\
and the propagated edge defect}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We complete the fixed-width scalar-only response sector of
X-B-ST on the supported host already prescribed by X-A-ST.
The new constructions are a second-vertex map and an
iterated first-vertex map on specified off-shell linear
insertions. Both are defined directly by a finite formal
test-level M{\o}ller expansion, without a time-ordered
product hierarchy. Their smooth compact output tests
satisfy retarded support and continuity bounds, and their
sum reproduces the inherited second scalar response.

The completed relational field is background invariant
through order \(\kappa^2\) in the unchanged free scalar
algebra on the gauge algebra frozen before the calculation.
The proof gives an explicit three-term variation:
a compact free-equation term, a dressed coordinate defect,
and a dressed switch defect. In particular, it retains
the propagation of the first-order edge gap into the
second coefficient. A complete pure-gauge two-jet closes
the remainder left in X-B-ST, while a massless conformal
example checks the iteration and its signs.

These coefficients contain no positive power of
\(\hbar\) in their tests and no Hadamard contraction
kernel. This tree-exactness concerns the specified linear
rooted prescription, not arbitrary interacting products.
The diagonal product boundary of X-B-ST remains unchanged.

The contribution is a supported realization and
normalization, sign, and defect audit of Cheung's X-C
construction, rather than a priority claim for the
underlying quadratic-field method. Its prospective value
is an explicit second-order baseline that separates
background covariance from singular-product admission
and finite readout. A complete destructive-clock
instrument encodes supplied evaluations and admits
reference-preserving regular transport. It does not
acquire a continuum field, evolve the host metric,
or establish spacetime emergence.
\end{abstract}

\statementbox{Counting rule: one first-order map and one second-order pair}{%
X-B-ST released the single first-vertex map
\(R_1(V_{1,g};\cdot)\), including its use on the first
bare dressing, and proved the completed identity through
\(\kappa^1\).

X-C-ST releases
\(R_1(V_{2,g};\cdot)\) and
\(\tfrac12R_2(V_{1,g},V_{1,g};\cdot)\).
Their sum on \(F_f\) represents the one inherited term
\(\int f\rho_{2,g}\).
A successor citing both files receives one first-order
map and one second-order pair, not two copies of the
second scalar completion.}

\tableofcontents

%=====================================================================
\section{Purpose, source attribution, and the fixed host}
\label{sec:host}
%=====================================================================

\subsection{The task and its limits}

The analytical source is Cheung's X-C, version 3
\cite{XC}. The supplied X-A-ST and X-B-ST manuscripts
fix the supported classical prescription and its first
scalar response \cite{XAST,XBST}. The first response
is recalled as an inherited input, not reconstructed
as a new achievement of this file.

Quadratic scalar perturbations admit particularly direct
linear-field constructions. Their relation to
perturbative agreement has a broader established
literature \cite{HW,DHP}. That context does not supply
the particular regulated coordinate prescription or
the supported host contract used here. Nor is a general
renormalized product family imported from that literature.

All coupling expansions below are finite formal expansions
over
\[
 \mathbb D_2=\RR[\kappa]/(\kappa^3),
\]
with complexification for tests and field coefficients.
An equality modulo \(\kappa^3\) means equality of the
three displayed coefficient slots. It is not a numerical
error estimate at a nonzero real coupling.

\subsection{Enter background}

The Enter distinction used here is operational:
destructive readout consumes the measured quantum names
and returns a classical record together with the
explicitly retained residual. A quantum-return decoder
is a different interface \cite{EnterI,EnterPhilosophy}.

This distinction is applied only after the analytical
objects have been defined. It prevents the following
identifications:
\begin{enumerate}
\item a finite table of evaluated numbers is not the
continuum field;
\item a complete channel identity is not a field detector
or a thermodynamic implementation;
\item a prescribed dependency schedule does not derive
a Lorentzian metric;
\item finite operations placed on a supplied spacetime
do not, merely by that placement, produce spacetime
emergence.
\end{enumerate}
I-B supplies the fixed-spacetime interpretation of the
finite clock and regular transport pattern
\cite{IB}. The finite channel identities actually used
below are proved in this file.

\subsection{The inherited host ledger}

\begin{assumption}[Same host, chart, and completion]
\label{ass:host}
Retain the X-A-ST instance without modification:
\begin{enumerate}[label=\textup{(H\arabic*)}]
\item the regular time-oriented conical product
\((\mathcal M^\circ,G)\), its prescribed worldsheet
\(\mathcal W\), and its supplied future relation;
\item the supported developing embedding
\[
 \iota:\mathcal L\longrightarrow\mathcal M^\circ
\]
and the regular compact reserve used by the predecessors;
\item the separate analytical 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 and time ledger
\[
 \iota^*G=-\eta,\qquad
 t=c_{\mathrm{host}}(T-T_\Sigma).
\]
The host signature remains \((-,+,+,+)\);
the analytical equations retain \((+,-,-,-)\);
\item the 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 host's regular part is not used as a globally
hyperbolic Cauchy spacetime. All global free fields and
Green operators in this manuscript belong to
\(\widehat M\).
\end{assumption}

This ledger does not state a third cone.
The chart does not supply a global Minkowski
representation, an intrinsic Green operator on the
conical product, or a global field-controlled unitary.

For \(f\in C_c^\infty(\mathcal O)\), write
\[
 K=\supp f,\qquad D_f=J_\eta^-(K)\cap M_{T_*}.
\]
The supported transfer concerns calculations retained
in \(D_f\subset D_*\). The new maps are additionally
defined on arbitrary compact completion tests, without
claiming that every such test has a supported host
interpretation.

\begin{definition}[Local Grade S]
\label{def:grade}
The local Grade S package in this file consists of:
\begin{enumerate}[label=\textup{(S\arabic*)}]
\item the inherited regular supported host;
\item explicit response maps with domains, normalization,
support, and background transformation laws;
\item one 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}
Grade E, metric emergence, and Einstein dynamics are
not included.
\end{definition}

\subsection{Dependency discipline}

Only the explicitly stated host and handoff data are
inherited assumptions. A bare citation is not an
additional hypothesis. In particular, the comparison
sheets listed later do not prove either
Theorem~\ref{thm:maps} or Theorem~\ref{thm:ward}.
The latter is proved by the three-term variation,
not by declaring that every identity in a predecessor
automatically transfers.

%=====================================================================
\section{Frozen analytical data and predecessor handoff}
\label{sec:data}
%=====================================================================

\subsection{Classical backgrounds and the frozen algebra}

Let \(h,k\) be arbitrary smooth real symmetric tensors
on the completion, with no spatial decay or metric
field equation imposed, and set
\[
 q=\eta+\kappa h+\kappa^2k.
\]
Their topology is the smooth compact-open topology.

Retain the same fixed \(\varepsilon>0\), smooth ramp
\(\sigma(t)=\chi(t/\varepsilon)\), and compact switch
\[
 g\in C_c^\infty((0,T_*)\times\RR^3;[0,1])
\]
as in the predecessors, where
\[
 \chi(s)=
 \begin{cases}
 0,&s\le0,\\[1mm]
 \displaystyle
 \frac{e^{-1/s}}{e^{-1/s}+e^{-1/(1-s)}},&0<s<1,\\[3mm]
 1,&s\ge1.
 \end{cases}
\]
The plateau and algebra are unchanged:
\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}

For \(\xi,\zeta\in\gp\), the formal pullback generated
by \(\kappa\xi+\kappa^2\zeta\) belongs to \(\Gp\).
The full background 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}
Outside the plateau, the same formula will be used only
to calculate defects for arbitrary compact interior
generators. Those generators are not added to \(\gp\).

\subsection{Scalar and metric-density coefficients}

Indices are raised with \(\eta\). Put \(H=\tr_\eta h\)
and define
\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:density}
\end{align}
Set
\[
 A_2[h,k]=a_1[k]+a_2[h],\qquad
 B_2[h,k]=b_1[k]+b_2[h].
\]
Thus
\begin{align}
 \sqrt{-\det q}\,q^{\mu\nu}
   &=\eta^{\mu\nu}+\kappa a_1^{\mu\nu}[h]
                        +\kappa^2A_2^{\mu\nu}[h,k],
 \nonumber\\
 \sqrt{-\det q}
   &=1+\kappa b_1[h]+\kappa^2B_2[h,k].
 \label{eq:densityexp}
\end{align}

The switched scalar density is
\[
 \mathscr L_g[q,\psi]
  =\frac12\left[
    \bigl(\eta^{\mu\nu}
       +g(\sqrt{-\det q}\,q^{\mu\nu}-\eta^{\mu\nu})\bigr)
                         \partial_\mu\psi\partial_\nu\psi
       -m^2\bigl(1+g(\sqrt{-\det q}-1)\bigr)\psi^2
           \right].
\]
It is switched before variation.
For \(P_m=\Box+m^2\),
\[
 \Box=\partial_t^2-\Delta_{\mathbf x},
\]
its positive Euler operator is
\begin{equation}
 E_g[q]=P_m+\kappa L_{1,g}[h]+\kappa^2L_{2,g}[h,k].
 \label{eq:Euler}
\end{equation}

The two local differential coefficients are
\begin{align}
 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,
 \nonumber\\
 L_{2,g}[h,k]u
   ={}&g\left[
        \partial_\mu(A_2^{\mu\nu}[h,k]\partial_\nu u)
                                      +m^2B_2[h,k]u\right]
       +(\partial_\mu g)A_2^{\mu\nu}[h,k]\partial_\nu u.
 \label{eq:operators}
\end{align}
Every displayed derivative of the switch is retained.

The corresponding vertices are
\begin{align}
 V_{1,g}[h;\psi]
   &=\frac12\int g\left(
       a_1^{\mu\nu}[h]\partial_\mu\psi\partial_\nu\psi
                                 -m^2b_1[h]\psi^2\right)\dd^4x,
 \nonumber\\
 V_{2,g}[h,k;\psi]
   &=\frac12\int g\left(
       A_2^{\mu\nu}[h,k]\partial_\mu\psi\partial_\nu\psi
                                 -m^2B_2[h,k]\psi^2\right)\dd^4x.
 \label{eq:vertices}
\end{align}
Both are quadratic in the scalar and have compact
coefficient support. Compact integration by parts gives
\begin{equation}
 S_0''=-P_m,\qquad
 V_{j,g}'[\psi]=-L_{j,g}\psi,\qquad
 V_{j,g}''=-L_{j,g},\quad j=1,2.
 \label{eq:Hessian}
\end{equation}
The positive Euler coefficients are not the action
Hessians.

\subsection{The completion Green operators and free algebra}

Write
\[
 \DD=C_c^\infty(\widehat M;\CC).
\]
It has its standard LF topology, with smooth seminorms
on each fixed compact-support stage.

The positive free Green operators satisfy
\begin{align}
 P_m\ERc=P_m\EAc=c,\qquad
 \ER P_ma=\EA P_ma=a,\nonumber\\
 \ER^{\mathrm T}=\EA,\qquad
 \supp\ERc\subset J_\eta^+(\supp c),\qquad
 \supp\EAc\subset J_\eta^-(\supp c).
 \label{eq:Green}
\end{align}
They act continuously from compact smooth sources to
smooth solutions. These are completion-space
Green-operator statements \cite{BGP}; no conical
Cauchy theorem is used.

Retain X-B-ST's scalar free Wick algebra and quotient
\[
 \Afree=\Aoff/\Jzero,
\]
where \(\Jzero\) is generated by \(F_{P_ma}\),
\(a\in\DD\). For
\[
 F_c(\psi)=\int c\psi\,\dd^4x,\qquad
 \field(c)=[F_c],
\]
the degree-one relations are
\begin{equation}
 \field(P_ma)=0,\qquad
 [\field(c),\field(d)]
    =i\hbar\pair{c}{(\EA-\ER)d}\,\mathbf1.
 \label{eq:free}
\end{equation}
The degree-one topology is the quotient test topology
on \(\DD/P_m\DD\).

