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\hypersetup{
  pdftitle={X-G-ST: Free causal BRST compatibility and a variational upgrade obstruction},
  pdfauthor={Ngo Cheung},
  pdfsubject={Supported fixed-width transfer, auxiliary--ghost Green matrix,
  switched-matter breaking, and source-preserving variational reciprocity}
}

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\title{\textbf{X-G-ST: Free causal BRST compatibility\\
and a variational upgrade obstruction}\\[5pt]
\large A supported fixed-width transfer on the inherited\\
conical host and declared Minkowski completion}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We give a supported realization of Cheung's Landmark X-G
on the host retained by I-B and X-A-ST, with all analytical
equations posed on its independently declared Minkowski
completion. The regulator width remains fixed and positive.

An independent auxiliary field and one normalized gauge
fermion produce the X-E/X-F Feynman-weight graviton operator.
We compute the auxiliary--ghost Green matrix and prove that
the off-shell nilpotent free differential preserves both
the free equations and the graded causal relations.
This constructs a free causal BRST derivation, not an
interacting quantum BRST operator.

Two classical obstacles are then separated. The switched
scalar action has an explicit natural gauge breaking,
including an off-shell equation term outside
\(\supp\dd g\), and a nonzero coefficient detected by a
free scalar and a compact odd test. Independently,
post-variation multiplication of gravitational sources by
the onset violates variational reciprocity. A compact
pure-gauge background in the onset layer gives the
strictly positive free-solution pairing
\[
 \frac{5M_\chi}{8}\int_0^\varepsilon
                         \zeta(t)(\sig''(t))^2\,\dd t.
\]
It excludes an exact source-preserving variational upgrade
on the unrestricted background class, even after
free-shell-zero changes of representatives. The witness
also admits free inputs with compact Cauchy data.
Algebraic auxiliary elimination and corrections beginning
at positive formal quantum order cannot remove it.

The direct three-tree background identity of X-F remains
valid on the unchanged plateau algebra. Its survival is
compatible with the upgrade obstruction because the two
statements test different backgrounds and different maps.
Local fish extensions, an exceptional scalar wavefront
configuration, and a conditional background-independent
loop addition are recorded without declaring the nonlocal
one-loop observable constructed.

The principal free construction and obstruction are
credited to X-G. The contribution here is their supported
proof, normalization and causal-sign audit, and explicit
separation of completion-level witnesses from laboratory
support claims. These results provide a reproducible
compatibility test for proposed quantum continuations,
without claiming a universal obstruction to perturbative
gravity. A finite supplied-evaluation interface has a
complete destructive clock and reference-preserving regular
transport. The release is local Grade S only.
\end{abstract}

\statementbox{The three verdicts are not interchangeable}{%
The free auxiliary--ghost causal algebra is constructed.

An exact variational upgrade preserving the entire
source-switched mixed metric map is obstructed in the
class defined below.

The nonlocal one-loop observable and an interacting quantum
BRST differential remain unconstructed. Neither the positive
free result nor the classical obstruction is renamed as
a theorem about those missing quantum operations.}

\tableofcontents

%=====================================================================
\section{Purpose, attribution, and the retained presentation}
\label{sec:host}
%=====================================================================

\subsection{Source and contribution}

The analytical source is N.~Cheung's
\emph{Landmark X-G: The normalized BRST multiplet and the
one-loop background gate: proved compatibility conditions,
an exact-upgrade obstruction, and the remaining nonlocal
extension problem}, version 1 \cite{XG}.
The supplied X-E-ST and X-F-ST are predecessor source
records \cite{XEST,XFST}. Their identifiers do not establish
registration or independent verification.

The present proof does not use the parent obstruction or
free-BRST theorem as a premise. Their required operators,
signs, and arguments are stated explicitly. Conversely,
the inherited scalar completion and three tree expressions
are not reconstructed as additional response maps.
Only the algebraic interface needed to explain their
survival is checked.

The principal normalization, free differential, reciprocity
argument, and positive witness are attributed to X-G.
The supported transfer and the audits in this manuscript
are not priority claims for those parent results.

The source is proposed as X-G-ST. No repository installation,
external registration, submission, acceptance, or successful
compilation is asserted.

\subsection{Enter and the refusal of identification}

The Enter distinction relevant here is destructive
classical readout versus quantum return. A measured clock
encoding is consumed from the returned quantum interface;
the complete output contains its classical code and the
unmeasured residual \cite{EnterI,EnterPhilosophy,En09}.

We keep separate:
\begin{enumerate}
\item an analytically defined quantity and acquisition of
unknown field data;
\item an outcome law and a complete instrument;
\item a reference-preserving transport channel and
entanglement generated by gravity;
\item a prescribed spacetime and a derivation of spacetime.
\end{enumerate}
I-B supplies the prescribed-host interpretation, finite
clock, calibration distinctions, and transport pattern
\cite{IB}. It does not supply the gravitational action,
the free multiplet, or the obstruction proved below.

Only explicit finite Born instruments are needed in the
operational part. Their complete channel identities are
proved directly. No shortened reconstruction of the
Enter characterization theorem is used.

\subsection{The unchanged host}

\begin{assumption}[Inherited supported presentation]
\label{ass:host}
Retain exactly the X-A-ST instance used by X-D-ST,
X-E-ST, and the supplied X-F-ST:
\begin{enumerate}[label=\textup{(H\arabic*)}]
\item the regular time-oriented conical product
\((\mathcal M^\circ,G)\), its stipulated worldsheet
\(\mathcal W\), and its supplied future relation;
\item the supported injective developing chart
\[
 \iota:\mathcal L\longrightarrow\mathcal M^\circ,
\]
with its retained compact reserve disjoint from
\(\mathcal W\);
\item the independently declared analytical completion
\[
 \widehat M=\RR^{1+3},\qquad
 \eta=\operatorname{diag}(1,-1,-1,-1),\qquad
 M_{T_*}=[0,T_*]\times\RR^3,\qquad
 \Sigma=\{t=0\};
\]
\item the signature and time ledger
\[
 \iota^*G=-\eta,\qquad
 t=c_{\mathrm{host}}(T-T_\Sigma);
\]
the host has signature \((-,+,+,+)\), whereas the
analytical equations use \((+,-,-,-)\);
\item a nonempty observation region
\[
 \mathcal O\Subset(0,T_*)\times\RR^3,\qquad
 D_*:=J_\eta^-(\overline{\mathcal O})\cap M_{T_*}
                                               \Subset\mathcal L.
\]
Compact laboratory tests are transported and extended
by zero on the completion.
\end{enumerate}
The regular conical product is not used as a globally
hyperbolic Cauchy spacetime. Every global free field and
Green operator below belongs to \(\widehat M\).
\end{assumption}

The defect remains a worldsheet, not a worldline.
There is no second cone, global conical inverse,
worldsheet gluing construction, or host-update map
\cite{IB,STRecords}.

\begin{definition}[Local Grade S]
\label{def:grade}
The package comprises the inherited supported host;
the named free identities and obstruction with their
normalization and support contracts; a finite dictionary
of supplied evaluations; a complete destructive-clock
instrument including untouched references; and scheduled
locality with a nonempty regular reference-preserving
route under declared finite permissions.

It does not include Grade E, spacetime emergence, or
Einstein dynamics derived from records.
\end{definition}

%=====================================================================
\section{Frozen operators, switches, and the predecessor interface}
\label{sec:frozen}
%=====================================================================

\subsection{Fixed width and supports}

Fix
\[
 m\ge0,\qquad
 f\in C_c^\infty(\mathcal O;\RR),\qquad
 g\in C_c^\infty((0,T_*)\times\RR^3;[0,1]).
\]
Let \(U\) be a nonempty relatively compact open plateau:
\[
 \overline U\Subset\mathcal L\cap\operatorname{int}\{g=1\},
 \qquad
 \gp=C_c^\infty(U;T\widehat M).
\]
Write
\[
 K_f=\supp f,\qquad
 S_f=\supp g\cap J_\eta^-(K_f),\qquad
 C_f=K_f\cup S_f,\qquad
 D_f=J_\eta^-(K_f)\cap M_{T_*}.
\]
The symbol \(K\), without a subscript, is reserved for
the Killing operator.

Choose \(\delta>0\) with
\begin{equation}
 K_f\cup\supp g\cup\overline U
                    \subset\{2\delta<t<T_*\},
 \qquad 0<\varepsilon<\delta.
 \label{eq:separation}
\end{equation}
The onset
\(\sig(t)=\chi(t/\varepsilon)\) is smooth, zero for
\(t\le0\), one for \(t\ge\varepsilon\), and flat at
both endpoints. The width is fixed. No limit
\(\varepsilon\to0\) is taken.

The classical coefficients \(h,k\) remain arbitrary
smooth real symmetric tensors. In particular, they
need not vanish in \(0<t<\varepsilon\).
The metric jets below are not updates of the host
metric \(G\).

\subsection{Signed operators and tensor types}

Set
\[
 \Box=\partial_t^2-\Delta,\qquad P_m=\Box+m^2.
\]
The positive equation inverses are
\[
 \Box\ERzero=\Box\EAzero=1,\qquad
 P_m\ER=P_m\EA=1.
\]
Their causal support, Cauchy uniqueness, finite
propagation, and continuity are used on the completion
\cite{BGP}.

For a covariant symmetric tensor \(r\), put
\[
 (\Ptr r)_{\mu\nu}
       =r_{\mu\nu}-\tfrac12\eta_{\mu\nu}\tr_\eta r,
 \qquad
 (\Cdiv r)^\mu=\partial_\nu(\Ptr r)^{\nu\mu},
\]
and
\[
 (K\xi)_{\mu\nu}=\partial_\mu\xi_\nu+\partial_\nu\xi_\mu.
\]
All component identifications in flat formulas use
\(\eta\) and the coordinate volume density.

The formal transpose is bilinear, not a
Hilbert-space adjoint. Compact integration by parts gives
\begin{equation}
 \Cdiv^*v=-\tfrac12\Ptr Kv,\qquad
 \Cdiv K=\Box,\qquad \Ptr^2=1.
 \label{eq:Cstar}
\end{equation}
Let
\begin{equation}
 \AK=\tfrac12G_{\mathrm{lin}},\qquad
 \AK K=0,\qquad
 \AK+\tfrac12\Cdiv^*\Cdiv
          =\Pgr=-\tfrac14\Ptr\Box.
 \label{eq:operators}
\end{equation}
These are the retained X-E/X-F normalization identities
\cite{XE,XF}. They can also be checked by substituting
the standard linearized Einstein tensor into
\eqref{eq:operators}. In particular,
\[
 \partial_\mu(\AK r)^{\mu\nu}=0,\qquad
 \partial_\mu(\Cdiv^*b)^{\mu\nu}
                              =-\tfrac12\Box b^\nu.
 \label{eq:dividentities}
\]
The metric Green inverses remain
\begin{equation}
 \DR=-4\Ptr\ERzero,\qquad
 \DA=-4\Ptr\EAzero.
 \label{eq:metricGreen}
\end{equation}
The scalar action Hessian is \(-P_m\), whose inverses
are \(-\ER,-\EA\). It is not the positive equation
operator \(P_m\).

Metric inputs are covariant tensors. Metric Euler
outputs are contravariant tensor densities. Denote their
Lie derivatives by \(\Liec_\xi,\Lied_\xi\). Thus
\begin{align}
 (\Liec_\xi r)_{\mu\nu}
   &=\xi^\lambda\partial_\lambda r_{\mu\nu}
        +r_{\lambda\nu}\partial_\mu\xi^\lambda
        +r_{\mu\lambda}\partial_\nu\xi^\lambda,
 \nonumber\\
 (\Lied_\xi A)^{\mu\nu}
   &=\partial_\lambda(\xi^\lambda A^{\mu\nu})
        -A^{\lambda\nu}\partial_\lambda\xi^\mu
        -A^{\mu\lambda}\partial_\lambda\xi^\nu.
 \label{eq:Lie}
\end{align}
Their compact pairing satisfies
\[
 \pair{r}{\Lied_\xi A}=-\pair{\Liec_\xi r}{A}.
\]

\subsection{Retained action and background gauge}

Keep the curvature convention and local action of X-E:
\begin{equation}
 S_{\mathrm{EH}}[q]=-\frac1{2\kappa^2}\int J_qR[q]\,\dd^4x,
 \qquad J_q=\sqrt{-\det q}.
 \label{eq:EH}
\end{equation}
Euler derivatives mean compact variations. Neither
global action integrability nor decay is assumed.
The Palatini variation and transverse-traceless sign
are retained from the stated action, not selected anew
\cite{XE,XEST}.

For \(\bar q=\eta+\kappa h\), retain the background
de Donder vector
\[
 C_{\bar q}^{\mu}[r]
   =\bar\nabla_\nu\left(
       \bar q^{\nu\alpha}\bar q^{\mu\beta}r_{\alpha\beta}
          -\tfrac12\bar q^{\nu\mu}
                         \bar q^{\alpha\beta}r_{\alpha\beta}\right)
\]
and the fluctuation gauge-fixing density
\[
 S_{\mathrm{gf}}[r;\bar q]
    =\tfrac14\int J_{\bar q}\bar q_{\mu\nu}
                      C_{\bar q}^\mu[r]C_{\bar q}^\nu[r]\,\dd^4x.
\]
Only the fluctuation is gauge fixed.