The insertion domain is instead
\[
 \Loff=\{F_c:c\in\DD\}\cong\DD.
\]
It consists of actual off-shell representatives.
Output classes are not used as unspecified new input
representatives.

The inherited degree-one separation property is
\begin{equation}
 \field(d)=0
 \quad\Longleftrightarrow\quad
 d\in P_m\DD
 \quad\Longleftrightarrow\quad
 \pair{d}{u}=0
 \ \text{for every smooth spatially compact free solution }u.
 \label{eq:separation}
\end{equation}
This is X-B-ST's normalization standard
\cite[Section~C3]{XBST}; it is also the
Green-operator exact-sequence mechanism
\cite[Section~3.4]{BGP}.

\subsection{Inherited coordinate data and bare tests}

Fix \(f\in C_c^\infty(\mathcal O;\RR)\).
The coordinate handoff is
\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 massless inverse here is the supplied zero-data
coordinate prescription at \(t=0\).
Smoothness without spatial decay, fixed-width continuity,
and retained-past dependence are inherited chart data.

On \(\Gp\), the supplied laws are
\begin{align}
 w[h']&=w[h]+\xi,\nonumber\\
 v[h',k']&=v[h,k]+X_\xi w[h]
                            +\zeta+\tfrac12X_\xi\xi,
 \qquad X_\xi=\xi^\mu\partial_\mu.
 \label{eq:coordinate-laws}
\end{align}
No proof of X-A-ST's coordinate covariance is claimed
again here.

Let
\[
 B^\mu=w^\nu\partial_\nu w^\mu-v^\mu.
\]
The inherited compact bare tests are
\begin{align}
 c_0&=f,\qquad c_1[h]=\partial_\mu(fw^\mu[h]),\nonumber\\
 c_2[h,k]
   &=-\partial_\mu(fB^\mu)
      +\frac12\partial_\mu\partial_\nu
                                      (fw^\mu[h]w^\nu[h]).
 \label{eq:bare}
\end{align}
They are smooth, supported in \(K\), and continuous
in the smooth backgrounds at fixed \(\varepsilon\).
Their scalar-only and joint admissions were established
in X-B-ST \cite{XBST}. The label test \(f\) remains
fixed.

Define
\[
 c[q]=f+\kappa c_1[h]+\kappa^2c_2[h,k].
\]
For every smooth scalar \(\psi\),
\[
 \pair{c[q]}{\psi}
   =\int f\,\psi\circ
        \bigl(x+\kappa w+\kappa^2v\bigr)^{-1}\dd^4x
                         \formal.
\]
This fixes the insertion representatives, including
their signs.

\subsection{The inherited first response and the unfilled slot}

The first map is recalled, not rebuilt:
\begin{equation}
 \Rone(V_{1,g}[h];F_c)
           =\field(-L_{1,g}[h]\EA c).
 \label{eq:first-map}
\end{equation}
X-B-ST supplies its support, continuity, tree-exactness,
and first-order completed identity, as well as
\[
 \widehat A_1[h]
       =\field(c_1[h]-L_{1,g}[h]\EA f).
\]

The classical target for the additional second-order
slot is the supplied expression
\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:rho2}
\end{equation}
For free \(u\), it is the second scalar Cauchy
coefficient of X-A-ST. For arbitrary smooth \(u\),
the same formula is a well-defined smooth linear
extension, because every source operator has compact
coefficients.

The predecessor's incomplete second coefficient was
\[
 \field(c_2[h,k])
       +\Rone(V_{1,g}[h];F_{c_1[h]}).
\]
It did not include \eqref{eq:rho2}.
The new pair below fills that one slot.

%=====================================================================
\section{Direct construction of the two new maps}
\label{sec:maps}
%=====================================================================

\subsection{Formal inverse on actual tests}

Put
\[
 L[q]=\kappa L_{1,g}[h]+\kappa^2L_{2,g}[h,k].
\]

\begin{definition}[Test-level retarded M{\o}ller map]
\label{def:moller}
On actual formal compact tests define
\begin{align}
 \Rq
   &=(1+L[q]\EA)^{-1}\nonumber\\
   &=1-\kappa L_{1,g}[h]\EA
      +\kappa^2\left(
         L_{1,g}[h]\EA L_{1,g}[h]\EA
                             -L_{2,g}[h,k]\EA\right).
 \label{eq:moller}
\end{align}
The inverse is purely formal over \(\mathbb D_2\).

The second-vertex map is
\begin{equation}
 \boxed{
 \Rone(V_{2,g}[h,k];F_c)
       =\field(s_c[h,k]),\qquad
 s_c[h,k]=-L_{2,g}[h,k]\EA c.}
 \label{eq:second-map}
\end{equation}
The iterated first-vertex coefficient is
\begin{equation}
 \boxed{
 \frac12\Rtwo(V_{1,g}[h],V_{1,g}[h];F_c)
       =\field(i_c[h]),\qquad
 i_c[h]=L_{1,g}[h]\EA L_{1,g}[h]\EA c.}
 \label{eq:iteration}
\end{equation}
The symbol \(\frac12R_2\) denotes this second Taylor
coefficient on the diagonal first-vertex family.
It does not name a general retarded product.
\end{definition}

The inner expression
\[
 a_c=L_{1,g}[h]\EA c
\]
is an actual smooth compact test. It is formed before
the next Green operation. In terms of the inherited
rule, \eqref{eq:iteration} can be written as applying
that rule to \(F_{-a_c}\), not to an unspecified
representative of \(\field(-a_c)\).

\begin{theorem}[Existence, support, continuity, and normalization]
\label{thm:maps}
The two maps in
\eqref{eq:second-map}--\eqref{eq:iteration} are defined
on every \(F_c\in\Loff\). Their output tests are smooth
and compact, and
\begin{equation}
 \supp s_c,\ \supp i_c
       \subseteq \supp g\cap J_\eta^-(\supp c).
 \label{eq:new-support}
\end{equation}
They are continuous in the compact-test topology and
the smooth-background topology.

For every smooth scalar \(u\),
\begin{align}
 \pair{s_c[h,k]}{u}
     &=-\pair{c}{\ER L_{2,g}[h,k]u},\nonumber\\
 \pair{i_c[h]}{u}
     &=\pair{c}{\ER L_{1,g}[h]\ER L_{1,g}[h]u}.
 \label{eq:individual-normalizations}
\end{align}
Consequently
\begin{equation}
 \pair{s_c+i_c}{u}
              =\pair{c}{\rho_{2,g}[h,k;u]}.
 \label{eq:sum-normalization}
\end{equation}
No singular product, diagonal extension, or subtraction
is used.
\end{theorem}

\begin{proof}
The function \(\EA c\) is smooth and supported in
\(J_\eta^-(\supp c)\). The coefficients of each
\(L_{j,g}\) are smooth and supported in the fixed
compact set \(S=\supp g\). This proves the assertion
for \(s_c\) and for the intermediate test \(a_c\).

The function \(\EA a_c\) is smooth and has support
in \(J_\eta^-(\supp a_c)\). Applying \(L_{1,g}[h]\)
gives
\[
 \supp i_c
   \subseteq S\cap J_\eta^-(\supp a_c)
   \subseteq S\cap J_\eta^-(\supp c),
\]
by transitivity of the completion causal relation.

For continuity, the maps
\[
 c\longmapsto\EA c,\qquad
 (h,u)\longmapsto L_{1,g}[h]u,\qquad
 (h,k,u)\longmapsto L_{2,g}[h,k]u
\]
are continuous into their respective smooth spaces;
the last two have range in the fixed test space
\(\DD_S\). The claim follows by composition, first
on each compact-support test stage and then on the
LF test space. The dependence on \(h,k\) is polynomial.

For normalization, formal self-adjointness and
\(\ER^{\mathrm T}=\EA\) give
\[
 \pair{-L_{2,g}\EA c}{u}
       =-\pair{c}{\ER L_{2,g}u}.
\]
Transposing the two Green operators in the iteration
successively gives
\[
 \pair{L_{1,g}\EA L_{1,g}\EA c}{u}
       =\pair{c}{\ER L_{1,g}\ER L_{1,g}u}.
\]
At every step, the differential source factor is
compactly supported. All propagations are applied to
smooth tests or smooth compact sources.
\end{proof}

This is the direct scalar-map construction of
X-C in the inherited supported presentation
\cite[\loc{def:moller}, \loc{thm:maps}]{XC}.
The equality to \(\rho_{2,g}\) is its normalization
against the classical handoff, not another copy of
that response.

\begin{corollary}[Causal vanishing and supported agreement]
\label{cor:support}
If
\[
 \supp g\cap J_\eta^-(\supp c)=\varnothing,
\]
both new maps vanish.

For \(c\) supported in \(\mathcal O\), all effective
interaction vertices lie in \(D_c\subset D_*\).
Two extensions of the background data on the same
declared flat completion that agree near \(D_c\)
give the same new output tests. The analogous
coordinate-data agreement for \(c[q]\) is the
inherited X-A-ST completion contract.
\end{corollary}

\begin{proof}
The vanishing follows from \eqref{eq:new-support}.
The second-vertex expression is local in its background
coefficients on \(S\cap J_\eta^-(\supp c)\).
For the iterated expression, every contributing chain
has
\[
 x\preceq_\eta y\preceq_\eta z,\qquad
 x,y\in S,\quad z\in\supp c.
\]
Since \(S\) lies in the open slab, these vertices and
their relevant causal intervals lie in \(D_c\).
The Green kernels are the same fixed completion kernels.
Thus agreement of the coefficients near \(D_c\) gives
agreement of the resulting tests.
\end{proof}

This is not independence from arbitrary global Green
realizations on different curved completions.
Transport through \(\iota\) is the rewriting of these
same retained compact pairings, not a construction
across the worldsheet.

\subsection{Factorial, signs, and diagonal nonadditivity}

For arbitrary smooth \(u\), define
\begin{equation}
 r_qu
  =u-\kappa\ER L_{1,g}[h]u
      +\kappa^2\left(
         \ER L_{1,g}[h]\ER L_{1,g}[h]u
                              -\ER L_{2,g}[h,k]u\right).
 \label{eq:scalar-map}
\end{equation}
The transpose of this scalar response is precisely
\eqref{eq:moller}. The second vertex therefore enters
with the same negative response sign as the first
vertex, while two first responses have positive sign.

For the first vertex alone,
\[
 \left.\frac{\dd^2}{\dd\lambda^2}
          (1+\lambda L_{1,g}[h]\EA)^{-1}c
                                      \right|_{\lambda=0}
       =2L_{1,g}[h]\EA L_{1,g}[h]\EA c.
\]
This explains the factor \(1/2\) in
\eqref{eq:iteration}. No factorial is assigned after
the gauge calculation.

The diagonal coefficient is quadratic, not additive,
in its Euler operator. For two specified admissible
operators \(L,M\), the direct identity is
\begin{align}
 &(L+M)\EA(L+M)\EA c
      -L\EA L\EA c-M\EA M\EA c\nonumber\\
 &\hspace{25mm}=L\EA M\EA c+M\EA L\EA c.
 \label{eq:cross-terms}
\end{align}
These mixed terms may not be discarded under the name
of polarization or diagonal additivity. Equation
\eqref{eq:cross-terms} is ordinary operator bookkeeping;
it does not install a general mixed retarded-product
family.

\subsection{The representative warning remains}

For \(a\in\DD\),
\begin{align}
 s_{P_ma}[h,k]&=-L_{2,g}[h,k]a,\nonumber\\
 i_{P_ma}[h]&=L_{1,g}[h]\EA L_{1,g}[h]a.
 \label{eq:equation-input}
\end{align}
Their classes need not vanish. At the full formal level,
\begin{equation}
 \Rq(P_ma)=P_ma-L[q]a+L[q]\EA L[q]a
                                      \formal.
 \label{eq:input-nondescent}
\end{equation}
Thus neither the formal recursion nor
tree-exactness changes the insertion domain into an
on-shell quotient.