Let \(\mathcal E[q]=J_qG^{\mu\nu}[q]\), with its Euler
density type, and define
\begin{equation}
 Q[\bar q]r
    =\tfrac12D\mathcal E[\bar q]\,r
                      +\frac{\delta S_{\mathrm{gf}}[r;\bar q]}{\delta r},
 \qquad
 Q[\eta+\kappa w]=\Pgr+\kappa Q_w+O(\kappa^2).
 \label{eq:Q}
\end{equation}
This fixes \(Q_w\) without an unspecified vertex.
It is a local formally symmetric differential
operator, linear in the two-jet of \(w\), with
\(\supp(Q_wr)\subset\supp w\).

Naturality of both terms in \eqref{eq:Q} gives
\begin{equation}
 Q_{K\xi}
       =\Lied_\xi\Pgr-\Pgr\Liec_\xi.
 \label{eq:commutator}
\end{equation}
Indeed, differentiate
\(Q[F_s^*\eta](F_s^*r)=F_s^*(Q[\eta]r)\).
This verifies the typed commutator used later; it
does not rebuild the three response kernels.

The frozen mixed source map is
\begin{equation}
 N_w=\sig Q_w,\qquad
 k_{\mathrm{mix}}[w;u]=-\DR N_wu.
 \label{eq:mixed}
\end{equation}
For general smooth inputs the source is prepared by its
smooth onset and interpreted in the past-compact
Green domain. For the compact backgrounds used in
the obstruction, it is a compact smooth source.

\subsection{The completed scalar and three-tree interface}

We record the insertion target to make the counting
unambiguous \cite{XC,STRecords,XFST}.
For
\(q=\eta+\kappa l+\kappa^2j\), define \(a_1[l]\),
\(A_2[l,j]\), \(b_1[l]\), and \(B_2[l,j]\) by
\[
 J_qq^{\mu\nu}
   =\eta^{\mu\nu}+\kappa a_1^{\mu\nu}[l]
                            +\kappa^2A_2^{\mu\nu}[l,j]+O(\kappa^3),
 \qquad
 J_q=1+\kappa b_1[l]+\kappa^2B_2[l,j]+O(\kappa^3).
\]
In particular \(a_1[l]=-\Ptr l\) and
\(b_1[l]=\frac12\tr_\eta l\).

The coordinate coefficients \(w[l],v[l,j]\) have zero
position and velocity on \(\Sigma\) and solve
\begin{align}
 \Box w^\rho[l]
   &=-\partial_\mu(\sig a_1^{\mu\rho}[l]),
 \nonumber\\
 \Box v^\rho[l,j]
   &=-\partial_\mu\left(
       \sig\{A_2^{\mu\rho}[l,j]
                +a_1^{\mu\nu}[l]\partial_\nu w^\rho[l]\}\right).
 \label{eq:coordinates}
\end{align}
Put
\begin{align}
 L_1[l]u&=\partial_\mu(g a_1^{\mu\nu}[l]\partial_\nu u)
                                           +gm^2b_1[l]u,
 \nonumber\\
 L_2[l,j]u&=\partial_\mu(g A_2^{\mu\nu}[l,j]\partial_\nu u)
                                           +gm^2B_2[l,j]u,
 \nonumber\\
 c_1[l]&=\partial_\mu(fw^\mu[l]),
 \nonumber\\
 c_2[l,j]&=-\partial_\mu\left(
     f(w^\nu[l]\partial_\nu w^\mu[l]-v^\mu[l,j])\right)
           +\tfrac12\partial_\mu\partial_\nu(fw^\mu[l]w^\nu[l]),
 \nonumber\\
 d_1[l]&=c_1[l]-L_1[l]\EA f,
 \nonumber\\
 d_2[l,j]&=c_2[l,j]-L_1[l]\EA c_1[l]
                       -L_2[l,j]\EA f+L_1[l]\EA L_1[l]\EA f.
 \label{eq:tests}
\end{align}
The affine decomposition is
\[
 d_2[l,j]=\mathcal Q[l,l]+d_1[j].
\]

With \(\Aone[l]=\field(d_1[l])\), the frozen three
tree coefficients are
\begin{align}
 T_{\mathrm{mat}}
    &=\Wick{\Aone[\DR(\tfrac12gT_0[\field])]},
 \nonumber\\
 T_{\mathrm{mix}}
    &=\Wick{\Aone[-\DR(\sig Q_h\rfield)]},
 \nonumber\\
 T_{\mathrm{self}}
    &=\Wick{\Aone[-\tfrac12\DR(\sig G_2[\rfield,\rfield])]}.
 \label{eq:trees}
\end{align}
Here \(G_2\) is the polarized quadratic coefficient of
\(J_qG^{\mu\nu}[q]\). These symbols mean the compact
distributional kernels of the fixed-width tree sheet,
not pointwise multiplication of singular fields.

The inherited expression is
\begin{align}
 \widehat\Phi_F[h,k]
    ={}&\field(f)+\kappa\Aone[h+\rfield]
 \nonumber\\
       &+\kappa^2\left(
          \Qhat[h+\rfield,h+\rfield]+\Aone[k]
               +T_{\mathrm{mat}}+T_{\mathrm{mix}}+T_{\mathrm{self}}
                    \right).
 \label{eq:PhiF}
\end{align}
The matter response equals
\(-2\Ptr\ERzero(gT_0[\field])\).
The scalar second-order pair in \eqref{eq:tests}
and the three coefficients in \eqref{eq:trees} are
retained once.

The inherited support distinctions remain:
the scalar slot lies in \(C_f\); the matter cubic
has support in \(C_f^3\); graviton slots of the other
trees can reach \(D_f\).
Full kernels are distributions. Suitable smooth
scalar-slot contractions are elements of \(\DD_{C_f}\).
No kernel-admission theorem is inferred from a
scalar-test topology.

\statementbox{What is new below}{%
The new free construction adds independent auxiliary and
ghost fields and proves compatibility of their causal
relations.

The new obstruction tests whether the fixed source map
\eqref{eq:mixed} can be the classical part of a variational
retarded map on all free inputs.

Neither task requires constructing the predecessor
trees again. Their admission is not used to prove the
reciprocity obstruction.}

%=====================================================================
\section{Stage 1: the normalized free auxiliary--ghost multiplet}
\label{sec:free}
%=====================================================================

\subsection{Independent generators and off-shell nilpotence}

Use independent fields
\[
 (r_{\mu\nu},\phi,c^\mu,\bar c_\mu,b_\mu).
\]
Their parities are \(0,0,1,1,0\), and their ghost
numbers are \(0,0,1,-1,0\).
The odd fields have coefficients in a Grassmann
algebra. All fields are ambient smooth configurations
before equations are imposed. Compactness belongs
to smearing tests and variations, not to a separate
antighost configuration space.

Define an odd derivation \(s_0\), commuting with
spacetime differentiation, by
\begin{equation}
 s_0r=Kc,\qquad
 s_0\phi=0,\qquad
 s_0c=0,\qquad
 s_0\bar c=b,\qquad
 s_0b=0.
 \label{eq:s0}
\end{equation}
The graded Leibniz rule is
\(s_0(FG)=(s_0F)G+(-1)^{|F|}F(s_0G)\).

\begin{lemma}[Off-shell free nilpotence]
\label{lem:s0nil}
The derivation \(s_0\) is nilpotent on the independent
fields and their graded polynomial differential algebra.
\end{lemma}

\begin{proof}
Its square is zero on each generator in
\eqref{eq:s0}. The square of an odd derivation is
an even derivation. It therefore vanishes on every
polynomial in generators and their derivatives.
No equation of motion is used.
\end{proof}

This is the free normalization of X-G, not the
differently scaled XI-B multiplet
\cite[\loc{lem:s0nil}, \loc{prop:gfnorm}]{XG}.

\subsection{Gauge fermion and ghost normalization}

Take the gauge fermion
\begin{equation}
 \Psi_0=\int\bar c_\mu\bigl((\Cdiv r)^\mu-b^\mu\bigr)\dd^4x.
 \label{eq:Psi0}
\end{equation}
The word ``fermion'' here denotes an odd gauge-fixing
functional; no physical spinor is introduced.
The free action is
\begin{equation}
 S_0
   =\tfrac12\pair{r}{\AK r}
        -\tfrac12\pair{\phi}{P_m\phi}
        +s_0\Psi_0,
 \qquad
 s_0\Psi_0
    =\int\left(b_\mu(\Cdiv r)^\mu-b_\mu b^\mu
                                      -\bar c_\mu\Box c^\mu\right)\dd^4x.
 \label{eq:S0}
\end{equation}
As elsewhere, these are local actions with compact
variations.

\begin{proposition}[Gauge-fixing normalization]
\label{prop:gfnorm}
The auxiliary equation gives
\[
 b=\tfrac12\Cdiv r.
\]
Eliminating \(b\) in the action produces exactly
\(\frac14\int(\Cdiv r)_\mu(\Cdiv r)^\mu\).
Thus the reduced metric operator is \(\Pgr\),
and the ghost term in this convention is
\(-\int\bar c_\mu\Box c^\mu\), without an extra
normalization factor.

The independent \(b\) is required for off-shell
nilpotence. After its elimination,
\(s_0^2\bar c=\frac12\Box c\) is only an on-shell
identity.
\end{proposition}

\begin{proof}
Complete the square:
\[
 b\cdot\Cdiv r-b^2
   =\tfrac14(\Cdiv r)^2
                      -\left(b-\tfrac12\Cdiv r\right)^2.
\]
Equation \eqref{eq:operators} gives the reduced
operator. The minus sign of the ghost term follows
from the odd parity of \(\bar c\) in the variation
of \eqref{eq:Psi0}. Finally,
\(s_0(\Cdiv r)=\Cdiv Kc=\Box c\).
\end{proof}

The free action is \(s_0\)-invariant modulo integrated
divergences: \(\AK K=0\) handles its metric term,
the scalar is \(s_0\)-inert, and \(s_0^2\Psi_0=0\).

\subsection{The metric--auxiliary Green operators}

The bosonic Hessian in the order \((r,b)\) is
\[
 \mathcal L_{rb}
      =\begin{pmatrix}
          \AK&\Cdiv^*\\
          \Cdiv&-2I
        \end{pmatrix}.
\]
For compact sources \((J,j)\), its equations are
\begin{equation}
 \AK r+\Cdiv^*b=J,\qquad
 \Cdiv r-2b=j.
 \label{eq:rbequations}
\end{equation}

\begin{proposition}[Auxiliary Green matrix]
\label{prop:Ghb}
For either retarded or advanced support,
\begin{align}
 \mathcal G_{rb}^{\mathrm R/\mathrm A}
  &=
  \begin{pmatrix}
   \Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}
       &\tfrac12\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}\Cdiv^*\\
   \tfrac12\Cdiv\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}
       &\tfrac14\Cdiv\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}
                                      \Cdiv^*-\tfrac12I
  \end{pmatrix}
 \nonumber\\
  &=
  \begin{pmatrix}
     -4\Ptr\mathsf E_0^{\mathrm R/\mathrm A}
          &K\mathsf E_0^{\mathrm R/\mathrm A}\\
     -2\Cdiv\Ptr\mathsf E_0^{\mathrm R/\mathrm A}&0
  \end{pmatrix}.
 \label{eq:Ghb}
\end{align}
Both inverse identities hold on compact tests.
The operators have the indicated causal support.
\end{proposition}

\begin{proof}
The second equation gives
\(b=(\Cdiv r-j)/2\). Substitution into the first
gives
\[
 \Pgr r=J+\tfrac12\Cdiv^*j.
\]
Apply the appropriate metric Green inverse and
substitute back for \(b\). This yields the first
matrix.

The identities
\[
 \Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}\Cdiv^*
                     =2K\mathsf E_0^{\mathrm R/\mathrm A},
 \qquad
 \Cdiv\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}\Cdiv^*=2I
\]
follow from \eqref{eq:Cstar} and
\(\Box\mathsf E_0^{\mathrm R/\mathrm A}=1\).
They give the second matrix. Substitution verifies
the right inverse; applying the same elimination
to a compact field pair verifies the left inverse.
Uniqueness reduces to that of the free metric wave
equation. Derivatives and the cancelled algebraic
block preserve causal support.
\end{proof}

This Green matrix is the auxiliary normalization of
X-G \cite[\loc{prop:Ghb}]{XG}.
It is not a Green matrix on the regular cone.

\subsection{Ghost matrix and full causal form}

Use left Grassmann variations, placing variations on
the left. The ghost Euler derivatives of
\(-\int\bar c\Box c\) are
\[
 E_c=\Box\bar c,\qquad E_{\bar c}=-\Box c.
\]
Consequently, in the order \((c,\bar c)\),
\begin{equation}
 \mathcal L_{\mathrm{gh}}
    =\begin{pmatrix}0&\Box\\-\Box&0\end{pmatrix},
 \qquad
 \mathcal G_{\mathrm{gh}}^{\mathrm R/\mathrm A}
    =\begin{pmatrix}
       0&-\mathsf E_0^{\mathrm R/\mathrm A}\\
       \mathsf E_0^{\mathrm R/\mathrm A}&0
      \end{pmatrix}.
 \label{eq:ghostGreen}
\end{equation}
Ordinary matrix multiplication verifies the two
inverse identities with the indicated fibre pairings.

For the complete order \((r,b,\phi,c,\bar c)\), use
\[
 \mathcal G^{\mathrm R/\mathrm A}
  =\operatorname{diag}\left(
      \mathcal G_{rb}^{\mathrm R/\mathrm A},
      -\mathsf E_m^{\mathrm R/\mathrm A},
      \mathcal G_{\mathrm{gh}}^{\mathrm R/\mathrm A}\right).
\]
Put
\[
 \delta_\mu=\mathsf E_\mu^{\mathrm R}
                            -\mathsf E_\mu^{\mathrm A},
 \qquad
 \Delta=\mathcal G^{\mathrm R}-\mathcal G^{\mathrm A}.
\]
Its nonzero blocks are
\begin{equation}
\begin{array}{c|c@{\qquad}c|c}
 AB&\Delta_{AB}&AB&\Delta_{AB}\\
 \hline
 rr&-4\Ptr\delta_0&\phi\phi&-\delta_m\\
 rb&K\delta_0&br&-2\Cdiv\Ptr\delta_0\\
 c\bar c&-\delta_0&\bar c c&\delta_0
\end{array}
 \label{eq:causalblocks}
\end{equation}
and \(\Delta_{bb}=0\).
The scalar sign comes from its action Hessian.
The ghost form is graded antisymmetric:
its odd--odd blocks are symmetric under simultaneous
exchange of species and spacetime arguments, because
\(\delta_0^{\mathrm T}=-\delta_0\).

\subsection{The free causal algebra}

Let \(\AA_0\) be the unital associative graded algebra
over \(\CC[[\hbar]]\) generated by compactly smeared
linear fields, modulo the free equations
\begin{equation}
 \AK r+\Cdiv^*b=0,\quad
 \Cdiv r-2b=0,\quad
 P_m\phi=0,\quad
 \Box c=\Box\bar c=0
 \label{eq:freeeq}
\end{equation}
and the causal relations
\begin{equation}
 \grbr{\Phi_A(a)}{\Phi_B(d)}
      =i\hbar\pair{a}{\Delta_{AB}d}\,\mathbf1,
 \qquad
 \grbr{F}{G}=FG-(-1)^{|F||G|}GF.
 \label{eq:CCR}
\end{equation}
No inverse of \(\hbar\) is adjoined.

The Green identities imply that the causal form
vanishes on equation tests, so it descends to their
linear quotient. Its central graded Lie algebra
has an enveloping algebra with the prescribed
central relation. This gives the associative
polynomial causal carrier. It does not require a
state or a time-ordered product.

\begin{theorem}[Free causal BRST differential]
\label{thm:freeBRST}
The derivation \(s_0\) preserves the free equation
ideal and all relations \eqref{eq:CCR}. It induces
a nilpotent graded derivation of \(\AA_0\).
\end{theorem}

\begin{proof}
First consider the equations. Their variations are
\[
 s_0(\AK r+\Cdiv^*b)=\AK Kc=0,\qquad
 s_0(\Cdiv r-2b)=\Box c.
\]
The antighost equation varies into \(\Box b\).
Taking the divergence of the first equation and
using \eqref{eq:dividentities} shows that
\(\Box b=0\) belongs to the equation ideal.
The scalar and ghost equations are otherwise inert.

For the causal relations, the only cancellation
between nonzero blocks occurs in the initially
zero \(r\)--\(\bar c\) relation. In distributional
component notation,
\[
 s_0\grbr{r}{\bar c}
   =\{Kc,\bar c\}+[r,b]
   =-i\hbar K\delta_0+i\hbar K\delta_0=0.
\]
The exchanged relation follows by graded
antisymmetry. Variations of the \(r\)--\(r\),
\(r\)--\(b\), and \(r\)--\(c\) relations involve
zero ghost cross blocks or the zero \(cc\) block.
Variations of the \(\bar c\)--\(\bar c\) and
\(b\)--\(\bar c\) relations use zero \(b\)-ghost
blocks and the zero \(bb\) block.
All scalar cross relations are inert.
These checks exhaust the generator pairs.

Thus \(s_0\) is well defined on the quotient.
Lemma~\ref{lem:s0nil} proves its nilpotence there.
\end{proof}

The nontrivial causal cancellation is the normalized
free mechanism of X-G \cite[\loc{thm:freeBRST}]{XG}.
The construction is a free graded causal statement,
not a time-ordered hierarchy.

\begin{remark}[No unproved identification of ordering presentations]
The supplied X-F-ST allows a stipulated Wick presentation.
The action-normalized causal algebra above is not identified
with that presentation by ignoring an antisymmetric-part
or scalar-sign difference. A smooth change of normal ordering
cannot change a causal commutator.

The predecessor tree identity is retained at
\(\hbar^0\), where it is an identity of normal symbols
modulo equations. Any later use at quantum order must
specify a compatible common product presentation and its
transport. No such identification is hidden in
Theorem~\ref{thm:freeBRST}.
\end{remark}

VII-I's prepared free algebra and XI-B's differently
normalized auxiliary multiplet remain comparisons,
not premises or identical carriers \cite{FreeComparisons}.

%=====================================================================
\section{Natural ghost equations without an interacting quantum differential}
\label{sec:ghost}
%=====================================================================

This section distinguishes formal gauge kinematics
from the released free causal derivation.

Write \(q=\bar q+\kappa r\), with \(\bar q\) held
fixed. Define the formal kinematical rule
\begin{equation}
 s_{\mathrm{kin}}q=\kappa\Lie_c q,\qquad
 s_{\mathrm{kin}}c^\mu=\kappa c^\nu\partial_\nu c^\mu,\qquad
 s_{\mathrm{kin}}\phi=\kappa c^\mu\partial_\mu\phi,\qquad
 s_{\mathrm{kin}}\bar c=b,\qquad s_{\mathrm{kin}}b=0.
 \label{eq:skin}
\end{equation}
Equivalently,
\(s_{\mathrm{kin}}r=\Lie_c\bar q+\kappa\Lie_c r\).
Odd coefficients are placed on the left.
Its leading flat rule is \eqref{eq:s0}.

\begin{lemma}[Kinematical nilpotence]
\label{lem:kin}
The rule \eqref{eq:skin} is nilpotent on the
off-shell graded tensor differential algebra.
This is not a claim that it acts on an interacting
quantum algebra.
\end{lemma}

\begin{proof}
For \(w^\mu=c^\nu\partial_\nu c^\mu\), expansion
with the odd Leibniz rule gives
\(\Lie_c^2=\Lie_w\). The terms with symmetric
second spacetime derivatives cancel against the
antisymmetric ghost coefficients.
Thus
\[
 s_{\mathrm{kin}}^2q
   =\kappa\Lie_{s_{\mathrm{kin}}c}q
                         -\kappa\Lie_c(s_{\mathrm{kin}}q)=0.
\]
The scalar calculation is the same.
In \(s_{\mathrm{kin}}^2c\), the two terms involving
first derivatives cancel after relabelling, and
the second-derivative term is zero by ghost
antisymmetry. The nonminimal pair is nilpotent
by definition.
\end{proof}

The natural background fermion is
\[
 \Psi_{\bar q}
   =\int J_{\bar q}\bar c_\mu
                 \left(C_{\bar q}^\mu[r]-\bar q^{\mu\nu}b_\nu\right)
                      \dd^4x.
\]
Since \(s_{\mathrm{kin}}\bar q=0\),
\[
 s_{\mathrm{kin}}\Psi_{\bar q}
  =\int J_{\bar q}\left(
       b_\mu C_{\bar q}^\mu[r]-b_\mu\bar q^{\mu\nu}b_\nu
       -\bar c_\mu C_{\bar q}^\mu[\Lie_c(\bar q+\kappa r)]
                     \right)\dd^4x.
\]
Auxiliary elimination gives the retained bosonic
background gauge fixing.

\begin{proposition}[Natural background ghost operator]
\label{prop:ghost}
Define the vector-density Euler operator
\[
 (P_{\mathrm{gh}}[\bar q]c)^\mu
       =-J_{\bar q}C_{\bar q}^\mu[\Lie_c\bar q].
\]
For the retained curvature convention,
\[
 C_{\bar q}^\mu[\Lie_c\bar q]
       =\bar\Box c^\mu+\bar R^\mu{}_\nu c^\nu.
\]
The operator is natural. Its first background
coefficient satisfies
\begin{equation}
 P_{\mathrm{gh}}^{(1)}[K\xi]
     =\Lie_\xi^{\mathrm{vd}}P_{\mathrm{gh}}^{(0)}
                    -P_{\mathrm{gh}}^{(0)}\Lie_\xi^{\mathrm v},
 \qquad P_{\mathrm{gh}}^{(0)}=-\Box,
 \label{eq:ghostcomm}
\end{equation}
where the two Lie derivatives act on vector
densities and vectors, respectively.
\end{proposition}

\begin{proof}
Trace reversal of \(\Lie_c\bar q\), with its indices
raised by \(\bar q\), is
\[
 \bar\nabla^\mu c^\nu+\bar\nabla^\nu c^\mu
                         -\bar q^{\mu\nu}\bar\nabla_\rho c^\rho.
\]
Take the divergence. The commutator of the two
covariant derivatives contributes the displayed
Ricci term. Every operation is natural under
simultaneous pullback, including the volume density.
Differentiate the finite naturality identity at
\(\eta\) to obtain \eqref{eq:ghostcomm}.
\end{proof}

This is an equation-level construction of X-G
\cite[\loc{prop:ghost}]{XG}.
For compactly supported \(\xi\), the retarded
inverse \(G_{\mathrm{gh}}^{\mathrm R}=-\ERzero\)
therefore gives
\begin{equation}
 -G_{\mathrm{gh}}^{\mathrm R}
                       P_{\mathrm{gh}}^{(1)}[K\xi]c
   =\Lie_\xi^{\mathrm v}c
       -G_{\mathrm{gh}}^{\mathrm R}
                    \Lie_\xi^{\mathrm{vd}}P_{\mathrm{gh}}^{(0)}c.
 \label{eq:ghostjet}
\end{equation}
Indeed \(\Lie_\xi c\) is compactly supported and the
retarded left inverse applies.
This identity does not construct a nonlinear
interacting differential or a quantum master operation.

%=====================================================================
\section{Stage 2A: the switched scalar has a classical gauge breaking}
\label{sec:matter}
%=====================================================================

\subsection{The reference action being varied}

For this calculation only, define the natural
reference action
\[
 S_{\mathrm{ref}}
     =S_{\mathrm{EH}}+S_{\mathrm{sc},g}
                            +s_{\mathrm{kin}}\Psi_{\bar q},
\]
where
\begin{align}
 S_{\mathrm{sc},g}
     &=\int\bigl((1-g)\mathcal L_\eta[\phi]
                              +g\mathcal L_q[q,\phi]\bigr)\dd^4x,
 \nonumber\\
 \mathcal L_q[q,\phi]
     &=\tfrac12J_q(q^{\mu\nu}\partial_\mu\phi\partial_\nu\phi
                                                   -m^2\phi^2).
 \label{eq:scalarAction}
\end{align}
It is used to evaluate a natural gauge variation.
Its Euler equation is not identified with the
post-variation source prescription \eqref{eq:mixed}.

The flat stress satisfies
\[
 T_0^{\mu\nu}[\phi]
       =\partial^\mu\phi\partial^\nu\phi
                             -\eta^{\mu\nu}\mathcal L_\eta[\phi],
 \qquad
 \partial_\mu T_0^\mu{}_\nu
                              =(P_m\phi)\partial_\nu\phi.
\]

\begin{proposition}[Full switched-scalar breaking]
\label{prop:breaking}
For compact ghost coefficient functions, modulo
integrated divergences,
\begin{align}
 s_{\mathrm{kin}}S_{\mathrm{ref}}
   &=s_{\mathrm{kin}}S_{\mathrm{sc},g}
 \nonumber\\
   &=\kappa\int\left(
      (1-g)T_0^{\mu\nu}\partial_\mu c_\nu
        +(\partial_\mu g)c^\mu(\mathcal L_\eta-\mathcal L_q)
                       \right)\dd^4x
 \nonumber\\
   &=\kappa\int c^\nu\mathfrak B_\nu\,\dd^4x,
 \label{eq:breaking}
\end{align}
where
\begin{equation}
 \mathfrak B_\nu
   =(\partial_\mu g)T_0^\mu{}_\nu
      -(1-g)(P_m\phi)\partial_\nu\phi
      +(\partial_\nu g)(\mathcal L_\eta-\mathcal L_q).
 \label{eq:B}
\end{equation}
The full off-shell breaking is not supported only
on \(\supp\dd g\).
\end{proposition}

\begin{proof}
The curved scalar density is natural:
\(s_{\mathrm{kin}}\mathcal L_q
=\kappa\partial_\mu(c^\mu\mathcal L_q)\).
For the flat density, \(\eta\) is fixed, so
\[
 s_{\mathrm{kin}}\mathcal L_\eta
   =\kappa\left(
       \partial_\mu(c^\mu\mathcal L_\eta)
                              +T_0^{\mu\nu}\partial_\mu c_\nu\right).
\]
Multiplying by \(g,1-g\) and integrating gives the
first integral in \eqref{eq:breaking}.
Integrating its stress term once more and using
the stress-divergence identity gives \eqref{eq:B}.