Non-descent already occurs for the inherited first map.
For example, in the massless case with \(h=r\eta\)
and a real bump \(r\) supported where \(g=1\), take
\(a=\partial_t r\). Then
\[
 \pair{-L_{1,g}[r\eta]a}{t}
                =-\int(\partial_t r)^2\,\dd^4x\ne0.
\]
This recalls the predecessor's representative test;
it is not a new construction of its map.

\subsection{Normalization and what it does not determine}

\begin{proposition}[Whole-domain normalization standard]
\label{prop:normalization}
The separate pairings in
\eqref{eq:individual-normalizations}, imposed for every
\(c\in\DD\) and every smooth spatially compact free
solution, determine the two degree-one output classes
uniquely.

If formal \(\hbar\)-dependent alternatives are allowed,
this statement requires equality of the full
normalization series, coefficientwise.

Normalization of the sum alone does not uniquely
specify a decomposition into two summands.
\end{proposition}

\begin{proof}
For each output, subtract two candidates with the same
individual pairing. Their compact test difference
annihilates every spatially compact free solution.
Equation \eqref{eq:separation} makes its class zero.
The argument applies to each \(\hbar\)-coefficient.

If only the sum is fixed, one may formally add
\(\field(n_c)\) to one candidate and subtract the same
class from the other. Individual normalization or
the coefficient recursion is needed to exclude such
redistribution.
\end{proof}

The direct inverse \eqref{eq:moller}, and not a choice
of contact after observing a defect, fixes the present
split. X-B-ST's sheet-compatible first-map contacts
remain zero classes. This file does not reopen VII-H's
different contact classification.

%=====================================================================
\section{Yang--Feldman characterization and quantum order}
\label{sec:yf}
%=====================================================================

\subsection{Formal Green operators}

Define, coefficientwise through \(\kappa^2\),
\begin{align}
 \ERq
   &=(1+\ER L[q])^{-1}\ER
        =\ER(1+L[q]\ER)^{-1},\nonumber\\
 \EAq
   &=(1+\EA L[q])^{-1}\EA
        =\EA(1+L[q]\EA)^{-1}
        =\EA\Rq.
 \label{eq:perturbed-green}
\end{align}
For instance,
\[
 \EAq=\EA-\kappa\EA L_{1,g}\EA
       +\kappa^2\left(
          \EA L_{1,g}\EA L_{1,g}\EA-\EA L_{2,g}\EA\right).
\]
These are finite formal compositions, not numerical
inverses of a background-dependent operator at finite
coupling.

Let
\[
 \field_q(c)=\field(\Rq c).
\]
This is a linear formal field in the inherited free
algebra. No state is part of this definition.

\begin{theorem}[Linear Yang--Feldman characterization]
\label{thm:yf}
The following statements hold modulo \(\kappa^3\):
\begin{enumerate}[label=\textup{(\roman*)}]
\item The test operator is the unique formal series
with leading term \(1\) satisfying
\[
 \Rq=1-\Rq L[q]\EA.
 \label{eq:YF-recursion}
\]
Equivalently,
\[
 \field_q(c)=\field(c)-\field_q(L[q]\EA c).
\]
\item The scalar map in \eqref{eq:scalar-map} obeys
\[
 (P_m+L[q])r_qu=P_mu
\]
for arbitrary smooth \(u\), and is the unique solution
with the same zeroth-order data and zero positive-order
data before the interaction.
\item For every compact \(a\),
\[
 \field_q(E_g[q]a)=0.
\]
\item If \(\supp g\cap J_\eta^-(\supp c)=\varnothing\),
then \(\field_q(c)=\field(c)\).
\item Its algebraic commutator is
\begin{equation}
 [\field_q(c),\field_q(d)]
       =i\hbar\pair{c}{(\EAq-\ERq)d}\,\mathbf1.
 \label{eq:perturbed-commutator}
\end{equation}
\end{enumerate}
\end{theorem}

\begin{proof}
The formal inverse identity
\[
 (1+A)^{-1}=1-(1+A)^{-1}A,\qquad A=L[q]\EA,
\]
proves (i). Coefficient comparison fixes the first
operator as \(-L_{1,g}\EA\) and the second as
\(L_{1,g}\EA L_{1,g}\EA-L_{2,g}\EA\).

The scalar-side recursion is
\[
 r_qu=u-\ER L[q]r_qu.
\]
Applying \(P_m\) gives (ii). Conversely, its
positive-order equations are ordinary free
inhomogeneous Cauchy problems with smooth compact
sources, so free Cauchy uniqueness determines them.

For compact \(a\), \(\EA P_ma=a\), and hence
\[
 E_g[q]a=(1+L[q]\EA)P_ma,\qquad
 \Rq E_g[q]a=P_ma.
\]
Equation \eqref{eq:free} proves (iii).
The support condition in (iv) makes
\(L[q]\EA c=0\), proving that assertion.

For (v), formal self-adjointness gives
\[
 \Rq^{\mathrm T}
       =(1+\ER L[q])^{-1}=1-\ERq L[q],
 \qquad
 \Rq=1-L[q]\EAq.
\]
Using the inherited free commutator, its kernel between
the original tests is
\[
 (1-\ERq L[q])(\EA-\ER)(1-L[q]\EAq).
\]
The advanced contribution is
\[
 \EAq-\ERq L[q]\EAq,
\]
because \(\EA(1-L[q]\EAq)=\EAq\).
The retarded contribution is
\[
 \ERq-\ERq L[q]\EAq,
\]
because \((1-\ERq L[q])\ER=\ERq\).
Their difference is \(\EAq-\ERq\).
Every composition is well defined on the stated
compact tests.
\end{proof}

The theorem characterizes the declared linear
construction \cite[\loc{thm:yf}]{XC}.
The commutator is not an interacting two-point
response or a construction of a state.
No assertion that \(\Rq\) is an automorphism of the
unchanged free algebra is made.

\subsection{Tree-exactness and the rooted leg count}

\begin{proposition}[Tree-exactness on the specified sector]
\label{prop:hbar}
The tests \(s_c,i_c\), and the completed tests constructed
below, contain no \(\hbar\) and no Hadamard contraction
kernel. They have no positive-\(\hbar\) remainder in
the directly defined linear rooted prescription.
\end{proposition}

\begin{proof}
All coefficients in \eqref{eq:moller} are compositions
of ordinary Green and differential operators on tests.
The same is true after substituting the bare test
polynomial \(c[q]\). This proves the assertion
algebraically.

The fixed Wick presentation provides a consistency
count, not an alternative construction. A quadratic
vertex and one linear insertion have three scalar
legs, permitting one rooted cross-contraction and
leaving one leg. Two quadratic vertices and a linear
insertion have five legs. A connected rooted
three-entry graph needs two edges, consumes four
legs, and leaves one leg. Since the insertion has
only one leg, that graph is a chain ending at the
insertion. Three contractions would require six
legs and are impossible.

The relevant contraction powers are removed by their
first- and second-order response normalizations.
A vertex-only vacuum component disconnected from
the insertion is not part of the rooted prescription;
self-contractions are absent in the fixed Wick
presentation \cite{FR}. The count explains the
quantum order of the direct formulas, but does not
define any missing singular product.
\end{proof}

The \(\hbar\) in \eqref{eq:free} and
\eqref{eq:perturbed-commutator} is unaffected.
Tree-exact tests are not a statement that the algebra
is classical or that every quantum correction in a
larger sector vanishes.

\subsection{Why the product boundary is unchanged}

The inherited X-B-ST nonextension statement remains
in force \cite[\loc{thm:nonextension}]{XBST}.
Its auxiliary kernel
\[
 \DF=\Dplus-i\EA,\qquad
 P_{m,x}\DF=-i\delta_{\mathrm{diag}},
\]
contains the full nonzero diagonal conormal in its
wavefront set. Thus at every diagonal point it contains
both
\[
 (x,x;\xi,-\xi)\quad\text{and}\quad
 (x,x;-\xi,\xi),\qquad \xi\ne0.
\]
The ordinary wavefront product criterion therefore
does not define \(\DF\cdot\DF\) there
\cite[Chapter~8]{Hormander}.

Equations \eqref{eq:second-map} and
\eqref{eq:iteration} never perform that operation.
The second Green propagation in \eqref{eq:iteration}
acts on the smooth compact test \(a_c\).
Sequential action on smooth tests is not an extension
of an identified-endpoint square of singular kernels.

No \(T_2(V,V)\), \(T_3\), or general nonlinear-insertion
\(R_2\) is supplied. Failure of the sufficient product
criterion is not a universal nonexistence theorem for
renormalized extensions. Such extensions and their
normalizations are additional analytical tasks
\cite{BF,DFreference}.

%=====================================================================
\section{The completed relational test}
\label{sec:completed}
%=====================================================================

\begin{definition}[Completed test and field]
\label{def:completed}
Set
\begin{align}
 e[q]&=\Rq c[q]=f+\kappa e_1[h]+\kappa^2e_2[h,k],
 \nonumber\\
 e_1[h]&=c_1[h]-L_{1,g}[h]\EA f,
 \nonumber\\
 e_2[h,k]
    &=c_2[h,k]-L_{1,g}[h]\EA c_1[h]
                  -L_{2,g}[h,k]\EA f
                  +L_{1,g}[h]\EA L_{1,g}[h]\EA f.
 \label{eq:completed}
\end{align}
Equivalently,
\begin{equation}
 e_2=c_2-L_{1,g}[h]\EA e_1-L_{2,g}[h,k]\EA f.
 \label{eq:completed-recursion}
\end{equation}
The completed field is
\[
 \widehat A[q]=\field(e[q])
        =\field(f)+\kappa\widehat A_1[h]
                           +\kappa^2\widehat A_2[h,k].
\]
\end{definition}

Its second coefficient contains exactly four terms:
\begin{equation}
\begin{split}
 \widehat A_2[h,k]
   ={}&\field(c_2[h,k])
       +R_1(V_{1,g}[h];F_{c_1[h]})\\
     &+R_1(V_{2,g}[h,k];F_f)
       +\tfrac12R_2(V_{1,g}[h],V_{1,g}[h];F_f).
\end{split}
\label{eq:fourterms}
\end{equation}

\begin{proposition}[Admission, polynomial scope, and normalization]
\label{prop:completed}
Every \(e_j\) is smooth and compact, with
\begin{equation}
 \supp e_j\subseteq
 C_f:=K\cup\bigl(\supp g\cap J_\eta^-(K)\bigr),
 \qquad j=0,1,2.
 \label{eq:completed-support}
\end{equation}
Their background dependence is continuous at fixed
\(\varepsilon\). The retained outputs have field
degree one, with zero allowed; \(e_1\) is linear in
\(h\), and \(e_2\) is quadratic in \(h\) plus linear
in \(k\).

For every smooth free solution \(u\),
\begin{align}
 \pair{e_1}{u}
  &=\int f\left(\rho_{1,g}[h;u]-w[h]\cdot\partial u\right)\dd^4x,
 \nonumber\\
 \pair{e_2}{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:completed-normalization}
\end{align}
The last line is the inherited full classical second
relational coefficient, counted once.
\end{proposition}

\begin{proof}
The bare tests are supported in \(K\).
The support and continuity of every response term
follow from Theorem~\ref{thm:maps} and the inherited
first-map theorem. Their finite sums have the stated
support and topology.

The coordinate coefficients satisfy
\(w[h]\) linear in \(h\) and
\(v[h,k]\) quadratic in \(h\) plus linear in \(k\).
The same degree statement follows for \(c_1,c_2\),
and then for \eqref{eq:completed}. Every output is
\(\field\) of a smooth compact test, so scalar-only
admission needs no further singular-kernel theorem.