The Einstein--Hilbert variation is a compact
divergence. Lemma~\ref{lem:kin} gives
\(s_{\mathrm{kin}}^2\Psi_{\bar q}=0\).
Neither contributes a breaking here.

Where \(g=0\) and \(\dd g=0\), the off-shell term
\(-(P_m\phi)\partial_\nu\phi\) can still be nonzero.
\end{proof}

This is Obstacle A of X-G
\cite[\loc{prop:breaking}]{XG}.
In particular, the noncompact set \(\supp(1-g)\)
has not been replaced by a compact source.

\subsection{An evaluated leading coefficient}

\begin{theorem}[Nonzero coefficient on a free scalar]
\label{thm:matter}
For every nonzero compact switch \(g\) with a plateau,
there are a smooth free scalar and a compact odd
ghost test on the completion for which the
coefficient of \(\kappa\) in \eqref{eq:breaking}
is nonzero.
\end{theorem}

\begin{proof}
Take
\[
 \phi_m(t)=
 \begin{cases}
   \sin(mt)/m,&m>0,\\
   t,&m=0.
 \end{cases}
\]
Then \(P_m\phi_m=0\),
\(T_0^{00}[\phi_m]=1/2\), and
\(T_0^{0i}[\phi_m]=0\).

A vertical line through a plateau point has \(g=0\)
at sufficiently early times and \(g=1\) at that point.
The fundamental theorem of calculus gives an open
region with \(\partial_tg>0\).
Choose a nonnegative nonzero
\(\beta\in C_c^\infty\) there and a nonzero odd
Grassmann generator \(\theta\). Put
\(c=\theta\beta\partial_t\).

At the leading metric \(q=\eta\), the density
difference in \eqref{eq:B} is zero, and the
free scalar equation term vanishes. Therefore
\[
 [s_{\mathrm{kin}}S_{\mathrm{ref}}]_\kappa
       =\frac{\theta}{2}\int\beta\,\partial_tg\,\dd^4x\ne0.
\]
The even coefficient of \(\theta\) is strictly positive.
\end{proof}

The witness is the parent mechanism
\cite[\loc{thm:matter}]{XG}.
The compact ghost coefficient is an off-shell test
of the action identity, not a nonzero compactly
supported free ghost solution.

\begin{remark}[Supported detection needs a support certificate]
Theorem~\ref{thm:matter} is a completion-level theorem
for every such switch. To call its detecting test
a retained laboratory test, one must also verify
\[
 \{\partial_tg>0\}\cap\mathcal L\ne\varnothing
\]
and choose \(\beta\) there. This follows, for example,
when the vertical segment from \(\Sigma\) to a
plateau point lies in the retained reserve.

The host assumption alone does not place every
transition of an arbitrary compact \(g\) inside
\(\mathcal L\). No such support claim is silently
added. The completion-level breaking and the
supported version are thereby kept distinct.
\end{remark}

\begin{corollary}[A necessary classical symmetry condition fails]
\label{cor:masternecessary}
An ordinary canonical BV extension generating
\eqref{eq:skin}, with \(S_{\mathrm{ref}}\) as its
antifield-independent action and with the switch
rule unchanged, cannot satisfy the classical
master equation for all these inputs.
A correction beginning at positive formal powers
of \(\hbar\) cannot cancel this classical breaking.
\end{corollary}

\begin{proof}
The antifield-independent coefficient of that
master equation requires
\(s_{\mathrm{kin}}S_{\mathrm{ref}}=0\).
Theorem~\ref{thm:matter} violates this necessary
condition. Formal powers of \(\hbar\) have
independent coefficients.
\end{proof}

This is a necessary-condition argument, not the
construction or evaluation of a quantum BV operator.
It does not establish that a particular observable's
one-loop Ward coefficient is nonzero.

For a compact ghost coefficient supported where
\(g=1\), both terms in the first integral of
\eqref{eq:breaking} vanish. Restricted plateau
symmetry and unrestricted breaking are compatible.

%=====================================================================
\section{Stage 2B: source switching is not density switching}
\label{sec:source}
%=====================================================================

For a fixed compact background \(w\), the unswitched
operator \(Q_w\) is formally symmetric. Hence
\begin{equation}
 N_w^{\mathrm T}=Q_w\sig,\qquad
 N_w-N_w^{\mathrm T}=[\sig,Q_w].
 \label{eq:adjoint}
\end{equation}
No equality between the two sides is automatic.

\begin{proposition}[The symmetric correction changes the prescription]
\label{prop:symmetric}
The quadratic functional
\[
 V_{\mathrm{sym},w}[r]
       =\tfrac12\pair{r}{\sig Q_wr}
\]
has Hessian
\begin{equation}
 V_{\mathrm{sym},w}''=
        \tfrac12(\sig Q_w+Q_w\sig)
        =N_w+\tfrac12[Q_w,\sig].
 \label{eq:symmetric}
\end{equation}
It is not in general the source operator \(N_w\).
Switching a chosen local density before variation
likewise produces derivatives of the onset, with
the result depending on that density representative.
\end{proposition}

\begin{proof}
Vary the two copies of \(r\) and transpose the
operator in the first variation. The two
second-variation contributions are \(N_w\) and
\(N_w^{\mathrm T}\).
For a density depending on derivatives of \(r\),
the Euler integration by parts differentiates its
switch as well as its original coefficients.
\end{proof}

This is Obstacle B, independent of
\eqref{eq:B} \cite{XG,XFST}.
The difference is classical, supported in the
onset layer for the commutator in
\eqref{eq:symmetric}, and can propagate retardedly.
It is not a loop counterterm.

Off-shell failure of symmetry alone would not exclude
a symmetric operator agreeing on the free shell.
The next section supplies precisely that stronger test.

%=====================================================================
\section{Stage 3: variational reciprocity and the positive witness}
\label{sec:obstruction}
%=====================================================================

\subsection{The excluded upgrade class}

\begin{definition}[Exact source-preserving variational upgrade]
\label{def:upgrade}
An upgrade in the class considered here must satisfy
all of the following:
\begin{enumerate}[label=\textup{(U\arabic*)}]
\item the free metric operator is \(\Pgr\), with the
retarded inverse \(\DR\) and its frozen causal relations;
\item the interaction is even and has a symmetric
bosonic second variation after any algebraic
auxiliary elimination;
\item for a compact background \(w\), its reduced mixed
metric Hessian \(L_w\) has compact coefficients,
or more generally a compact two-point kernel,
so its bilinear pairing on smooth free inputs
is defined and symmetric;
\item its classical retarded mixed field response obeys
\begin{equation}
 -\DR L_wu=-\DR N_wu
 \label{eq:exactresponse}
\end{equation}
for every smooth free metric solution
\(\Pgr u=0\) and every allowed smooth background,
including compact backgrounds in the onset layer;
\item quantum corrections are formal series in
nonnegative powers of \(\hbar\), with the classical
response in (U4) unchanged.
\end{enumerate}
Equality in (U4) is equality of fields, equivalently
equality after every compact smooth linear metric
smearing. It is stronger than equality of one chosen
relational observable.
\end{definition}

The compact pairing condition makes explicit the
domain required by the reciprocity proof.
It is satisfied by local mixed interactions with
compact background coefficients. No exclusion is
claimed for a proposed object whose bilinear
pairing is undefined or whose boundary data change
the prescribed response.

\begin{lemma}[Necessary reciprocity on free inputs]
\label{lem:reciprocity}
For an upgrade in Definition~\ref{def:upgrade},
\begin{equation}
 \mathcal D_w(u,v)
   :=\int\left(u_{\mu\nu}(N_wv)^{\mu\nu}
                   -v_{\mu\nu}(N_wu)^{\mu\nu}\right)\dd^4x
       =0
 \label{eq:reciprocity}
\end{equation}
for every pair of smooth free metric solutions and
every compact background \(w\).
\end{lemma}

\begin{proof}
Apply \(\Pgr\) to \eqref{eq:exactresponse}.
The right-inverse identity yields
\(L_wu=N_wu\) for each free \(u\).
Symmetry of the bosonic second variation gives
\[
 \pair{u}{L_wv}=\pair{v}{L_wu}.
\]
Both pairings exist by (U3). Substitute the equalities
on the free inputs to obtain
\eqref{eq:reciprocity}.
\end{proof}

This is a necessary free-shell test, not just a
test on arbitrary off-shell variations
\cite[\loc{lem:reciprocity}]{XG}.

\subsection{A compact background inside the onset}

A nonconstant smooth onset with the stated endpoint
extensions has
\(\sig''\not\equiv0\) on \((0,\varepsilon)\).
Otherwise it would be affine there, and smoothness
at the endpoints would force zero slope.

Choose
\[
 0\ne\zeta\in C_c^\infty((0,\varepsilon)),\qquad
 \zeta\ge0,\qquad
 \supp\zeta\subset\{\sig''\ne0\},
 \qquad
 \psi=\zeta\sig''.
\]
Choose
\[
 \chi_0\in C_c^\infty(\RR^3),\qquad
 \chi_0\ge0,\qquad
 M_\chi=\int\chi_0\,\dd^3x>0.
\]
Set
\begin{equation}
 \xi_{\mathrm{on}}
       =\psi(t)\chi_0(\mathbf x)\partial_t,
 \qquad
 w=K\xi_{\mathrm{on}},\qquad
 u=e_{00},\qquad v=te_{00},
 \label{eq:witnessdata}
\end{equation}
where \(e_{00}\) has only covariant component \(00\)
equal to one. Both \(u,v\) solve \(\Pgr u=\Pgr v=0\).

The vector field \(\xi_{\mathrm{on}}\) parametrizes
an allowed background. It is not asserted to lie
in the plateau algebra \(\gp\).

\begin{proposition}[Evaluated reciprocity defect]
\label{prop:Dsigned}
For \eqref{eq:witnessdata},
\begin{equation}
 \boxed{\displaystyle
 \mathcal D_w(u,v)
       =\frac{5M_\chi}{8}
             \int_0^\varepsilon\sig''(t)\psi(t)\dd t
       =\frac{5M_\chi}{8}
             \int_0^\varepsilon\zeta(t)(\sig''(t))^2\dd t>0.}
 \label{eq:positive}
\end{equation}
\end{proposition}

\begin{proof}
The typed commutator \eqref{eq:commutator}, evaluated
on a free input, gives
\(Q_wu=-\Pgr\Liec_{\xi_{\mathrm{on}}}u\), and the
same formula for \(v\).

For these tensors the nonzero Lie components are
\begin{align*}
 (\Liec_{\xi_{\mathrm{on}}}u)_{00}
      &=2\psi'\chi_0,
 &(\Liec_{\xi_{\mathrm{on}}}u)_{0i}
      &=\psi\,\partial_i\chi_0,\\
 (\Liec_{\xi_{\mathrm{on}}}v)_{00}
      &=(\psi+2t\psi')\chi_0,
 &(\Liec_{\xi_{\mathrm{on}}}v)_{0i}
      &=t\psi\,\partial_i\chi_0.
\end{align*}
There are no spatial--spatial components.
For any tensor \(a\) of this form,
\[
 (\Pgr a)^{00}=-\tfrac18\Box a_{00}.
\]
Only the \(00\) output is paired with \(u,v\).
Therefore
\[
 \mathcal D_w(u,v)
  =\frac18\int\sig\left[
      \Box\bigl((\psi+2t\psi')\chi_0\bigr)
                      -t\Box(2\psi'\chi_0)\right]\dd^4x.
\]
The spatial Laplacian terms integrate to zero,
since \(\chi_0\) is compactly supported. The
remaining temporal coefficient is
\[
 (\psi+2t\psi')''-t(2\psi')''=5\psi''.
\]
Thus
\(\mathcal D_w=(5M_\chi/8)\int\sig\psi''\,\dd t\).
Two integrations by parts have no boundary terms
because \(\psi\) is compactly supported inside the
onset interval. They give \eqref{eq:positive}.
Strict positivity follows from the choices of
\(\zeta,\chi_0\).
\end{proof}

The numerical factor and witness are those of
X-G \cite[\loc{prop:Dsigned}]{XG}; their calculation
has been included to make the sign auditable.

\subsection{Supported realization and compact Cauchy inputs}

\begin{proposition}[The witness fits the retained regular reserve]
\label{prop:localwitness}
The spatial support in \eqref{eq:witnessdata} can
be chosen so that
\(\supp w\Subset\mathcal L\cap\{0<t<\varepsilon\}\),
inside the retained causal reserve.
Moreover \(u,v\) can be replaced by smooth free
solutions with compact Cauchy data without changing
\eqref{eq:positive}.
\end{proposition}

\begin{proof}
Choose an event \((t_o,\mathbf x_o)\in\mathcal O\).
By \eqref{eq:separation}, \(t_o>2\delta>2\varepsilon\).
A sufficiently small spatial ball \(B\) around
\(\mathbf x_o\) has
\[
 [0,\varepsilon]\times\overline B
        \subset I_\eta^-((t_o,\mathbf x_o))\cap M_{T_*}
        \subset D_*.
\]
Choose \(\chi_0\) supported in \(B\). The temporal
support of \(\psi\) is strictly inside
\((0,\varepsilon)\), so \(\supp w\) is a compact
subset of the regular chart.

For the second assertion, take a compact
neighbourhood of \(\supp w\) and its initial causal
shadow. On \(\Sigma\), choose smooth compact Cauchy
data equal to those of \(u\), respectively \(v\),
on a ball containing that shadow. Flat finite
propagation makes the resulting free solutions
equal to \(u,v\) near \(\supp w\).
The local operator \(Q_w\) is supported there,
so the pairing is unchanged.

For either replaced metric solution \(r\), set
\(b=\Cdiv r/2\) and take scalar and ghost fields
zero. Equation \eqref{eq:operators} shows that
this is a solution of the full free multiplet.
\end{proof}

Multiplying the \emph{solution itself} by a
spacetime cutoff would generally destroy its free
equation. The proof instead cuts off its Cauchy data.

\subsection{The obstruction theorem}

\begin{theorem}[No exact source-preserving variational upgrade]
\label{thm:obstruction}
There is no upgrade satisfying
Definition~\ref{def:upgrade} on the unrestricted
smooth background class of the frozen X-F source law.

The contradiction persists if off-shell representatives
are changed by terms that vanish on every free
configuration. It is not removed by algebraic
auxiliary elimination, ghost loops, or counterterms
beginning at positive formal powers of \(\hbar\).
\end{theorem}

\begin{proof}
Linearize the claimed reduced interaction at zero
fluctuation, zero scalar, and zero ghosts in the
compact background \(w\) of
\eqref{eq:witnessdata}. Its mixed Hessian must
satisfy Lemma~\ref{lem:reciprocity}.
Proposition~\ref{prop:Dsigned} contradicts that
necessary condition.

A change annihilating every free configuration
does not change \(N_wu,N_wv\) on the detecting pair,
so it does not change their reciprocity defect.

Eliminating a nondegenerate algebraic auxiliary
field from an action gives another action.
At quadratic order its reduced Hessian is the
symmetric Schur complement
\[
 L_{rr}-L_{rb}L_{bb}^{-1}L_{br}.
\]
It still satisfies the reciprocity condition.
The explicit witness therefore survives auxiliary
elimination.

Finally, at zero ghosts the classical bosonic
Hessian used in the contradiction is unchanged
by ghost loops or positive-quantum-order terms.
For arbitrary coefficients \(a_n\),
\[
 \mathcal D_w(u,v)+\sum_{n\ge1}\hbar^na_n
             \ne0\quad\text{in }\CC[[\hbar]],
\]
because its constant coefficient is strictly
positive.
\end{proof}

This transfers the parent obstruction with its actual
quantifiers \cite[\loc{thm:obstruction}]{XG}.
It does not exclude a new classical correction that
changes the source map, a weaker assignment of selected
observable symbols, or a separately specified
non-variational quantum prescription.

No statement about numerical cancellation at a
chosen nonzero value of \(\hbar\) is inferred from
the formal-series argument.

%=====================================================================
\section{Why the direct tree identity survives}
\label{sec:survival}
%=====================================================================

The obstruction is not another calculation of the
X-E-ST one-tree defect. That defect was cancelled
by the mixed tree on the X-F background-gauge sheet.

Let \(\xi,\zeta_p\in\gp\), and transform only the
classical coefficients:
\[
 h'=h+K\xi,\qquad
 k'=k+\Liec_\xi h+K\zeta_p+\tfrac12\Liec_\xi K\xi.
 \label{eq:backgroundlaw}
\]
The free fields and ordering data remain fixed.

\begin{proposition}[Plateau response law]
\label{prop:plateau}
For every smooth \(r\),
\begin{equation}
 k_{\mathrm{mix}}[K\xi;r]
       =\Liec_\xi r-\DR\Lied_\xi\Pgr r.
 \label{eq:metricjet}
\end{equation}
No onset commutator occurs in this particular variation.
\end{proposition}

\begin{proof}
The coefficients of \(Q_{K\xi}\) have support in
\(\supp\xi\subset U\), where \(\sig=1\).
Consequently
\[
 -\DR\sig Q_{K\xi}r
    =\DR\Pgr\Liec_\xi r-\DR\Lied_\xi\Pgr r.
\]
The tensor \(\Liec_\xi r\) has compact support.
The retarded left-inverse identity gives
\eqref{eq:metricjet}.
\end{proof}

For a compact metric test \(a\), the second term
has the equation representative
\[
 \pair{a}{\DR\Lied_\xi\Pgr r}
        =-\pair{\Pgr\Liec_\xi\DA a}{r}.
\]
The preimage is smooth and compactly supported in
\(\supp\xi\).
For the mixed observable, the corresponding
distributional preimage is obtained by applying
\(-\Liec_{\xi,y}\) to its transposed insertion--Green
kernel. Compact generator coefficients cut off
the source argument. Differentiation preserves
the inherited sum-zero wavefront bound, so this
is the graviton-slot equation image of the frozen
tree carrier \cite{XF,XFST}.

For clarity, the completed scalar identity used in
that carrier is
\begin{align}
 d_1[h']-d_1[h]
   &=P_m(X_\xi\EA f),
 \nonumber\\
 d_2[h',k']-d_2[h,k]
   &=P_m\left(X_\xi\EA d_1[h]
             +(X_{\zeta_p}+\tfrac12X_\xi^2)\EA f\right).
 \label{eq:scalaridentity}
\end{align}
One verifies it by setting
\(e=(1+L\EA)^{-1}c\), as in \eqref{eq:tests}.
The coordinate intertwiner on the plateau gives
\(c'=F^\sharp c\).
Naturality of the switched scalar Euler operator
there gives \(s'=F^*s\) for its advanced solution
\(s=\EA e\). Applying \(P_m\) and expanding proves
\eqref{eq:scalaridentity}. The generator coefficients
make the displayed preimages compact.

\begin{corollary}[The unchanged tree identity is not refuted]
\label{cor:survival}
In the frozen admitted tree carrier, at
\(\hbar^0\),
\[
 \widehat\Phi_F[h',k']-\widehat\Phi_F[h,k]
                    \equiv0\pmod{\JJ,\kappa^3}.
 \label{eq:treesurvive}
\]
This statement and Theorem~\ref{thm:obstruction}
are compatible.
\end{corollary}

\begin{proof}
For the first metric coefficient \(h+r\), a complete
second-slot transformation would contain
\(k'+\Liec_\xi r\). The affine identity in
\eqref{eq:tests} and \eqref{eq:scalaridentity}
therefore leave the scalar-completed expression
with defect
\(-\kappa^2\Aone[\Liec_\xi\rfield]\).

Equation \eqref{eq:metricjet} makes the mixed
tree vary by
\(+\kappa^2\Aone[\Liec_\xi\rfield]\), modulo the
explicit graviton-slot equation image.
The matter and self trees are background independent
and inert. These are exactly the inherited
cancelling terms, not new copies of them.

The obstruction instead chooses the background
\(K\xi_{\mathrm{on}}\) inside the onset layer and
requires agreement of the entire metric field map
on every free input. That background is not a
plateau transformation. The two claims have
different quantifiers.
\end{proof}

Survival of \eqref{eq:treesurvive} does not supply
a variational map. The obstruction does not invalidate
that polynomial identity.

\begin{center}
\small
\begin{tabularx}{\linewidth}{@{}p{.28\linewidth}p{.32\linewidth}X@{}}
\toprule
Operation & What is held fixed & Released statement\\
\midrule
Free \(s_0\) &
Classical backgrounds; independent free generators &
Nilpotent causal derivation\\
Natural \(s_{\mathrm{kin}}\) &
\(\bar q,g,\eta\) &
Kinematical nilpotence; reference-action breaking\\
Plateau background change &
Both free fields and the tree prescription &
Direct tree identity modulo equations\\
Exact variational upgrade test &
Free operator and the entire source response &
Reciprocity contradiction on free inputs\\
\bottomrule
\end{tabularx}
\end{center}

%=====================================================================
\section{Stage 4: graph inventory and the remaining quantum gate}
\label{sec:loops}
%=====================================================================

No nonlocal one-loop coefficient is constructed in
this section. The inventory states what would have
to be defined; it is not a substitute for that operation.

\subsection{Bare versus completed counting}

For a separately specified variational theory, let
\[
 V=\kappa V_1+\kappa^2V_2+\cdots,\qquad
 O=F_f+\kappa O_1+\kappa^2O_2+\cdots .
\]
Its formal retarded expansion would have second
coefficient
\begin{equation}
 O_2+R_1(V_1;O_1)+R_1(V_2;F_f)
                         +\tfrac12R_2(V_1,V_1;F_f).
 \label{eq:combinatorics}
\end{equation}
This is a counting identity, not existence of the
maps on the nonlocal insertion \cite{FR,XG}.

The bare \(O_1\) contains \(c_1\), not \(d_1\)
followed by a second generation of its scalar
completion. Starting from completed
\eqref{eq:tests} and applying the same full scalar
retarded correction again would duplicate terms.

\subsection{A restricted non-tadpole inventory}

For the following ledger only, assume:
\begin{enumerate}[label=\textup{(G\arabic*)}]
\item auxiliary fields are eliminated algebraically;
\item contractions join matching species, except for
the ghost--antighost pairing;
\item a fixed normal-ordering convention removes
self-contractions at a single vertex;
\item connected graphs are counted in the flat free
carrier, with no added explicit quantum counterterm.
\end{enumerate}
These are inventory conventions, not an interaction
construction for the source-switched map.

At order \(\kappa\), the relevant quantum leg types are
\[
 r\phi\phi,\quad
 \phi\phi\text{ with background coefficient},\quad
 rr\text{ with background coefficient},\quad
 rrr,\quad
 r\bar c c,\quad
 \bar c c\text{ with background coefficient}.
\]

\begin{proposition}[Two contraction types under the stated convention]
\label{prop:inventory}
The connected \(\kappa^2\hbar\) inventory of
\eqref{eq:combinatorics} contains two types:
\begin{enumerate}[label=\textup{(\roman*)}]
\item two lines between the \(r\phi\) part of \(O_1\)
and one \(r\phi\phi\) vertex;
\item one scalar line from \(F_f\) to a matter vertex,
and a scalar line plus a metric line between that
vertex and a second \(r\phi\phi\) vertex.
\end{enumerate}
Each has one external scalar leg and no explicit
classical-background coefficient.
There is no higher positive quantum power from
these contraction graphs at order \(\kappa^2\).
\end{proposition}

\begin{proof}
With \(n\) interaction vertices and \(E\) contraction
lines, the retarded quantum power is \(E-n\).
For \(n=1\), quantum power one requires \(E=2\).
Both lines must end on the two-leg order-one
insertion. Only a matter vertex has the required
metric and scalar legs.

For \(n=2\), the insertion is \(F_f\), with one
scalar leg, and quantum power one requires \(E=3\).
After that root line is attached, two further
lines must join the vertices. The attached
vertex must retain a metric and scalar leg.
Thus both vertices are matter vertices.
A background quadratic vertex, a gravitational
cubic, or a ghost vertex cannot match these species
and leg counts.

A single \(V_2\) attached to the one-leg root has
at most one connecting line in this convention.
An insertion without vertices has no
self-contractions. The same degree bounds give
\(E\le2\) for \(n=1\) and \(E\le3\) for \(n=2\);
\(n\ge3\) requires higher coupling order.
\end{proof}

This is the scheme-qualified inventory of X-G
\cite[\loc{prop:inventory}]{XG}.
It constructs neither nonlocal retarded operation.
Additional finite quantum normalization terms fall
outside its hypotheses.

In this convention a background-dependent metric
one-point response needs two vertices; inserting it
through an order-\(\kappa\) metric slot first reaches
the scalar observable at order \(\kappa^3\).
Changing normal ordering can add tadpole
normalizations and changes that ledger.
No one-point expectation value in an unspecified
state is meant.