Transpose the response terms using
\eqref{eq:individual-normalizations}. For the mixed
term,
\[
 \pair{-L_{1,g}[h]\EA c_1[h]}{u}
       =\pair{c_1[h]}{\rho_{1,g}[h;u]}
       =-\int f w[h]\cdot\partial\rho_{1,g}[h;u]\,\dd^4x.
\]
The bare terms give \(O^{(j)}\), and the new sum gives
\(\rho_{2,g}\).
\end{proof}

The proposition concerns the retained degree-one
outputs. The vertices are quadratic, their auxiliary
products need not have degree one, and the finite
instruments below are channels rather than field
elements. No universal background-degree bound for
arbitrary further compositions is asserted.

%=====================================================================
\section{Formal duality and the two defects}
\label{sec:defects}
%=====================================================================

The variation is calculated before imposing the plateau.
Let \(\xi,\zeta\) now be arbitrary smooth compact
interior vector fields, and let \(q'=F^*q\) have the
full law \eqref{eq:metriclaw}.

Write
\begin{align}
 F^*&=1+\kappa X_\xi
                +\kappa^2\left(X_\zeta+\tfrac12X_\xi^2\right),
 \nonumber\\
 F^\sharp&=1+\kappa D_\xi
                +\kappa^2\left(D_\zeta+\tfrac12D_\xi^2\right),
 \qquad
 D_\xi a=\partial_\mu(\xi^\mu a).
 \label{eq:pullbacks}
\end{align}
The first acts on scalar functions. The second is
density pullback, including the Jacobian.

\begin{lemma}[Formal scalar--density duality]
\label{lem:duality}
For compact formal tests \(a\) and smooth formal
scalars \(u\),
\begin{equation}
 \pair{F^\sharp a}{F^*u}=\pair{a}{u}\formal.
 \label{eq:duality}
\end{equation}
\end{lemma}

\begin{proof}
Integration by parts gives \(D_\xi^{\mathrm T}=-X_\xi\).
Thus
\[
 (F^\sharp)^{\mathrm T}
  =1-\kappa X_\xi
       +\kappa^2\left(-X_\zeta+\tfrac12X_\xi^2\right),
\]
the inverse of the scalar pullback through this order.
Multiplication by \(F^*\) proves the identity.
\end{proof}

\begin{definition}[Coordinate and switch defects]
\label{def:defects}
For arbitrary compact generators in the open slab,
define
\begin{equation}
 \Theta_F[q]=c[q']-F^\sharp c[q],\qquad
 \Sigma_F[q]=E_g[q']F^*-F^\sharp E_g[q].
 \label{eq:defects}
\end{equation}
Write
\[
 \Theta_F=\kappa\theta_1+\kappa^2\theta_2,\qquad
 \Sigma_F=\kappa S_1+\kappa^2S_2.
\]
The \(\theta_j\) are tests; the \(S_j\) are
differential operators.
\end{definition}

\subsection{The coordinate defect}

\begin{lemma}[Coordinate defect coefficients]
\label{lem:theta}
The coordinate defect has compact coefficients with
\(\supp\theta_j\subset K\), and
\begin{align}
 \theta_1
   &=\partial_\mu\bigl(f(w[K\xi]-\xi)^\mu\bigr),
 \nonumber\\
 \theta_2
   &=c_2[h',k']-c_2[h,k]
       -D_\xi c_1[h]
       -\left(D_\zeta+\tfrac12D_\xi^2\right)f.
 \label{eq:theta}
\end{align}
For \(F\in\Gp\), \(\Theta_F=0\).
\end{lemma}

\begin{proof}
Expansion of \eqref{eq:defects} gives
\eqref{eq:theta}. Formal pullback coefficients are
differential operators and do not enlarge coefficient
supports. This proves the compact-support statement.

On \(\Gp\), substitute the inherited coordinate laws
\eqref{eq:coordinate-laws}. They give
\[
 c_1[h']-c_1[h]=D_\xi f,\qquad
 c_2[h',k']-c_2[h,k]
     =D_\xi c_1[h]
           +\left(D_\zeta+\tfrac12D_\xi^2\right)f.
\]
For example, with \(B=w\cdot\partial w-v\),
\[
 B'=B+w\cdot\partial\xi
                  +\tfrac12\xi\cdot\partial\xi-\zeta.
\]
Insertion in \eqref{eq:bare}, followed by compact
integration by parts, gives the second test identity.
Both defect coefficients vanish.
\end{proof}

The proof uses the supplied coordinate transformation
data. It is not a new derivation of classical ramp
covariance. The support assertion is coefficientwise;
it is not a claim that an unexpanded numerical
pullback leaves the support of a test fixed.

\subsection{Switch derivatives before restriction}

Let the unswitched coefficients be
\[
 Q_1[h]=L_{1,1}[h],\qquad
 Q_2[h,k]=L_{2,1}[h,k],
\]
and put \(N_j=Q_j-L_{j,g}\).
Explicitly,
\begin{align}
 N_1[h]u
   &=(1-g)Q_1[h]u
          -(\partial_\mu g)a_1^{\mu\nu}[h]\partial_\nu u,
 \nonumber\\
 N_2[h,k]u
   &=(1-g)Q_2[h,k]u
          -(\partial_\mu g)A_2^{\mu\nu}[h,k]\partial_\nu u.
 \label{eq:N}
\end{align}
The \(N_j\) need not have compact coefficients.
Only their transformation combinations used below
have compact coefficients.

The unswitched scalar Euler operator
\[
 E_{\mathrm{unsw}}[q]u
  =\partial_\mu\bigl(\sqrt{-\det q}\,
                       q^{\mu\nu}\partial_\nu u\bigr)
                     +m^2\sqrt{-\det q}\,u
\]
maps scalars to densities. The chain rule and change
of variables give its naturality
\[
 E_{\mathrm{unsw}}[F^*q]F^*
                 =F^\sharp E_{\mathrm{unsw}}[q].
\]
For formal jets this is an identity of Taylor
coefficients; no global numerical inverse of \(q\)
is required.

\begin{lemma}[Switch defect with all retained derivatives]
\label{lem:sigma}
Let \(K_F=\supp\xi\cup\supp\zeta\).
The coefficients of \(S_1,S_2\) are smooth and
supported in \(K_F\). They are
\begin{align}
 S_1&=-N_1[K\xi]
      =(g-1)Q_1[K\xi]
            +(\partial_\mu g)a_1^{\mu\nu}[K\xi]\partial_\nu,
 \nonumber\\
 S_2&=-\bigl(N_2[h',k']-N_2[h,k]\bigr)
                -N_1[h']X_\xi+D_\xi N_1[h].
 \label{eq:sigma}
\end{align}
If \(g=1\) on a neighbourhood of \(K_F\), then
\(\Sigma_F=0\).
\end{lemma}

\begin{proof}
Write
\[
 E_g[q]=E_{\mathrm{unsw}}[q]
                          -\kappa N_1[h]-\kappa^2N_2[h,k].
\]
Unswitched naturality cancels its contribution to
\eqref{eq:defects}. The remainder is
\[
 -\bigl(\kappa N_1[h']+\kappa^2N_2[h',k']\bigr)F^*
       +F^\sharp\bigl(\kappa N_1[h]+\kappa^2N_2[h,k]\bigr).
\]
Expansion gives \eqref{eq:sigma}, using the linearity
of \(N_1\). Outside \(K_F\), every transformation
coefficient is locally the identity, and the
differences vanish.

To check the switch dependence without suppressing
derivatives, put
\[
 X=X_\xi,\quad D=D_\xi,\quad
 \Delta A_2=A_2[h',k']-A_2[h,k],\quad
 \Delta Q_2=Q_2[h',k']-Q_2[h,k].
\]
Equation \eqref{eq:sigma} gives
\begin{align}
 S_2u={}&(1-g)\left(
          -\Delta Q_2u-Q_1[h']Xu+DQ_1[h]u\right)
 \nonumber\\
 &+(\partial_\mu g)\Delta A_2^{\mu\nu}\partial_\nu u
   +(\partial_\mu g)a_1^{\mu\nu}[h']\partial_\nu(Xu)
 \nonumber\\
 &-(Xg)Q_1[h]u
   -\partial_\rho\left(
       \xi^\rho(\partial_\mu g)
                    a_1^{\mu\nu}[h]\partial_\nu u\right).
 \label{eq:sigma-expanded}
\end{align}
In the final line, the last divergence includes
\begin{align*}
 &-(\partial_\rho\xi^\rho)(\partial_\mu g)
                      a_1^{\mu\nu}[h]\partial_\nu u
   -\xi^\rho(\partial_\rho\partial_\mu g)
                      a_1^{\mu\nu}[h]\partial_\nu u\\
 &\quad
   -\xi^\rho(\partial_\mu g)
                      (\partial_\rho a_1^{\mu\nu}[h])\partial_\nu u
   -\xi^\rho(\partial_\mu g)
                      a_1^{\mu\nu}[h]\partial_\rho\partial_\nu u.
\end{align*}
Thus the possible second derivatives of \(g\) have
not been dropped.

When \(g=1\) near \(K_F\), \(1-g\) and all its
positive-order derivatives vanish there. The displayed
operators vanish there, and already vanish outside
\(K_F\).
\end{proof}

Neither \(\supp(1-g)\) nor \(\supp(g-\sigma)\) is
assumed compact. Compactness in the variation proof
comes from compact generator support, from \(f\),
and from the compact coefficients of \(L_{j,g}\).

%=====================================================================
\section{Three-term variation and the completed identity}
\label{sec:variation}
%=====================================================================

\subsection{Advanced uniqueness}

\begin{lemma}[Formal advanced solution]
\label{lem:advanced}
For a formal compact test \(d\), the equation
\[
 E_g[q]s=d
\]
has a unique smooth advanced-type formal solution,
where advanced-type means that each coefficient is
supported in \(J_\eta^-(K_0)\) for a compact \(K_0\).
It is
\begin{equation}
 s=\EAq d=\EA\Rq d,\qquad P_ms=\Rq d.
 \label{eq:advanced}
\end{equation}
\end{lemma}

\begin{proof}
The coefficient equations are
\[
 P_ms_0=d_0,\qquad
 P_ms_1=d_1-L_{1,g}s_0,\qquad
 P_ms_2=d_2-L_{1,g}s_1-L_{2,g}s_0.
\]
Each right-hand side is smooth and compact because
the \(d_j\) and interaction coefficients are compact.
Successive application of \(\EA\) gives the displayed
solution and its causal support.

A homogeneous advanced-type free solution vanishes
on a sufficiently late complete flat Cauchy surface,
so it vanishes by free Cauchy uniqueness. Induction
proves formal uniqueness. Finally
\(P_m\EA\Rq d=\Rq d\), since \(\Rq d\) is a formal
compact test.
\end{proof}

\subsection{Proof of record}

For the bare insertion \(c[q]\), set
\[
 s[q]=\EAq c[q]=\EA e[q].
\]
In particular,
\[
 s_0=\EA f,\qquad s_1=\EA e_1.
\]
The notation \(s[q]\) is distinct from the coordinate
vector \(w[h]\).

\begin{proposition}[Three-term variation]
\label{prop:variation}
For arbitrary smooth classical backgrounds and arbitrary
compact interior generators,
\begin{equation}
 \boxed{
 e[q']-e[q]
   =P_m\bigl((F^*-1)s[q]\bigr)
      +\mathcal R_{q'}\Theta_F[q]
      -\mathcal R_{q'}\bigl(\Sigma_F[q]s[q]\bigr)
       \quad\formal.}
 \label{eq:variation}
\end{equation}
The first term is coefficientwise in \(P_m\DD\).
The other two are the dressed coordinate and switch
defects.