\subsection{The local fish is a separate extension problem}

For specified flat Feynman kernels
\(D_0^{\mathrm F},D_m^{\mathrm F}\), the ordinary
local scalar--metric fish has component kernels
\[
 t_{\alpha\beta}(z)
    =D_0^{\mathrm F}(z)
                  \partial^\alpha\partial^\beta D_m^{\mathrm F}(z),
 \qquad z\ne0,\qquad |\alpha|+|\beta|\le2.
\]

\begin{proposition}[Local extension interface]
\label{prop:fish}
These component distributions are defined off the
origin and have scaling degree at most six in four
dimensions. They admit extensions with that bound.
Two such extensions differ by
\[
 \sum_{|\gamma|\le2}a_\gamma\partial^\gamma\delta_0,
\]
with smaller ambiguity for components with lower
scaling degree.
No extension is selected here.
\end{proposition}

\begin{proof}
Off the origin, the singular Feynman factors at a
null separation have the same covector orientation,
so their product meets the wavefront product test.
The scalar Feynman singularity has scaling degree
two; at most two derivatives occur in the displayed
component. The flat short-distance expansion gives
the upper bound six.
The finite-scaling-degree extension theorem then
gives an extension without raising that bound.
A difference is supported at one point, and
\(\sd(\partial^\gamma\delta_0)=4+|\gamma|\),
which gives the ambiguity bound \cite{BF}.
\end{proof}

This is the local statement recorded in
X-G \cite[\loc{prop:fish}]{XG}.
The ambiguity bound is for the internal distribution.
External differentiated scalar factors need not
give a quadratic local density of total derivative
order at most two.

A choice of local fish extension is not a definition
of the nonlocal relational loop.

\subsection{An exceptional scalar wavefront configuration}

A scalar model appearing in the coordinate part of a
time-ordered contraction is
\begin{equation}
 R(x-z)D_0^{\mathrm F}(z-y)D_m^{\mathrm F}(x-y).
 \label{eq:triangle}
\end{equation}
It is only a model for the product criterion.
It is not asserted to be the full tensor coefficient.

\begin{proposition}[Failure of the sufficient product test]
\label{prop:nullWF}
At a strictly ordered null-collinear triple
\(z\prec y\prec x\), the ordinary sufficient
wavefront product criterion for
\eqref{eq:triangle} fails.
This does not prove nonexistence of an extension,
a cancellation in the complete sum, or a normalized
retarded observable.
\end{proposition}

\begin{proof}
In slot order \((x,z,y)\), the three covectors can
have forms
\[
 (k,-k,0),\qquad(0,l,-l),\qquad(p,0,-p).
\]
Their sum is zero precisely when \(l=k=-p\).
At the stated null triple, the two Feynman factors
supply the opposing orientations at the earlier
and later endpoints, while the retarded kernel has
both signs of its null conormal. The displayed
cancellation is therefore allowed.
It violates a hypothesis of the usual product
theorem \cite{Hormander}; a failed sufficient
hypothesis is not a nonexistence theorem.
\end{proof}

This is the exceptional scalar audit of X-G
\cite[\loc{prop:nullWF}]{XG}.
For an actual weighted tensor graph, its weights,
derivatives, tensor contractions, and complete
retarded sum must still be evaluated. An exceptional
set away from the total diagonal is not repaired
merely by extending the local fish at coincidence.

\subsection{The conditional identity has an existence hypothesis}

\begin{theorem}[Conditional fixed-background-input addition]
\label{thm:conditional}
Suppose, on one explicitly specified common carrier
and ordering presentation:
\begin{enumerate}[label=\textup{(C\arabic*)}]
\item the inherited tree expression has its stated
equation-ideal identity;
\item an admitted one-loop coefficient
\(\field(\ell)\), with \(\ell\in C_c^\infty(\widehat M)\),
has actually been constructed;
\item \(\ell\), including its finite insertion and
normalization terms, is independent of \(h,k\);
\item no untracked ordering or background-transport
correction is present at the compared order.
\end{enumerate}
Then
\[
 \widehat\Phi_{\mathrm{cond}}
       =\widehat\Phi_F+\kappa^2\hbar\field(\ell)
                         \pmod{\kappa^3,\hbar^2}
\]
has the same fixed-free-input background identity
through that order.
\end{theorem}

\begin{proof}
The tree difference belongs to its equation ideal
by (C1). The additional term has identically zero
background difference by (C3).
Conditions (C2) and (C4) ensure that these are
defined elements of the same comparison problem.
\end{proof}

This is the conditional logic of X-G
\cite[\loc{thm:conditional}]{XG}, not an existence
proof for \(\ell\). It does not identify the
conditional expression with the Bogoliubov image
of an unchanged source-switched interaction.

General perturbative gravity and locally covariant
BV constructions require their own domains,
renormalization conditions, and comparison maps
\cite{BFR}. They are not automatic completion
theorems for \eqref{eq:mixed} and its nonlocal
insertion.

\subsection{The actual remaining gate}

An unconditional continuation must supply either:
\begin{enumerate}
\item a fully specified non-variational quantum prescription
for the unchanged equation-level source map, with
its products, transport, ideals, and normalization
proved; or
\item a newly declared variational sheet, including its
classical onset corrections, followed by the
required nonlocal retarded construction and quantum
background comparison.
\end{enumerate}
Neither option is inserted here.

Interacting quantum BRST, positivity, physical
cohomology, Kugo--Ojima analysis, whole-Lagrangian
switching, spurions, compensators, and an adiabatic
construction remain outside the release.

%=====================================================================
\section{Support, topology, and comparison boundaries}
\label{sec:topology}
%=====================================================================

For compact \(A\subset\widehat M\), let
\[
 \DD_A=\{a\in C_c^\infty(\widehat M;\CC):\supp a\subset A\},
 \qquad
 p_N(a)=\max_{|\alpha|\le N}\sup|\partial^\alpha a|.
\]
The scalar degree-one quotient is
\[
 \Qm=\DD/P_m\DD,\qquad \DD=C_c^\infty(\widehat M;\CC).
\]

\begin{lemma}[The inherited scalar topology]
\label{lem:closed}
The subspace \(P_m\DD\) is closed in \(\DD\).
Consequently \(\Qm\) is Hausdorff and the map
\(\DD_{C_f}\to\Qm\) is continuous.
\end{lemma}

\begin{proof}
The continuous map
\(\EA-\ER:\DD\to C^\infty(\widehat M)\)
has kernel \(P_m\DD\). If
\(\EA a=\ER a=u\), then \(u\) has support in the
compact causal diamond of \(\supp a\).
Hence \(u\in\DD\) and \(P_mu=a\).
The converse follows from the Green identities.
\end{proof}

This is the scalar topology retained from X-D-ST,
not its width-removal theorem \cite{STRecords}.
It does not topologize a complete cubic as a
degree-one scalar class.

The Green operators of Stage 1 are continuous
test-to-smooth maps on compact observation reserves.
Their matrix entries are finite compositions with
constant differential operators. Compactly smeared
Green and nilpotence identities therefore depend
continuously on the fixed test data \cite{BGP}.
This is not an operator-norm estimate in a physical
representation.

For the obstruction, the right-hand side of
\eqref{eq:positive} is continuous in the smooth
data on fixed compact supports. More explicitly,
if \(|\sig''|\ge a>0\) on \(\supp\zeta\), then
\[
 \mathcal D_w(u,v)
      \ge\frac{5M_\chi a^2}{8}\int\zeta\,\dd t>0.
\]
This gives a local stability certificate when the
stated positive lower bounds are supplied. It is
not a theorem uniform in shrinking widths or
arbitrary moving profiles.

\begin{proposition}[Supported transport of the new results]
\label{prop:patch}
The compactly smeared free identities and the
supported reciprocity witness are unchanged by
completion extensions agreeing near their relevant
finite causal reserves. Chart transport through
\(\iota\) preserves their pairings and fixed-support
continuity.
\end{proposition}

\begin{proof}
Finite propagation localizes the smooth free
solutions and Green responses needed by a compact
test to its finite causal shadow.
The witness calculation is local in \(w,u,v\);
Proposition~\ref{prop:localwitness} places its
support in the retained regular reserve.

Distribution transport is defined by transposition
on tests and densities. On the fixed compact
reserve, the chart and inverse have bounded
derivatives of every required finite order.
The corresponding test seminorms therefore give
equivalent local descriptions.
\end{proof}

No global conical field algebra or boundary condition
around the worldsheet follows.

\paragraph{VII-C is not applied.}
Its sharp insertion, fixed Wick matrix, normal-topology
approximating family, and mandatory
\(\hbar^3\) coefficient of
\(B_f\star_H W_g\star_H B_{f'}\)
are not the free causal differential or the classical
reciprocity pairing here \cite{VIIC}.
Its word is not formed.

\paragraph{Negative controls remain separate.}
IX-A-ST's boundary-sensitive separator is not this
multiplet. IX-D-ST's support interface is not time
ordering. X-A-conv is a different prescription, and
X-Enter's conditional readout does not repair its
classical defect. XI-G's frozen-class obstruction
is not repaired \cite{ComparisonRecords}.

\paragraph{Other loop records do not fill the gate.}
A fixed smooth multiplier does not enlarge a
wavefront set, but that fact is neither a subtraction
nor a loop regulator \cite{Hormander,XVIA}.
Em01-B's polynomial growth and Em01-C's aligned-null
scalar family do not construct the present nonlocal
one-loop observable \cite{EmRecords}.

The stated XVI-O kernel subsector, under its own
prescription, is not bulk BV restoration or the
observable missing here. Its scoped statement is
not used to close O.1 \cite{XVIO}.

\paragraph{Historical and full characteristic stops.}
This manuscript discharges neither historical H2
nor the missing full I3/J2 maps. XIV-I's narrower
internal-amplitude result is not withdrawn or
identified with those full maps
\cite{BoundaryRecords}. A free causal differential
does not define the absent characteristic quantum
coefficient.

%=====================================================================
\section{Stage 5: supplied evaluations and a complete Enter instrument}
\label{sec:instrument}
%=====================================================================

\subsection{A finite installed dictionary}

A dictionary entry specifies its smooth data,
compact support contract, and report. Examples are:
\begin{enumerate}
\item a compactly tested component of \(s_0^2\Phi=0\);
\item a pairing of
\(\mathcal L\mathcal G^{\mathrm R/\mathrm A}a-a=0\)
with a fixed compact test;
\item a supplied value or certified enclosure of
\eqref{eq:positive}.
\end{enumerate}
The zero identities and the strictly positive
obstruction coefficient make this dictionary
nonempty without a continuum acquisition claim.

An entry is a supplied mathematical evaluation.
It is not measurement of Grassmann field data,
a physical ghost detector, or an algorithm for
evaluating arbitrary smooth descriptions.
Rational reports or certified intervals are
installed; arbitrary real numbers are not assumed
to have finite exact descriptions.

Oai279-A's finite reversible lookup and Oai283-A's
finite-channel or oracle pilots are comparisons
only \cite{FiniteComparisons}. A frozen finite
lookup does not establish a uniform all-input
field-theoretic family; a constant-error certificate
does not give an all-orders completion.
No such circuit is needed as a premise here.

\subsection{Complete destructive clock}

Let \(C_{\mathrm{clk}}\cong\CC^{2^{n_C}}\), \(n_C\ge1\),
have a calibrated orthonormal basis
\(\{\ket{\tau_j}\}\).
Let \(A\) contain every surviving finite register,
and set
\[
 D_j=\bra{\tau_j}_{C_{\mathrm{clk}}}\otimes I_A,
 \qquad P_j=\proj{\tau_j}.
\]

\begin{theorem}[Complete clock instrument]
\label{thm:clock}
The map
\begin{equation}
 \mathcal M(X)
    =\sum_j\proj j_R\otimes D_jXD_j^\dagger
    =\sum_j\proj j_R\otimes
       \Tr_{C_{\mathrm{clk}}}
          [(P_j\otimes I_A)X(P_j\otimes I_A)]
 \label{eq:clock}
\end{equation}
is CPTP and satisfies
\[
 \Tr_R\mathcal M(X)=\Tr_{C_{\mathrm{clk}}}X.
\]
These identities hold on all operators and after
adjoining every untouched reference.
The measured clock names are consumed; no quantum
clock factor is returned.
\end{theorem}

\begin{proof}
The recorded Kraus maps are \(\ket j_R\otimes D_j\).
Their squared moduli sum to
\(\sum_jP_j\otimes I_A=I\).
In a clock-basis expansion
\(X=\sum_{j,l}\kb{\tau_j}{\tau_l}\otimes X_{jl}\),
both recorded \(j\)-blocks equal \(X_{jj}\).
Their sum is the partial trace.
The same calculation includes an arbitrary
reference factor. Consumption is the specified
destructive interface, whose quantum codomain is
\(A\), not the measured clock.
\end{proof}

This is the I-B/En09-A complete-channel contract
\cite{IB,En09}. Conditional reference states can
be steered by outcome selection, while the
unconditional reference marginal is unchanged.
A positive branch is normalized by its trace;
a zero branch receives no normalized state.

The clock consumes a finite encoding, not geometric
time. If distinct labels must all equal the time
of one fixed readout event on positive branches,
only one label can occur. Multi-outcome history
calibration is a different map.
The extraction identity alone also does not imply
conditional unitary evolution.