Writing
\[
 Z=X_\zeta+\tfrac12X_\xi^2,\qquad
 L'_1=L_{1,g}[h'],
\]
the coefficient formulae are
\begin{align}
 e'_1-e_1
   ={}&P_m(X_\xi\EA f)+\theta_1-S_1\EA f,
 \label{eq:var1}\\
 e'_2-e_2
   ={}&P_m\left(X_\xi\EA e_1+Z\EA f\right)
           +\theta_2-S_2\EA f-S_1\EA e_1
 \nonumber\\
      &-L'_1\EA\bigl(\theta_1-S_1\EA f\bigr).
 \label{eq:var2}
\end{align}
\end{proposition}

\begin{proof}
Let \(s'=s[q']\). By Lemma~\ref{lem:advanced},
\[
 E_g[q]s=c[q],\qquad
 E_g[q']s'=c[q'],\qquad
 P_ms=e[q],\qquad P_ms'=e[q'].
\]
The defect definitions give
\[
 E_g[q']F^*s
     =F^\sharp c[q]+\Sigma_Fs
     =c[q']-\Theta_F+\Sigma_Fs.
\]
Subtracting,
\[
 E_g[q'](s'-F^*s)=\Theta_F-\Sigma_Fs.
\]
The right-hand side is a compact formal source:
\(\Theta_F\) has support in \(K\), and the coefficients
of \(\Sigma_F\) have compact generator support.

The difference \(s'-F^*s\) is advanced-type.
Indeed, \(s,s'\) are advanced-type, and each coefficient
of \((F^*-1)s\) has compact generator support.
No assumption that an unexpanded diffeomorphism preserves
the completion causal relation is needed.

Advanced uniqueness therefore yields
\[
 s'-F^*s
        =\mathsf E_{q'}^{\mathrm A}
                      (\Theta_F-\Sigma_Fs).
\]
Subtract \(s\) and apply \(P_m\).
Using \(P_m\mathsf E_{q'}^{\mathrm A}
=\mathcal R_{q'}\) proves \eqref{eq:variation}.

Every coefficient of \((F^*-1)s\) is a smooth compact
function, so its \(P_m\)-image lies in \(P_m\DD\).
Finally expand
\[
 \mathcal R_{q'}=1-\kappa L'_1\EA+O(\kappa^2),
 \qquad s_0=\EA f,\quad s_1=\EA e_1.
\]
The degree-one and degree-two coefficients are exactly
\eqref{eq:var1} and \eqref{eq:var2}.
\end{proof}

The final term in \eqref{eq:var2} is essential.
In particular,
\[
 -L_{1,g}[h']\EA\theta_1
\]
propagates the first coordinate defect through the
response. Dropping it would give an incorrect
second-order off-plateau variation.
This is the proof-of-record structure of X-C
\cite[\loc{prop:variation}]{XC}.

\subsection{The completed background identity}

\begin{theorem}[Background identity through second order]
\label{thm:ward}
For every \(F\in\Gp\), with the free field held fixed,
\begin{equation}
 \widehat A[F^*q]=\widehat A[q]
                    \quad\text{in }\Afree\otimes\mathbb D_2.
 \label{eq:ward}
\end{equation}
Equivalently,
\[
 \widehat A_1[h']=\widehat A_1[h],\qquad
 \widehat A_2[h',k']=\widehat A_2[h,k].
\]
Off shell, the compact tests obey
\begin{equation}
 e[q']-e[q]=P_m\bigl((F^*-1)s[q]\bigr),
 \label{eq:ideal-representative}
\end{equation}
not necessarily equality as tests.
\end{theorem}

\begin{proof}
Lemma~\ref{lem:theta} gives \(\Theta_F=0\) on the
inherited coordinate plateau.
Equation \eqref{eq:plateau} gives \(g=1\) on a
neighbourhood of both generator supports, so
Lemma~\ref{lem:sigma} gives \(\Sigma_F=0\).
Proposition~\ref{prop:variation} reduces to
\eqref{eq:ideal-representative}.
Every coefficient on its right belongs to
\(P_m\DD\), and therefore has zero free-field class.
\end{proof}

The first coefficient agrees with the already released
X-B-ST identity. The new conclusion is its correctly
completed second coefficient.
The proof neither deletes the bare second dressing nor
changes the originally frozen algebra.

This is background covariance at a fixed free field.
A general scalar pullback is not asserted to act on the
unchanged free solution space. No BRST differential,
quantum master equation, or quantum metric dynamics is
defined.

\subsection{Route II: a pairing cross-check}

\begin{corollary}[Classical pairing confirmation]
\label{cor:routeII}
The normalization to the inherited classical relational
coefficient, together with X-A-ST's on-shell classical
identity, independently confirms
\(\field(e'_2-e_2)=0\) on \(\Gp\).
This is not the proof of record for
Theorem~\ref{thm:ward}.
\end{corollary}

\begin{proof}
By Proposition~\ref{prop:completed}, pairing \(e_2\)
with a smooth free solution gives the complete
classical second coefficient. The inherited classical
identity therefore makes \(e'_2-e_2\) annihilate every
such solution.

To recall the relevant separation argument, let
\(d\in\DD\) annihilate every spatially compact free
solution. Since \((\EA-\ER)a\) is one for every
\(a\in\DD\),
\[
 \pair{d}{(\EA-\ER)a}=0
       \quad\Longrightarrow\quad(\EA-\ER)d=0.
\]
Then \(\EA d=\ER d=:v\), whose support lies in the
compact causal diamond
\[
 J_\eta^-(\supp d)\cap J_\eta^+(\supp d).
\]
Thus \(v\in\DD\) and \(P_mv=d\). Applying this to
\(d=e'_2-e_2\) proves the claimed free-algebra equality.
\end{proof}

The pairing argument does not display the edge and
switch terms. Those terms, and their propagated
second-order contribution, are supplied by
Proposition~\ref{prop:variation}.

%=====================================================================
\section{The excluded edge gap and switch dependence}
\label{sec:edge}
%=====================================================================

For arbitrary compact interior generators, the free
class of \eqref{eq:variation} is
\begin{equation}
 \widehat A[q']-\widehat A[q]
   =\field\left(
      \mathcal R_{q'}
                (\Theta_F-\Sigma_Fs[q])\right).
 \label{eq:class-defect}
\end{equation}
The coordinate and switch mechanisms remain separate.

The inherited preflight gap for an initial-trivial
compact \(\xi\) is
\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}
For the X-A-ST witness
\[
 \xi=\alpha(t)\beta(\mathbf x)\partial_1,\qquad
 0\ne\alpha\ge0,\quad
 \supp\alpha\subset(0,\varepsilon),\quad
 \int\beta\,\dd^3x=1,
\]
the retained integrated gap is
\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 \varepsilon\le t\le T_*.
 \label{eq:edge-integral}
\end{equation}
This is supplied classical chart data
\cite[\loc{prop:edge}]{XAST}; the new maps do not
alter the coordinate Cauchy prescription.

\begin{proposition}[Propagation without edge repair]
\label{prop:edge}
Suppose an excluded edge transformation has generators
supported in an already existing interaction plateau,
so \(g=1\) near \(K_F\). Then
\begin{align}
 \delta_F\widehat A_1&=\field(\theta_1),\nonumber\\
 \delta_F\widehat A_2
       &=\field\bigl(\theta_2
                         -L_{1,g}[h']\EA\theta_1\bigr).
 \label{eq:propagated-edge}
\end{align}
The second term in the latter expression is the
propagated first-order gap. The independent term
\(\theta_2\) is still the full second-coordinate test
difference in \eqref{eq:theta}.
\end{proposition}

\begin{proof}
Under the stated interaction-plateau condition
\(S_1=S_2=0\). Pass
\eqref{eq:var1}--\eqref{eq:var2} to free-field classes.
The compact \(P_m\)-terms vanish and the displayed
formulae remain.
\end{proof}

The condition in Proposition~\ref{prop:edge} is
conditional: not every frozen switch has an interaction
plateau in the ramp zone.
If it does not, the switch terms in
\eqref{eq:var1}--\eqref{eq:var2} must also be retained.

A nonzero integrated coordinate gap does not prove
a nonzero total smeared free class for every fixed
\(f,g\). A test may miss it, its resulting representative
may be an equation zero, or switch contributions may
also enter. Likewise,
\(-L_{1,g}[h']\EA\theta_1\) need not be nonzero on
every datum and is not the whole second defect.

If a genuinely nonzero order-\(\kappa\),
order-\(\hbar^0\) total defect has been established,
a correction beginning at \(\kappa^2\) or at positive
\(\hbar\)-order cannot remove it while that lower-order
prescription remains fixed. The coefficientwise
reason is the same logical pattern as VI-B's order
obstruction, not an identification with its defect
\cite{VIB}.

\subsection{Observation plateaux do not determine past response}

The operators in \eqref{eq:operators} depend on the
switch throughout the contributing causal past.
Agreement of two switches only near \(\supp f\) does
not imply equality of either map.

For a comparison with different switch data, keep the
bare coordinate prescription fixed and denote the two
formal Euler perturbations by \(L_g[q]\) and
\(L_{\widetilde g}[q]\). The finite inverse identity gives
\[
 \mathcal R_{\widetilde g}
       -\mathcal R_g
  =\mathcal R_{\widetilde g}
           (L_g[q]-L_{\widetilde g}[q])\EA\mathcal R_g
                         \formal.
\]
Only agreement on the complete contributing causal
region forces this difference to vanish.
This observation follows from the present test operators.
VII-A-ST's analogous switch ledger concerns a different
coordinate equation space \cite{VIIAST}.

\subsection{Zero switch}

In the separate diagnostic specialization \(g=0\),
both \(L_{1,g}\) and \(L_{2,g}\) vanish, so both new
maps vanish and
\[
 \Rq=1,\qquad \widehat A[q]=\field(c[q]).
\]
The coordinate dressing remains. The interaction
plateau is empty, so this specialization supplies no
new nonempty Ward algebra.

%=====================================================================
\section{Diagnostics of the completed coefficient}
\label{sec:diagnostics}
%=====================================================================

\subsection{Complete pure-gauge cancellation}

\begin{proposition}[Pure-gauge two-jet closes the predecessor remainder]
\label{prop:puregauge}
Let \(\xi\in\gp\), set \(\zeta=0\), and take
\[
 h=K\xi,\qquad k=\tfrac12\Lie_\xi^2\eta.
\]
Write
\[
 X=X_\xi,\qquad D=D_\xi,\qquad
 L=L_{1,g}[K\xi],\qquad
 M=L_{2,g}[K\xi,\tfrac12\Lie_\xi^2\eta].
\]
Then the new pair has the representative identity
\begin{equation}
 L\EA L\EA f-M\EA f
   =\tfrac12D^2f
       +P_m\left(
          \tfrac12X^2\EA f-X\EA Df\right).
 \label{eq:pg-new}
\end{equation}
Consequently, in the free algebra,
\[
 \widehat A_2[K\xi,\tfrac12\Lie_\xi^2\eta]
      =\left(\tfrac12-1+\tfrac12\right)
                     \field(D^2f)=0.
 \label{eq:pg-cancel}
\]
The bare second dressing is retained.
\end{proposition}

\begin{proof}
On the flat starting background, plateau Euler
covariance gives, by expansion,
\[
 L=DP_m-P_mX,\qquad
 M=\tfrac12D^2P_m+\tfrac12P_mX^2-DP_mX.
 \label{eq:pg-operators}
\]
These operator identities can also be obtained directly
by differentiating the density naturality formula.
They are identities on arbitrary smooth scalars.

Using \(P_m\EA f=f\), and
\(\EA P_ma=a\) for compact \(a\), one obtains
\begin{align*}
 L\EA f
    &=Df-P_m(X\EA f),\\
 L\EA L\EA f
    &=D^2f-P_m(X\EA Df)
                -DP_m(X\EA f)+P_m(X^2\EA f),\\
 M\EA f
    &=\tfrac12D^2f+\tfrac12P_m(X^2\EA f)
                                      -DP_m(X\EA f).
\end{align*}
The uses of \(\EA P_m=1\) are legitimate because
\(X\EA f\) and its further \(X\)-derivatives have
compact generator support. Subtraction proves
\eqref{eq:pg-new}.

The supplied coordinate laws give
\(w=\xi\), \(v=\tfrac12X\xi\), and hence
\[
 c_1=Df,\qquad c_2=\tfrac12D^2f.
\]
The mixed inherited term has representative
\[
 -L\EA Df=-D^2f+P_m(X\EA Df).
\]
Adding it and the bare second test to
\eqref{eq:pg-new} gives the stronger test identity
\[
 e_2=P_m\left(\tfrac12X^2\EA f\right).
\]
Taking its free class proves \eqref{eq:pg-cancel}.
\end{proof}

With
\[
 h=K\xi,\qquad k=0,
\]
one is not testing a second-order pure-gauge jet.
That substitution cannot be used to reject or verify
the complete cancellation.

\subsection{A sign-reversal diagnostic}

Reversing only the second-vertex sign would add
\[
 2\field(M\EA f)
\]
to the correctly completed pure-gauge coefficient.
It is not generally legitimate to replace this by a
fixed multiple of \(\field(D^2f)\), because the
term \(DP_m(X\EA f)\) in
\eqref{eq:pg-operators} is not generally an equation
zero.

A concrete classical check is available on admissible
instances with a sufficiently large plateau cylinder
and causal reserve. Let \(m=0\), and choose
\(\xi=a(t)\partial_t\) near the relevant causal compact,
with spatial cutoff outside it and
\(a\in C_c^\infty((\varepsilon,t_0))\), \(a\ne0\).
Take the observation times after \(t_0\).
The time reparametrization gives locally
\[
 M t=\partial_t\left(\tfrac12(a')^2-\tfrac12aa''\right),
\]
and zero-data retarded integration gives
\[
 \ER M t=\int_0^t(a'(s))^2\,\dd s-\tfrac12a(t)a'(t).
\]
After the support of \(a\), sign reversal therefore
produces the strictly positive evaluation
\[
 2\int(a')^2\,\dd t
\]
against a nonnegative observation density of integral
one. All required reserve and plateau conditions are
conditions on this diagnostic subfamily, not changes
made to a frozen instance after discovering a defect.

\subsection{Massless conformal check}

\begin{proposition}[Conformal diagnostic]
\label{prop:conformal}
Let \(m=0\), \(h=\psi\eta\), \(k=0\).
Then
\[
 a_1[h]=\psi\eta^{-1},\qquad
 b_1[h]=2\psi,\qquad
 b_2[h]=\psi^2,\qquad a_2[h]=0,
\]
and therefore \(L_{2,g}[h,0]=0\).

Suppose \(\psi=\psi(t)\) and \(g=g_0(t)\) near the
complete contributing causal compact, with spatial
cutoffs outside that compact. Define
\begin{align*}
 \Psi(t)&=\int_0^t\sigma(s)\psi(s)\,\dd s,
 &
 I_\sigma(t)&=\int_0^t(\sigma(s)\psi(s))^2\,\dd s,\\
 \Gamma(t)&=\int_0^t g_0(s)\psi(s)\,\dd s,
 &
 I_g(t)&=\int_0^t(g_0(s)\psi(s))^2\,\dd s.
\end{align*}
On the smooth free solution \(u=t\),
\begin{equation}
 w^0=-\Psi,\qquad v^0=I_\sigma,\qquad
 \rho_{1,g}=-\Gamma,\qquad \rho_{2,g}=I_g.
 \label{eq:conformal-coeff}
\end{equation}
The corresponding classical evaluations are
\begin{align}
 \pair{e_1}{t}
      &=\int f(\Psi-\Gamma)\,\dd^4x,\nonumber\\
 \pair{e_2}{t}
      &=\int f\left[
          I_g-I_\sigma+\Psi\psi(\sigma-g_0)\right]\dd^4x.
 \label{eq:conformal-value}
\end{align}
\end{proposition}

\begin{proof}
In four dimensions the conformal metric satisfies
\[
 \sqrt{-\det((1+\kappa\psi)\eta)}
                       = (1+\kappa\psi)^2,\qquad
 \sqrt{-\det q}\,q^{-1}=(1+\kappa\psi)\eta^{-1}.
\]
Thus the kinetic density has no quadratic term, and
the quadratic mass density does not contribute when
\(m=0\).

The coordinate first equation gives
\(w^0=-\Psi\), with zero spatial components on the
retained calculation. The second equation gives
\[
 \partial_t^2v^0
       =-\partial_t(\sigma\psi\,\partial_tw^0)
       =\partial_t(\sigma\psi)^2,
\]
hence \(v^0=I_\sigma\).
The scalar equations similarly give
\[
 \rho_{1,g}=-\Gamma,\qquad
 P_0\rho_{2,g}
      =\partial_t(g_0\psi)^2,
\]
and hence \(\rho_{2,g}=I_g\).

For \(u=t\), the bare second term is
\(\Psi\sigma\psi-I_\sigma\), the mixed term is
\(-\Psi g_0\psi\), and the new iteration is \(I_g\).
Adding gives \eqref{eq:conformal-value}.
\end{proof}

These numbers vanish when \(g_0\psi=\sigma\psi\)
throughout the contributing past, although the individual
response terms can be nonzero.
For instance, if \(\psi\) is supported in a post-ramp
interaction plateau and observations are later than
that support, then \(I_g=I_\sigma\), while the
iteration can be strictly positive.

The notation
\[
 \ev_u\field(d)=\pair{d}{u}
\]
denotes the degree-one classical evaluation justified
by \eqref{eq:separation}. It is not a state on the Wick
algebra or a quantum expectation value.
The diagnostics check signs and counting; they do not
replace the variation proof.

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

\begin{proposition}[The host is not evolved]
\label{prop:fixedhost}
The two maps, a regulator-description replacement,
and an installed-table update do not change
\(G,\mathcal W,T\), or the supplied future relation.
A different regulator may change the auxiliary
analytical prescription and its values.
\end{proposition}

\begin{proof}
The host components were fixed in
Assumption~\ref{ass:host}. The constructed maps return
free-field classes or finite operational outputs.
No map returning a new host metric has been defined.
\end{proof}

The following are comparison records, not inputs to
the response construction.

\begin{itemize}
\item \textbf{IX-C-ST.}
Its truncated partial product and selected wordwise
coherence through \(\hbar^3\) do not define the ramp
M{\o}ller coefficients. Wordwise coherence is not a
causal lift \cite{IXCST}.

\item \textbf{IX-D-ST and IX-E variants.}
A commutator consistency identity, an algebraic
extension criterion, a strongly separated vanishing
statement, or a conditional null-sphere bridge is
not a sharp time-ordered product. The boundary-trace
certificate for the weak inherited adjoint is not
verified by this scalar construction.
The ramp coordinate kernel is not identified with
IX-E-2's locked kernel \(K\) \cite{IXDST,IXERecords}.

\item \textbf{VII-A-ST.}
Its switch ledger illustrates why switch derivatives
must be displayed before imposing a plateau.
Its harmonic-coordinate intertwiner is not
\(\Rq\), and common observation-plateau agreement
does not establish equality of arbitrary past
responses \cite{VIIAST}.

\item \textbf{VI-B.}
The coefficientwise no-higher-order repair principle
is a logical comparison. Its Einstein--scalar
switch defect is not \(\Theta_F\) or \(\Sigma_F\)
\cite{VIB}.

\item \textbf{VII-H and VII-I.}
The former's susceptibility contact ambiguity is
not reopened here; the latter's finite-width free
algebra and changed sharp transmission data are not
a theorem about the present M{\o}ller maps
\cite{VIIH,VIII}.

\item \textbf{V-B.}
Its Newton--wave cubic form is neither the second
scalar response nor the inherited characteristic
gravitational coefficient \cite{VB}.

\item \textbf{X-A-conv and X-Enter.}
The convolution prescription and its conditional
readout are not inputs. The ramp integral
\eqref{eq:edge-integral} is not the convolution
quantity \(c_\varepsilon\) \cite{ConvRecords}.
\end{itemize}

VII-C's original hypotheses are not assumed or
repaired \cite{VIIC}. In particular, a fixed-width
scalar-linear chain calculation is not an extension
of its sharp mixed-kernel word.

Branch B remains unopened. No gravitational action,
gauge-fixed graviton Green operator, quantum metric
two-point kernel, or mixed contraction domain is
introduced. Neither existence nor a universal
obstruction for a future graviton tree or triangle
follows from these scalar maps.

%=====================================================================
\section{Finite installed encoding and the clock instrument}
\label{sec:instrument}
%=====================================================================

\subsection{What the table can encode}

Use the same finite clock and regular-route pattern
as X-B-ST. A finite installed catalogue may contain
reports or independently supplied enclosures for
\[
 \ev_{u_i}R_1(V_{2,g}[h_i,k_i];F_{c_i}),\qquad
 \ev_{u_i}\tfrac12R_2(V_{1,g}[h_i],V_{1,g}[h_i];F_{c_i}),
\]
and, when selected, the completed evaluation
\(\ev_{u_i}\widehat A_2[h_i,k_i]\).

Each report records its map identifier, background
description, insertion representative, evaluation
solution, and normalization convention.
The coupling-order label is part of that description;
a formal nilpotent coupling is not treated as a
nonzero numerical experimental parameter.

These are finite encodings of already defined
evaluations. They are not preparation channels from
unknown continuum fields. Arbitrary smooth functions
and arbitrary real numbers are not assumed to have
universal finite exact descriptions.

A nonempty two-map catalogue exists without numerical
execution: at \(h=k=0\), both evaluations are exactly
zero. On the diagnostic subfamily of
Proposition~\ref{prop:conformal}, one may choose a
post-ramp profile with \(\int\psi^2\,\dd t=1\), an
observation test of integral one later than its
support, and \(g_0=1\) on that support. Then the
second-vertex value is zero and the iteration value
is one, while the completed second evaluation is
zero. Spatial cutoffs are admissible under the
same causal-reserve condition as that diagnostic.
This example is a mathematical evaluation, not field
acquisition.

\subsection{Complete destructive readout}

Let the inherited live clock be
\(C\cong\CC^{2^{n_C}}\), \(n_C\ge1\), with its
orthonormal calibration basis \(\{\ket{\tau_k}\}\).
The finite dictionary is chosen to fit its codes,
or any additional description register is charged
separately. Let \(S\) contain all surviving finite
registers, and put
\[
 D_k=\bra{\tau_k}_C\otimes I_S,\qquad
 P_k=\proj{\tau_k}.
\]

\begin{theorem}[Complete clock instrument]
\label{thm:instrument}
The map
\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
\[
 \Tr_R\mathcal M_C(X)=\Tr_CX.
\]
All identities hold after adjoining any untouched
reference. The clock names are consumed; there is
no quantum \(C\)-output.
\end{theorem}

\begin{proof}
The recorded Kraus maps are
\(\ket{k}_R\otimes D_k\), and
\[
 \sum_k(\ket{k}_R\otimes D_k)^\dagger
                (\ket{k}_R\otimes D_k)
       =\sum_kP_k\otimes I_S=I_{CS}.
\]
This proves complete positivity and trace preservation.

For an arbitrary input operator,
\[
 X=\sum_{k,\ell}\kb{\tau_k}{\tau_\ell}\otimes X_{k\ell},
\]
the \(k\)-th output block is \(X_{kk}\).
It is also the indicated clock partial trace.
Summing the blocks proves the discard identity.
The calculation is unchanged with a reference tensor
factor retained.
\end{proof}

This is the finite I-B/En09-A interface
\cite{IB,En09}, proved here rather than inferred from
the field identities. Conditional reference states
can be steered by outcome selection; the unchanged
marginal statement is unconditional.

\begin{proposition}[Fixed-event and recovery limits]
\label{prop:fixed-event}
If distinct calibrated labels \(t_k\) must all equal
the time of one fixed readout event whenever their
codes occur, at most one code has positive probability.