\subsection{One explicit report encoding}

For normalized reports \(v_j\in[-1,1]\), put
\[
 p_y(v)=\frac{1+yv}{2},\qquad y=\pm1,
\]
and
\[
 \mathcal I_v(X)
   =\sum_{j,y}\proj{j,y}_R\otimes p_y(v_j)D_jXD_j^\dagger.
\]

\begin{proposition}[Complete report error]
\label{prop:report}
The channel \(\mathcal I_v\) is CPTP and
\[
 \ddia(\mathcal I_v,\mathcal I_{\widetilde v})
      =\tfrac12\max_j|v_j-\widetilde v_j|,
 \qquad
 \ddia(\Lambda,\Gamma)=\tfrac12\|\Lambda-\Gamma\|_\diamond.
\]
The comparison includes all untouched references.
\end{proposition}

\begin{proof}
A complete Kraus family is
\(\sqrt{p_y(v_j)}\ket{j,y}_R\otimes D_j\).
On an arbitrary input with reference, the difference
has orthogonal blocks
\(y(v_j-\widetilde v_j)X_{jj}/2\).
Its trace norm is bounded by
\[
 \max_j|v_j-\widetilde v_j|
                 \sum_j\|X_{jj}\|_1
       \le\max_j|v_j-\widetilde v_j|\,\|X\|_1.
\]
The last step is pinching contraction.
A basis input at a maximizing \(j\) attains the
bound. Divide by two.
\end{proof}

This continuity is a property of the chosen
probabilistic encoding. Distinct deterministic
orthogonal words need not obey it.
En10-A's source-weight estimation requires a
separate access contract, not merely an installed
evaluation table \cite{En10}.

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

At each tick let the complete finite channel factor
over a prescribed partition into finite-site blocks.
Include all records, controls, resets, selectors,
and resource wires. For terminal sites \(Y\), define
\[
 N_{t,t}(Y)=Y,\qquad
 N_{s,t}(Y)=
    \bigcup\{B:B\text{ is a block at tick }s+1,\
                       B\cap N_{s+1,t}(Y)\ne\varnothing\}.
\]

\begin{theorem}[Complete scheduled locality]
\label{thm:locality}
The network obeys
\[
 (\Phi_{s+1}^*\circ\cdots\circ\Phi_t^*)(\mathcal A(Y))
                              \subseteq\mathcal A(N_{s,t}(Y)).
\]
Equal input reductions on that region give equal
corresponding output reductions.
For joint outputs with a reference \(E\), equality
is required on the joint ancestor--reference input
reduction.

A deterministic intervention outside the complete
backwards region cannot change the corresponding
joint output law. If every dependency edge has a
certified regular future-causal host route,
scheduled ancestry is contained in the prescribed
metric causal relation.
\end{theorem}

\begin{proof}
A channel adjoint is unital. A block disjoint from
the current observable support therefore acts on
an identity factor. A meeting block enlarges
support only within itself. Backwards induction
proves the inclusion. Testing all joint output
observables proves the reduction and intervention
statements, with the reference carried unchanged.
Concatenation of the separately certified routes
proves the geometric containment.
\end{proof}

The theorem is an ancestry bound, not equality with
the physical causal past \cite{IB,VIIE,En08A}.
Postselection is not a deterministic intervention.
Entangled inputs do not invalidate the theorem,
but separate marginals do not determine a joint
ancestor--reference input.

Choose a straight developed spatial segment of
length \(\ell>0\) in the inherited regular convex
tube. After readout, lookup, and a positive
processing buffer, depart at \(t_{\mathrm d}\).
For \(n\ge1\), place stations
\[
 \mathbf x_j=\mathbf x_0+
            \frac jn(\mathbf x_n-\mathbf x_0),
       \qquad j=0,\ldots,n.
\]
Each station has a \(p\)-qubit register \(A_j\).
Initialize \(A_1,\ldots,A_n\) in independent blanks
\(\omega\). At hop \(j\), SWAP \(A_{j-1}\) and
\(A_j\) across developed duration \(\ell/n\).

The tick-boundary SWAP is a declared finite
permission, not a derived microscopic gravitational
Hamiltonian. Its cross-port dependencies are null;
same-station dependencies are timelike.
Classical report wires have their own route
certificates.

\begin{theorem}[Regular reference-preserving route]
\label{thm:route}
For every input \(\rho_{A_0E}\), after hop \(j\),
\[
 \rho_{A_jE}\otimes
    \bigotimes_{\substack{0\le i\le n\\i\ne j}}\omega_{A_i}
\]
is the joint state under the register identifications.
The logical channel is the identity including
every reference. Classical reports can be
forwarded with their entire joint law.
Arrival is at
\[
 t_{\mathrm d}+\ell,\qquad
 T_{\mathrm d}+\ell/c_{\mathrm{host}},
\]
within the chosen regular time reserve.
\end{theorem}

\begin{proof}
A SWAP moves the complete payload factor, including
its reference correlations, into the next blank
and leaves the old factor blank. Induction gives
the state formula. Forwarding diagonal classical
records preserves their joint correlations.
The station spacing and tick duration give the
certified null cross-port routes inside the
regular tube.
\end{proof}

This preserves correlations already present; it
does not establish gravity-generated entanglement
\cite{IB,En08B}.

A conservative preallocated reserve is
\[
 n_C+(n+1)p+(n+1)b_{\mathrm{rec}}
                         +w_{\mathrm{lookup}}+1,
\]
including clock, station payloads, report reserves,
workspace, and an optional binary-report probe.
Descriptions, tables, references, supplier storage,
resets, and returned factors are separately charged.
The route uses \(n\) register SWAPs and \(n\)
classical transmissions; all basis changes,
readout, lookup, and buffers have declared
durations.

For compatible complete implementations,
\[
 \ddia(\widetilde\Gamma,\Gamma)
   \le\min\left\{1,\
      \epsilon_{\mathrm{enc}}+\epsilon_{\mathrm{dec}}
      +\epsilon_{\mathrm{table}}+\epsilon_{\mathrm{clock}}
                                      +\sum_j\epsilon_j\right\}.
\]
Replace one channel factor at a time and use
diamond-norm contraction and the triangle inequality
\cite{Watrous}. Arrival or failure probabilities
are controlled only when their records belong to
the compared complete output.

No field detector, gate-synthesis rate, or exact
causality of a faulty implementation follows from
this error estimate.

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

\begin{theorem}[X-G-ST local Grade S release]
\label{thm:release}
On the retained supported host, with fixed positive
width:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the independent free auxiliary--ghost multiplet,
normalized fermion, and causal Green matrix are
constructed on \(\widehat M\);
\item \(s_0\) is off-shell nilpotent and descends to a
nilpotent derivation of the free causal algebra;
\item the natural switched-scalar breaking is evaluated,
with its full off-shell support and a nonzero
completion-level leading coefficient;
\item the source-switching prescription is distinguished
from density switching;
\item the exact variational upgrade class of
Definition~\ref{def:upgrade} is obstructed by the
supported positive free-solution witness
\eqref{eq:positive};
\item the inherited direct tree identity survives on
the unchanged plateau algebra, without being
promoted to a variational or interacting-BRST
identity;
\item the quantum inventory has the explicitly
restricted and conditional status stated in
Section~\ref{sec:loops};
\item the finite supplied-evaluation interface has a
complete destructive clock, scheduled locality,
regular reference-preserving transport, and its
resource/error ledger.
\end{enumerate}
For the arbitrary-switch matter witness, laboratory
detection additionally requires the support
certificate stated after
Theorem~\ref{thm:matter}.
This is local Grade S only.
\end{theorem}

\begin{proof}
Lemmas~\ref{lem:s0nil} and
Proposition~\ref{prop:gfnorm}, together with
Proposition~\ref{prop:Ghb} and
\eqref{eq:ghostGreen}, prove (i).
Theorem~\ref{thm:freeBRST} proves (ii).
Proposition~\ref{prop:breaking} and
Theorem~\ref{thm:matter} prove (iii).
Proposition~\ref{prop:symmetric} proves (iv).
Lemma~\ref{lem:reciprocity},
Propositions~\ref{prop:Dsigned} and
\ref{prop:localwitness}, and
Theorem~\ref{thm:obstruction} prove (v).
Proposition~\ref{prop:plateau} and
Corollary~\ref{cor:survival} give (vi).
The stated hypotheses of
Propositions~\ref{prop:inventory},
\ref{prop:fish}, \ref{prop:nullWF}, and
Theorem~\ref{thm:conditional} give exactly (vii).
Sections~\ref{sec:instrument}--\ref{sec:transport}
prove (viii). The support qualification was
stated explicitly and is not removed by this
summary.
\end{proof}

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.34\linewidth}p{.20\linewidth}X@{}}
\toprule
Object & Status & Boundary\\
\midrule
Free auxiliary--ghost multiplet &
Constructed &
Completion, not a global conical inverse\\
Off-shell \(s_0^2=0\) &
Proved &
Independent auxiliary field retained\\
Free causal compatibility &
Proved &
Not a time-ordered or interacting differential\\
Switched-scalar breaking &
Evaluated &
Not a quantum master operation\\
Exact source-preserving variational upgrade &
Obstructed &
All-input class of Definition~\ref{def:upgrade}\\
Direct X-F tree identity &
Survives &
Different quantifiers; not a repair\\
Local fish extensions &
Local interface &
No extension selected for the nonlocal insertion\\
Nonlocal one-loop observable &
Unresolved &
Neither admissible prescription option supplied\\
Interacting quantum BRST &
Not constructed &
No master equation or anomaly classification\\
Physical state and positivity &
Not opened &
No physical cohomology or Kugo--Ojima analysis\\
Sharp limit as metric admission &
Not inherited &
Fixed width throughout\\
Spacetime emergence &
Not constructed &
Host and action are supplied inputs\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Significance and novelty boundary}

The parent free construction and source-preserving
upgrade obstruction are Cheung's X-G results
\cite{XG}. The supported transfer does not present
them as a new universal no-go theorem.

Its useful contribution is an explicit compatibility
audit:
\[
 \text{free nilpotence},\qquad
 \text{causal compatibility},\qquad
 \text{variational reciprocity},\qquad
 \text{nonlocal quantum admission}
\]
are different obligations.
The first two are established here; the stated
source-preserving variational upgrade fails its
classical test; the nonlocal quantum construction
is not thereby decided.
The positive witness remains meaningful even for
compact-Cauchy free inputs in the regular supported
presentation.

\subsection{Successor package}

A successor receives:
\begin{enumerate}
\item the unchanged host, chart, completion, and signature ledger;
\item the normalized free multiplet and Green matrix;
\item the free nilpotence and causal-compatibility theorem;
\item the natural ghost equation and its typed retarded identity;
\item the two separated classical obstacles;
\item the reciprocity criterion, explicit positive witness,
and exact-upgrade obstruction;
\item the qualification preserving the direct tree identity;
\item the local and conditional quantum ledger;
\item the finite supplied-evaluation instrument contract.
\end{enumerate}

It receives no interacting quantum BRST differential,
nonlocal one-loop observable, physical-state selection,
variational switched-action identification, metric
admission of the X-D-ST limit, \(\mathsf T_2\) family,
or VII-C word.

\subsection{Standing non-claims}

In the inherited documentary problem ledger,
Problems 1, 2, and 8 stay unfinished, Problem 12
stays unresolved, and Problems 3--7 and 9--11 are
not reached by this release. These labels are not
new mathematical problems solved here.

O.1 stays \loc{NOT_CLOSED} in the inherited scope.
XV-B is not opened as the downstream quantum-admission
exit. XVI-N is not discharged.
Historical H2 and the missing full I3/J2 maps are
not discharged; XIV-I's narrower amplitude is not
withdrawn \cite{BoundaryRecords,XVIO}.

No quantum coefficient of the inherited
characteristic graph is admitted. This does not
deny the free carrier or the three tree coefficients
already admitted on their stated earlier sheets.

The formal metric jet does not evolve the host.
This is not spacetime emergence, Einstein dynamics
derived from records, a Page curve, or holographic
identification.