The clock channel has no left inverse on all inputs,
and its record alone does not determine its residual.
No finite clock matrices satisfy
\([T_C,H_C]=iI_C\).
\end{proposition}

\begin{proof}
Distinctness proves the first assertion.
The inputs
\[
 \frac{\ket{\tau_0}+\ket{\tau_1}}{\sqrt2},
 \qquad
 \frac{\ket{\tau_0}-\ket{\tau_1}}{\sqrt2}
\]
with the same surviving state have identical outputs,
so no left inverse recovers both.
Inputs \(P_0\otimes\rho\) and
\(P_0\otimes\widetilde\rho\) have the same record and
different residuals. Finally, the trace of a finite
matrix commutator is zero, whereas \(\Tr(iI_C)\ne0\).
\end{proof}

A multi-outcome evaluation catalogue therefore uses
programme/history calibration unless a separate
event-time contract is supplied. Clock consumption
does not consume \(T\), \(t\), or a spacetime dimension.

\subsection{One explicit finite-table error topology}

Choose scaled installed reports \(v_k\in[-1,1]\),
with their map and data identifiers carried by \(k\).
For \(y=\pm1\), put
\[
 p_y(v_k)=\frac{1+yv_k}{2},
\]
and define the recorded channel
\[
 \mathcal I_v(X)
     =\sum_{k,y}\proj{k,y}_R\otimes
                           p_y(v_k)D_kXD_k^\dagger.
\]

\begin{proposition}[Fixed-interface table error]
\label{prop:table}
The channel \(\mathcal I_v\) is CPTP and, on the
same complete interfaces,
\begin{equation}
 d_\diamond(\mathcal I_v,\mathcal I_{\widetilde v})
       =\frac12\max_k|v_k-\widetilde v_k|,
 \qquad
 d_\diamond(\Lambda,\Gamma)
                  =\tfrac12\|\Lambda-\Gamma\|_\diamond.
 \label{eq:table-error}
\end{equation}
The equality includes arbitrary untouched references.
\end{proposition}

\begin{proof}
Use the complete Kraus family
\[
 \sqrt{p_y(v_k)}\,\ket{k,y}_R\otimes D_k.
\]
With a reference included, the difference on an
operator \(X\) has orthogonal 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 pinching contraction: pinching
is an average of unitary conjugations.
A clock-basis input at an index attaining the maximum
saturates the bound. Division by two proves the formula.
\end{proof}

This is a specified probabilistic encoding topology.
Numerically nearby values encoded as distinct
deterministic orthogonal words need not have nearby
channels. No such false continuity assertion is used.

The abstract finite instruments can be implemented
under the inherited local-unitary and destructive-readout
permissions, with a calibrated basis change and an
optional binary probe rotation. This is not a hardware
compilation theorem.

%=====================================================================
\section{Scheduled locality and the regular finite route}
\label{sec:transport}
%=====================================================================

\subsection{Locality with all records retained}

Let a finite site set carry finite local matrix algebras.
At tick \(j\), the complete channel is a tensor product
over a prescribed partition \(\mathcal B_j\).
All classical records used for control or reported at
the end are included in the network.

For a final set \(Y\), define
\[
 N_{t,t}(Y)=Y,\qquad
 N_{s,t}(Y)=
    \bigcup\{B\in\mathcal B_{s+1}:
                       B\cap N_{s+1,t}(Y)\ne\varnothing\}.
\]

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

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

\begin{proof}
A channel adjoint preserves the identity.
Blocks disjoint from the current observable support
therefore see only identity factors. A meeting block
can enlarge support only within itself.
Backward induction proves the support inclusion.
Testing every output observable gives the reduction
and intervention statements. Untouched references
are retained in the same argument.

Concatenating the individually certified host routes
gives the stated causal containment.
\end{proof}

This is an upper bound on dependency, not a derivation
or identification of the full physical causal past.
Scheduler consistency alone does not select a Lorentz
cone \cite{En08A}.

\subsection{The unchanged target-adapted transport pattern}

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

Install \(n+1\) stations
\[
 \mathbf x_j=\mathbf x_0+
                 \frac jn(\mathbf x_n-\mathbf x_0),
 \qquad j=0,\ldots,n.
\]
Each has a payload factor \(A_j\cong\CC^{2^p}\).
Initialize \(A_1,\ldots,A_n\) in independent blank
states \(\omega\). At hop \(j\), exchange
\(A_{j-1}\) and \(A_j\), with developed tick duration
\(\ell/n\).

The finite SWAP on these tick-boundary ports is an
explicit inherited operational permission.
Its cross-port dependencies are null and its
same-station dependencies are timelike.
A continuous-time implementing Hamiltonian is not
derived. 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 is
\[
 \rho_{A_jE}\otimes
       \bigotimes_{\substack{0\le i\le n\\i\ne j}}\omega_{A_i},
\]
under the declared register identifications.
The logical payload channel is the identity,
including arbitrary reference correlations.

The finite classical report can be forwarded with
its complete joint law. Arrival is at
\[
 t_{\mathrm d}+\ell,\qquad
 T_{\mathrm d}+\ell/c_{\mathrm{host}}.
\]
All dependency routes remain in the regular part.
\end{theorem}

\begin{proof}
Each SWAP exchanges the entire current payload
factor, including all correlations with \(E\),
with the next independent blank. It leaves the
previous station blank. Induction gives the formula.

Classical forwarding copies only diagonal record
data and therefore preserves its joint classical law;
it is not cloning an arbitrary quantum state.
The prescribed spacings and durations make the
cross dependencies null in the inherited developing
chart. No worldsheet crossing is used.
\end{proof}

The reference \(E\) may include the retained clock
record after extraction. A joint output formula does
not imply instantaneous delivery of that record:
receiver control based on it requires the declared
causal classical wire.

This is finite register transport relative to
prescribed geometry \cite{IB,En08B}, not propagation
of an unknown continuum field or generation of
entanglement by gravity.

\subsection{Resources and complete-output errors}

For an inherited \(n_C\)-qubit clock, \(p\)-qubit payloads,
\(b\)-bit report reserves, and lookup workspace
\(w_{\mathrm{lookup}}\), a conservative preallocated
quantum reserve is
\[
 n_C+(n+1)p+(n+1)b+w_{\mathrm{lookup}}+1.
\]
Any additional description register is separately
added to the ledger. The final qubit is an optional
table probe. Classical table storage, external
reference storage, gate counts, and durations are
separately declared.

The route uses \(n\) register SWAPs and \(n\) classical
report transmissions. Readout, lookup, and the buffer
are charged before departure.
The consumed clock names are not returned. All
retained record copies and any discarded blank
factors are specified in the complete interface.

For compatible implemented complete channels,
factorwise replacement gives
\begin{equation}
 d_\diamond(\widetilde\Gamma,\Gamma)
  \le\min\left\{1,\epsilon_{\mathrm{enc}}
       +\epsilon_{\mathrm{dec}}+\epsilon_{\mathrm{table}}
                                  +\sum_j\epsilon_j\right\}.
 \label{eq:error-ledger}
\end{equation}
Indeed, telescope the difference of the two
compositions and use diamond-norm contraction under
pre- and postcomposition by channels
\cite{Watrous}. An arrival or failure probability
is controlled only when its flag belongs to the
compared complete output. A channel-error bound is
not itself exact causal containment of an erroneous
implementation.

%=====================================================================
\section{Release and successor handoff}
\label{sec:release}
%=====================================================================

\begin{theorem}[X-C-ST local Grade S release]
\label{thm:release}
Relative to the explicitly stated inherited host,
classical handoff, and scalar free-field presentation:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the second-vertex map
\(R_1(V_{2,g};F_c)\) and the diagonal iterated coefficient
\(\tfrac12R_2(V_{1,g},V_{1,g};F_c)\) are constructed
on all specified off-shell compact linear
representatives;
\item both have smooth compact retarded output tests,
continuous background and test dependence, and no
positive-\(\hbar\) remainder in the stated linear
rooted prescription;
\item their sum represents the inherited second scalar
response exactly once;
\item the completed relational field is background
invariant through \(\kappa^2\) on the unchanged
\(\Gp\), with the explicit equation-ideal
representative \eqref{eq:ideal-representative};
\item outside that algebra the coordinate and switch
defects retain the three-term formula, including
the propagated first-order coordinate gap;
\item the complete pure-gauge two-jet closes the
X-B-ST remainder with the bare second dressing
retained; the conformal diagnostic has the stated
classical values;
\item the new finite installed encoding has a complete
destructive-clock instrument, scheduled locality,
reference-preserving regular transport, and a
declared resource/error ledger.
\end{enumerate}
This is local Grade S only.
\end{theorem}

\begin{proof}
Definition~\ref{def:moller} and
Theorem~\ref{thm:maps} prove (i)--(iii), with
Theorem~\ref{thm:yf} and
Proposition~\ref{prop:hbar} supplying the restricted
recursion and quantum-order characterization.
Proposition~\ref{prop:variation} and
Theorem~\ref{thm:ward} prove (iv)--(v).
Section~\ref{sec:diagnostics} proves (vi).
Theorems~\ref{thm:instrument},
\ref{thm:locality}, and \ref{thm:transport}, with
Proposition~\ref{prop:table} and
\eqref{eq:error-ledger}, prove (vii).
These provide 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 & Verdict & Exact scope\\
\midrule
Inherited first-vertex map &
Retained &
Not reconstructed or counted twice\\
Second-vertex scalar map &
Constructed &
Actual off-shell linear representatives\\
Iterated first-vertex coefficient &
Constructed &
Second Taylor coefficient, not a general \(R_2\)\\
Completed background identity &
Proved &
Through \(\kappa^2\), on the frozen plateau group\\
Pure-gauge remainder &
Cancelled &
Complete two-jet and all four terms retained\\
Edge and switch defects &
Retained &
Including first-gap propagation at second order\\
Whole-domain normalization &
Matched &
Separate slots fixed by the direct recursion\\
General \(T_2,T_3\) family &
Not constructed &
Inherited diagonal extension boundary unchanged\\
Finite Enter interface &
Constructed under permissions &
Supplied evaluations, not field acquisition\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Contribution and prospective use}

The central achievement is an explicit completion of
one second-order scalar-linear sector, with an
equation-ideal representative proving its background
identity and an off-plateau formula retaining the
failed hypotheses. The source result is credited to
X-C \cite{XC}; the supported transfer and its domain,
sign, and counting audit are the contribution of this
file.

Such an explicit baseline can be useful when deciding
which later difficulty is a regulator defect, which
is an input-representative issue, and which is a
genuinely new singular-product problem. It is not
an established quantum-gravity prediction or a
new general theory of renormalized retarded products.

\subsection{Handoff toward X-D-ST}

A successor receives:
\begin{enumerate}
\item the same supported host and fixed-width prescription;
\item the two new maps with their actual insertion domain,
support, continuity, and separate normalizations;
\item the completed test \(e[q]\), counted as in
\eqref{eq:fourterms};
\item the background identity through \(\kappa^2\);
\item the full three-term variation and propagated
edge formula;
\item the diagnostics and the statement that no
time-ordered family was used.
\end{enumerate}

It receives no sharp-width limit, no uniform
\(\varepsilon\)-dependent estimate, no quantum metric,
and no product hierarchy. A later scalar prescription
must not add \(\int f\rho_{2,g}\) again after including
the two maps that already represent it.

\subsection{Standing analytical boundaries}

The scalar free-field classes and the two response
coefficients are admitted in the senses proved above.
The separate inherited characteristic gravitational
quantum coefficient is not admitted by this release.

XV-B is not opened as a downstream quantum-admission
exit. XVI-N is not discharged. O.1 remains
\loc{NOT_CLOSED} in its inherited scope.
This file passes neither the historical H2
requirement nor the full I3/J2 characteristic-map
obligations. This does not retract XIV-I's narrower
constructed internal-amplitude result
\cite{BoundaryRecords}.