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

\begin{longtable}{@{}p{.36\linewidth}p{.59\linewidth}@{}}
\toprule
Potential error & Correction\\
\midrule
\endfirsthead
\toprule
Potential error & Correction\\
\midrule
\endhead

Using Cauchy theory on the regular cone &
All global free equations are on the declared
Minkowski completion.\\[4pt]

Replacing the worldsheet by a worldline &
The inherited product defect and laboratory
separation are unchanged.\\[4pt]

Importing XI-B's factors &
The fermion \(\int\bar c(C-b)\) gives
\(-\bar c\Box c\) and the X-E/X-F operator
\(-\Ptr\Box/4\).\\[4pt]

Eliminating \(b\) before claiming off-shell nilpotence &
The independent field is retained for that theorem.
The eliminated antighost rule squares to
\(\Box c/2\).\\[4pt]

Making the antighost compact but its variation unrestricted &
All configurations are ambient smooth; only tests
and variations are compact.\\[4pt]

Confusing the scalar equation inverse and action inverse &
They are \(\mathsf E_m^{\mathrm R/\mathrm A}\)
and \(-\mathsf E_m^{\mathrm R/\mathrm A}\),
respectively.\\[4pt]

Treating causal compatibility as time ordering &
The theorem concerns generator equations and graded
causal brackets only.\\[4pt]

Reusing an unmatched Wick presentation &
No smooth ordering change is claimed to repair a
different antisymmetric causal part.\\[4pt]

Confining all switched breaking to \(\supp\dd g\) &
The full off-shell coefficient includes
\(-(1-g)(P_m\phi)\partial\phi\).\\[4pt]

Treating a compact ghost test as a free ghost solution &
The matter witness tests an off-shell action identity.\\[4pt]

Locating every switch transition in the laboratory &
Supported matter detection requires its explicit
transition-support certificate.\\[4pt]

Using a ramp vector as a plateau transformation &
\(\xi_{\mathrm{on}}\) defines the obstructing
background; it is not in \(\gp\).\\[4pt]

Using only an off-shell adjoint mismatch &
The positive reciprocity pairing is evaluated on
genuine free metric inputs.\\[4pt]

Cutting off the free solutions themselves &
Compact Cauchy data are used to retain the free
equations near the detecting support.\\[4pt]

Declaring a variational map by renaming a source response &
The upgrade has an explicit all-input equality
requirement and fails reciprocity.\\[4pt]

Repairing a classical coefficient by positive powers
of \(\hbar\) &
Formal coefficients are compared order by order.\\[4pt]

Re-cancelling the X-E-ST truncation defect &
The existing mixed tree supplies that cancellation
once; X-G-ST retains it.\\[4pt]

Calling a scalar product-criterion failure nonexistence &
Only failure of a sufficient ordinary criterion
is proved.\\[4pt]

Promoting a local fish extension to the nonlocal loop &
Their extension loci and required operations differ.\\[4pt]

Using the two-graph inventory as a normalization theorem &
Its ordering convention excludes tadpoles and explicit
quantum counterterms; it constructs no missing map.\\[4pt]

Equating a supplied table with field acquisition &
The evaluations and enclosures are installed inputs.\\[4pt]

Comparing only reference marginals &
Joint output claims require the joint
ancestor--reference input reduction.\\[4pt]

Calling reference transport gravity-generated entanglement &
The route preserves correlations already present.\\[4pt]

Reading a source identifier as registration evidence &
This is a proposed manuscript; no installation or
test execution is asserted.\\

\bottomrule
\end{longtable}

%=====================================================================
\section{Documentary catalogue and finite-interface comparisons}
\label{app:comparisons}
%=====================================================================

The supplied OpenAI catalogue descriptions are retained
only as documentary comparisons \cite{OpenAICatalogue}.
Their proofs are not verified or used as premises.

Families 282, 362, and 376 are not, respectively,
this background identity, this auxiliary--ghost
multiplet, or a loop regulator.
The mentions of families 265 and 271 supply no
Page-curve theorem for the present model.

Oai272-A, Oai276-A, Oai119-A, Oai212-A, and Oai365-A
concern separately specified finite or operational
interfaces \cite{FiniteComparisons}.
They do not construct a continuum interaction,
a quantum characteristic coefficient, or the
missing nonlocal retarded product here.

A finite circuit can encode a supplied evaluation
or status certificate. It cannot replace an
all-input analytic construction merely by being
exact on a frozen finite table. Conversely, the
existence of a restricted uniform circuit pilot
does not make it a uniform construction of the
field-theoretic objects in this manuscript.

%=====================================================================
\section{Proposed integration contract}
\label{app:integration}
%=====================================================================

This appendix is a proposed repository contract,
not a record of executed changes.

A maintainer may install
\loc{content/manuscripts/X-G-ST.tex}, add the actual
source import and versioned metadata, and add
separation edges to X-G, the supplied X-F-ST,
X-E-ST, I-B, and the comparison boundaries.
Actual hashes, schemas, and tests must come from
the repository, not from this proposal.

\paragraph{Proposed routing note.}
X-G-ST is a local Grade S supported transfer of the
normalized free auxiliary--ghost causal multiplet
and an exact source-preserving variational upgrade
obstruction. Its free BRST derivation is not an
interacting quantum differential. Its positive
classical reciprocity witness is not a loop anomaly.
The existing three-tree plateau identity survives,
but no nonlocal one-loop observable or spacetime
emergence construction is supplied.

\begin{verbatim}
identifier: X-G-ST
sourceStatus: proposed
grade: local Grade S
primarySourceRecord: X-G@1
host: unchanged I-B / X-A-ST instance
widthStatus: fixed-positive-width
constructed:
  - normalized independent auxiliary--ghost multiplet
  - retarded and advanced free Green matrix
  - off-shell free nilpotence
  - compatibility with free graded causal relations
  - natural switched-scalar breaking and leading witness
  - supported reciprocity witness
provedObstructed:
  - exact source-preserving variational upgrade
retained:
  - direct X-F tree plateau identity at tree order
unresolved:
  - nonlocal one-loop observable
notConstructed:
  - interacting quantum BRST
  - physical-state selection or positivity
  - time-ordered hierarchy
  - sharp metric admission
  - spacetime emergence
registryEditsExecuted: false
repositoryTestsExecuted: false
compilationAsserted: false
\end{verbatim}

Registration, if later performed, would change a
documentary source status. It would not change
the analytical verdicts above.

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

\paragraph{Authorship and attribution.}
The author name follows the supplied predecessor
metadata. The principal free construction and exact
upgrade obstruction are credited to N.~Cheung's X-G.
The predecessor action, scalar completion, and three
trees retain their separate attribution.

\paragraph{Source status.}
The supplied ST files are source records, not
registration certificates. The parent X-G is cited
by its versioned rendered record and internal labels.
A source quotation or restatement is not independent
proof verification. No internal publication date,
DOI, or registration is invented.

\paragraph{Analytical scope.}
The proof constructs a free causal graded algebra
and proves a classical obstruction within a named
upgrade class. It does not equate free nilpotence
with an interacting quantum BRST theorem.
The compact pairing hypothesis of the obstruction
is explicit.

\paragraph{Operational scope.}
The instrument encodes supplied finite evaluations.
It is not a continuum field detector or an acquisition
theorem. Clock consumption does not consume geometric
time, and the regular route does not derive
microscopic gravity.

\paragraph{Verification.}
No successful \loc{pdflatex} execution,
proof-assistant verification, symbolic-program
execution, numerical evaluation, hardware synthesis,
repository integration, independent execution,
experiment, independent referee review, or journal
acceptance is asserted.
The mathematical release consists of the definitions
and written proofs in this source.

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

\bibitem{XG}
N.~Cheung,
\emph{Landmark X-G: The normalized BRST multiplet and
the one-loop background gate: proved compatibility
conditions, an exact-upgrade obstruction, and the
remaining nonlocal extension problem}.
\loc{X-G.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-G}.
Especially \loc{lem:s0nil}, \loc{prop:gfnorm},
\loc{prop:Ghb}, \loc{thm:freeBRST},
\loc{prop:ghost}, \loc{prop:breaking},
\loc{thm:matter}, \loc{def:upgrade},
\loc{lem:reciprocity}, \loc{prop:Dsigned},
\loc{thm:obstruction}, \loc{prop:inventory},
\loc{prop:fish}, \loc{prop:nullWF}, and
\loc{thm:conditional}.

\bibitem{XF}
N.~Cheung,
\emph{Landmark X-F: The second-order quantum metric
as the retarded M{\o}ller response, and the closure
of the \(\kappa^2\) background identity}.
\loc{X-F.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-F}.
The three source coefficients, typed commutator,
sum-zero admission, and direct tree identity.
Its variational interpretation is separately
qualified by X-G.

\bibitem{XE}
N.~Cheung,
\emph{Landmark X-E: A specified gravitational action
for the ramp relational scalar, and the first
graviton tree}.
\loc{X-E.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-E}.
Einstein--Hilbert normalization, free graviton
operator, matter tree, and one-tree truncation defect.

\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}.
Completed scalar insertion and its counting.

\bibitem{XFST}
N.~Cheung,
\emph{X-F-ST: Three fixed-width retarded trees and
the background identity}.
Supplied \loc{X-F-ST.tex}.
Supported predecessor source, including the
equation-level response interpretation, typed
commutator, compact graviton-slot representative,
and variational audit.
No registration is inferred.

\bibitem{XEST}
N.~Cheung,
\emph{X-E-ST: Einstein--Hilbert normalization and
one matter-induced tree}.
Supplied \loc{X-E-ST.tex}.
Action normalization, supported matter tree,
corrected scalar support, and truncation audit.

\bibitem{STRecords}
N.~Cheung,
\emph{X-A-ST: Supported classical covariance of a
fixed-ramp relational scalar};
\emph{X-C-ST: Second-order scalar completion on the
inherited X-A-ST/X-B-ST host};
and \emph{X-D-ST: The sharp-width limit of the
completed scalar tests}.
Version-specific predecessor source records.
The X-B-ST adaptation is a separate predecessor record.
Host instance, completed representatives, and scalar
compact-test topology are retained without importing
a sharp metric-admission theorem.

\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 host, worldsheet, clock calibration,
complete locality, and regular transport.

\bibitem{EnterI}
N.~Cheung,
\emph{Manuscript I: Finite operational characterization
of destructive measurement in the Enter calculus}.
\loc{I.tex}, version 2.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/I}.
Characterization within its stated structural class;
contextual attribution only.

\bibitem{EnterPhilosophy}
N.~Cheung,
\emph{The Philosophy of Enter: Destructive Readout,
Quantum Recovery, and the Refusal of Identification}.
Introductory essay, Enter Calculus Companion.
\url{https://enter.grok.me/enter}.
Interpretive background, not a gravitational
or BRST premise.

\bibitem{En09}
N.~Cheung,
\emph{En09-A: Realization of Relational Time by
Destructive Enter on a Clock Register}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/En09-A}.
Complete reference-stable instrument, consumption,
and separate dynamical hypotheses.

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

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

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

\bibitem{VIIE}
N.~Cheung,
\emph{VII-E}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-E}.
Complete scheduled-support and observation interfaces.

\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}.
Cauchy theory, finite propagation, and Green operators;
applied here 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 and retarded-product frameworks.
No missing nonlocal interacting map is imported.

\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}.
Finite-scaling-degree extension framework, used only
for the stated local fish interface.

\bibitem{BFR}
R.~Brunetti, K.~Fredenhagen, and K.~Rejzner,
``Quantum gravity from the point of view of locally
covariant quantum field theory,''
\emph{Communications in Mathematical Physics}
\textbf{345} (2016), 741--779.
\url{https://arxiv.org/abs/1306.1058}.
External perturbative-gravity and BV framework;
not an automatic completion theorem for the
source-switched nonlocal observable.

\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.
Distributional kernels and wavefront product,
pullback, and pushforward calculus.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Finite instruments, completely bounded trace norm,
and channel comparison.

\bibitem{FreeComparisons}
N.~Cheung,
\emph{VII-I}, version 1, and
\emph{XI-B}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-I};
\url{https://enter.grok.me/browse/XI-B}.
Different prepared free-algebra and auxiliary--ghost
constructions; comparison only.

\bibitem{VIIC}
N.~Cheung,
\emph{VII-C: The mixed-Wick obstruction for the
sharp Cauchy sector}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/VII-C}.
Its exact sharp insertion, Wick matrix, topology,
and mandatory-word hypotheses remain distinct.

\bibitem{XVIA}
N.~Cheung,
\emph{XVI-A}.
Version 1, Enter Calculus Companion.
\url{https://enter.grok.me/browse/XVI-A}.
Conditional retarded-slot wavefront interface
and formal noncancellation audit; no subtraction
or missing coefficient is supplied.

\bibitem{XVIO}
N.~Cheung,
\emph{XVI-O}.
Revision o70-sync, version-specific supplied record,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/XVI-O}.
Kernel-subsector result under its stated prescription,
not bulk BV restoration or closure of O.1.

\bibitem{ComparisonRecords}
N.~Cheung,
\emph{IX-A}, version 1;
\emph{IX-D}, version 3;
\emph{XI-A}, \emph{XI-B}, \emph{XI-G},
\emph{X-A-conv}, and \emph{X-Enter}, version 1;
with the version-specific \emph{IX-A-ST} and
\emph{IX-D-ST} adaptation records.
Enter Calculus Companion.
\url{https://enter.grok.me/browse}.
Documentary domain, normalization, and product
separation controls; not construction premises.

\bibitem{EmRecords}
N.~Cheung,
\emph{Em01-B} and \emph{Em01-C}, version 3.
Version-specific supplied records,
Enter Calculus Companion.
\url{https://enter.grok.me/browse}.
Separate smoothing-growth and aligned-null
diagnostic 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 records,
Enter Calculus Companion.
\url{https://enter.grok.me/browse}.
Historical H2, the narrower internal amplitude,
missing full I3/J2 maps, and downstream
admission boundaries remain distinct.

\bibitem{FiniteComparisons}
N.~Cheung,
\emph{Oai279-A}, \emph{Oai283-A},
\emph{Oai272-A}, \emph{Oai276-A},
\emph{Oai119-A}, \emph{Oai212-A},
and \emph{Oai365-A}.
Version 1, version-specific supplied records,
Enter Calculus Companion.
\url{https://enter.grok.me/browse}.
Finite circuit, channel, information, and operational
comparison interfaces; no proof transfer into the
continuum multiplet or nonlocal loop.

\bibitem{OpenAICatalogue}
OpenAI,
\emph{Mathematics catalogue}.
Supplied documentary descriptions of families
282, 362, 376, 265, and 271.
Separate catalogue attribution only.
No catalogue proof is independently verified
or used as a premise.

\end{thebibliography}

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