The technical XV-B version-2 record and the
downstream admission exit are not silently identified.
No full characteristic contraction, interacting
BRST/BV differential, quantum master equation,
sharp product, loop triangle, or two-point response
is constructed here.

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

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

Declaring a third cone &
The X-A-ST host, worldsheet, chart, signature ledger,
and future relation are unchanged inputs.\\[4pt]

Using conical global hyperbolicity &
All Green operators, free fields, and causal support
calculations belong to the Minkowski completion.\\[4pt]

Counting the old first map again &
Its formula is inherited. The new release consists
of the second vertex and the iterated first vertex.\\[4pt]

Adding a further copy of \(\rho_{2,g}\) &
The sum of the new pair already represents that one
classical coefficient.\\[4pt]

Using the positive Euler operator as the action Hessian &
The Hessians are \(-P_m\) and \(-L_{j,g}\).
The second-vertex response has sign \(-L_{2,g}\EA\).\\[4pt]

Ignoring \(\partial g\) or its second derivatives &
The operator and variation ledgers retain every
switch derivative before imposing the plateau.\\[4pt]

Applying iteration to an unspecified on-shell class &
The intermediate expression \(L_{1,g}\EA c\) is
formed as an actual smooth compact test.\\[4pt]

Inferring input descent from tree-exactness &
Equation \eqref{eq:input-nondescent} shows that an
equation-zero input can have nonzero response.\\[4pt]

Interpreting the diagonal \(R_2\) label as a hierarchy &
It denotes a second Taylor coefficient of one
quadratic first-vertex family only.\\[4pt]

Using polarization to claim diagonal additivity &
The cross terms in \eqref{eq:cross-terms} remain.
No mixed hierarchy is installed by that identity.\\[4pt]

Fixing a pair from normalization of its sum alone &
Separate normalization or the stated recursion
fixes the split.\\[4pt]

Proving quantum uniqueness from only a classical limit &
A formal-series uniqueness statement uses full
coefficientwise normalization.\\[4pt]

Calling sequential propagation a diagonal extension &
Each Green operator acts on a smooth compact
intermediate test. No Feynman square is assigned.\\[4pt]

Treating a failed sufficient criterion as nonexistence &
Only non-extension by the present tools is asserted,
not universal failure of renormalized extensions.\\[4pt]

Dropping the propagated first defect &
The term
\(-L_{1,g}[h']\EA(\theta_1-S_1\EA f)\)
is part of \eqref{eq:var2}.\\[4pt]

Assuming \(N_j\) or \(g-\sigma\) has compact support &
Only the defect combinations have compact generator
support; no such global compactness is assumed.\\[4pt]

Inferring off-shell equality from a quotient identity &
The test difference is the explicit compact
\(P_m\)-image in \eqref{eq:ideal-representative}.\\[4pt]

Using \(k=0\) as a quadratic pure-gauge test &
The correct two-jet includes
\(k=\tfrac12\Lie_\xi^2\eta\).\\[4pt]

Asserting a universal simple reversed-sign remainder &
Sign reversal adds \(2\field(M\EA f)\);
a \(DP_m\)-term cannot generally be dropped.\\[4pt]

Claiming the edge witness forces every total smearing
to be nonzero &
Smearing, equation-zero classes, switch terms, and
possible cancellations must be checked on the actual
data.\\[4pt]

Claiming a second-order term repairs a nonzero first
coefficient &
Formal orders cannot cancel that lower coefficient
while the prescription is fixed.\\[4pt]

Keeping a nonempty main plateau at \(g=0\) &
Zero switch is a separate diagnostic specialization
with a trivial plateau algebra.\\[4pt]

Using a classical evaluation as a quantum state &
\(\ev_u\) is a degree-one solution pairing, not a
state or quantum expectation value.\\[4pt]

Turning a finite table into a field detector &
Only supplied evaluations or enclosures are encoded.\\[4pt]

Equating every history label to one event time &
At one fixed event, exact matching of distinct
labels is deterministic.\\[4pt]

Identifying scheduled ancestry with a derived cone &
Physical containment uses separately certified
regular dependency routes.\\[4pt]

Calling endpoint SWAP a derived local Hamiltonian &
The finite tick-boundary permission is an explicit
operational input.\\[4pt]

Preserving stale empty-author declarations &
The title and PDF metadata identify Ngo Cheung,
as in the supplied predecessor documents.\\

\bottomrule
\end{longtable}

%=====================================================================
\section{Comparison-only operational and catalogue records}
\label{app:comparisons}
%=====================================================================

En08-A distinguishes scheduler consistency from an
additional geometric representation. En08-B treats
finite transport relative to a prescribed null boundary.
En09-A treats the destructive relational clock, and
En10-A supplies a separate finite source-weight
estimation contract \cite{En08A,En08B,En09,En10}.
None derives \(\EA\), acquires the present field, or
evolves the host metric.

The OpenAI-labelled components named in the requested
plan are comparison suggestions only:
\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.24\linewidth}XX@{}}
\toprule
Record & Possible interface comparison & Not supplied here\\
\midrule
Oai283-A &
Complete-channel error accounting &
Either scalar map or characteristic admission\\
Oai272-A &
A separately declared finite support witness &
An all-orders completion\\
Oai276-A &
Finite clock-outcome information analysis &
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 in this construction &
A Page curve, lapse, or Einstein dynamics\\
\bottomrule
\end{tabularx}
\end{center}

This table does not verify or apply those catalogue
proofs. Their hypotheses, constants, and normalization
classes are not imported. The finite channel
identities needed for X-C-ST were proved directly
in this file.

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

\paragraph{Authorship and attribution.}
The author is Ngo Cheung. The analytical source
results are attributed by manuscript identifier and
version. The supplied X-A-ST and X-B-ST documents
are cited as the predecessor manuscripts actually
used, without retaining their inconsistent
empty-author boilerplate.

\paragraph{Novelty.}
The underlying scalar response construction is
credited to X-C and situated within the established
quadratic-field and perturbative-agreement literature.
The contribution claimed here is the supported
completion, explicit inherited-domain ledger, and
normalization, sign, defect, and finite-interface audit.

\paragraph{Analytical scope.}
The quantum field is the scalar on the flat completion.
Metric coefficients remain classical and off shell.
The completed background identity is through
\(\kappa^2\) on the frozen group. No general
time-ordered product family is constructed.

\paragraph{Operational scope.}
Enter consumes a finite clock encoding of supplied
evaluations. No continuum-field acquisition,
microscopic relativistic gate Hamiltonian, hardware
synthesis, or thermodynamic implementation is derived.

\paragraph{Bibliographic status.}
Internal source records are cited without invented
publication dates or DOIs. Comparison records do
not become premises through shared terminology.
The standing analytical stops are version-qualified,
not a claim to have rescanned or verified the entire
series.

\paragraph{Verification.}
No successful \loc{pdflatex} execution, proof-assistant
check, symbolic-program execution, numerical evaluation,
hardware synthesis, experiment, independent referee
review, or journal acceptance is asserted.
The claimed mathematical results are the explicit
definitions and written proofs above.

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

\bibitem{XC}
N.~Cheung,
\emph{Landmark X-C: The missing second-order retarded
maps for the ramp relational scalar, with the metric
kept classical}.
\loc{X-C-v3.tex}, version 3.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-C}.
Especially \loc{def:moller}, \loc{thm:maps},
\loc{thm:yf}, \loc{prop:hbar}, \loc{lem:duality},
\loc{def:defects}, \loc{prop:variation},
\loc{thm:ward}, \loc{cor:routeII},
and \loc{thm:release}.

\bibitem{XAST}
N.~Cheung,
\emph{X-A-ST: Supported classical covariance of a
fixed-ramp relational scalar}.
Supplied manuscript \loc{X-A-ST.tex}.
Host, supported chart, flat completion, signature
ledger, plateau, classical handoff, and finite
operational permissions.

\bibitem{XBST}
N.~Cheung,
\emph{X-B-ST: One scalar retarded map on the X-A-ST host}.
Supplied manuscript \loc{X-B-ST.tex}.
First scalar response, fixed-width admission,
first-order identity, normalization, and
one-contraction nonextension boundary.

\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}.
Classical source record; the present inherited
supported data are stated explicitly.

\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.
\url{https://enter.grok.me/browse/X-B}.

\bibitem{XBii}
N.~Cheung,
\emph{Landmark X-B-ii: Scope closure for the scalar-only
retarded map}.
\loc{X-B-ii-v1.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-B-ii}.
One-contraction boundary, whole-domain normalization,
and unopened quantum-metric branch.

\bibitem{IB}
N.~Cheung,
\emph{I-B: A Finite Enter Spacetime:
Calibrated Destructive Time Readout on a Prescribed
Conical Product}.
\loc{I-B-v1.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/I-B}.
Prescribed geometry, complete clock instrument,
calibration, scheduled locality, and regular
target-adapted 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 within its specified
strategy class; contextual, not an analytical premise.

\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, Enter Calculus Companion.
\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, used on the Minkowski 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 functional products and contraction bookkeeping;
no higher interacting family is imported here.

\bibitem{HW}
S.~Hollands and R.~M.~Wald,
\emph{Conservation of the stress tensor in perturbative
interacting quantum field theory in curved spacetimes}.
arXiv:gr-qc/0404074.
\url{https://arxiv.org/abs/gr-qc/0404074}.
Perturbative-agreement and quadratic-interaction
context, not a selected product prescription for
the present sheet.

\bibitem{DHP}
N.~Drago, T.-P.~Hack, and N.~Pinamonti,
\emph{The generalised principle of perturbative
agreement and the thermal mass}.
arXiv:1502.02705.
\url{https://arxiv.org/abs/1502.02705}.
Quadratic M{\o}ller-map context; its hypotheses and
larger conclusions are not transferred here.

\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}.
General retarded-product framework, distinct from
the two directly defined coefficients.

\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}.
Wavefront product, pullback, and pushforward criteria.

\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}.
Extension and renormalization framework;
not an extension selected in this file.

\bibitem{IXCST}
N.~Cheung,
\emph{IX-C-ST}.
Version 1 supplied manuscript record,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/IX-C-ST}.
Selected truncated partial product and wordwise
coherence; comparison only.

\bibitem{IXDST}
N.~Cheung,
\emph{IX-D-ST}.
Version 1 supplied manuscript record,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/IX-D-ST}.
Ordered interface and bridge stages, especially
\loc{thm:Drelease}, \loc{thm:Erelease},
\loc{thm:E3release}, and \loc{thm:release}.
No sharp time-ordered family is imported.

\bibitem{IXERecords}
N.~Cheung,
\emph{IX-E}, \emph{IX-E-2}, and \emph{IX-E-3}.
Version 1 supplied manuscript records,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/IX-E};
\url{https://enter.grok.me/browse/IX-E-2};
\url{https://enter.grok.me/browse/IX-E-3}.
Algebraic consistency, locked-kernel evaluation,
and conditional boundary-trace bridge; comparisons only.

\bibitem{VIIAST}
N.~Cheung,
\emph{VII-A-ST}.
Version 1 supplied manuscript record,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-A-ST}.
Switch ledger and second-order classical intertwiner
on its own equation space.

\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 classical switch defect and coefficientwise
order-obstruction comparison.

\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 susceptibility and its
contact ambiguity.

\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 changed sharp
transmission data; not the present M{\o}ller map.

\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}.
Newton--wave cubic-form and finite-interface
comparisons.

\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 original extension hypotheses.

\bibitem{ConvRecords}
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 conditional
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.

\bibitem{En08B}
N.~Cheung,
\emph{En08-B}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En08-B}.
Finite resource-controlled 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
scope.

\bibitem{En10}
N.~Cheung,
\emph{En10-A}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En10-A}.
Source-weight estimation for a separately declared
finite access model.

\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 coefficient or downstream admission
is supplied by the present scalar construction.

\end{thebibliography}

\end{document}