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\hypersetup{
  pdftitle={X-H-ST: Relative classical BV structure and complete instruments for supplied residual reports},
  pdfauthor={Ngo Cheung},
  pdfsubject={Fixed-width plateau complex, free-shell non-descent,
  variational reciprocity obstruction, and destructive Enter instruments}
}

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\title{\textbf{X-H-ST: Relative classical BV structure\\
and complete instruments for supplied residual reports}\\[5pt]
\large A fixed-width plateau construction, a free-shell
non-descent test,\\
and the unchanged variational upgrade obstruction}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We construct a relative classical Batalin--Vilkovisky
structure and finite destructive-readout interfaces on the
supported host retained by I-B and X-A-ST. All analytical
fields and Green operators remain on the declared
Minkowski completion, at fixed positive regulator width.

On the plateau, the X-G-ST field multiplet admits an explicit
nonlinear differential
\(s=s_0+\kappa s_1+\kappa^2s_2\) that is nilpotent off shell.
A parity-checked antifield lift and the natural
Einstein--scalar reference action give a classical master
action through order \(\kappa^2\), as a local variational
identity modulo divergences. The auxiliary field remains
independent. The construction retains background tadpoles
rather than deleting terms needed by the action identity.

Two corrections determine the scope. First, nonlinear
nilpotence does not imply descent to the unchanged free
equation quotient. A compactly tested free-solution
witness shows that the natural first-order scalar
transformation does not preserve that quotient.
Consequently the relative classical complex is not an
interacting differential on X-G-ST's free causal algebra.
Second, the positive reciprocity pairing of X-G is a
ghost-number-zero obstruction to an exact source-preserving
variational upgrade, not the ghost-number-one classical
master-equation residual. Both distinctions are proved
rather than imposed terminologically.

We define finite dictionaries of supplied coefficient
evaluations and construct separate complete destructive
instruments for classical residual reports and for the
reciprocity witness. Their output includes the classical
code and the unmeasured finite residual, with arbitrary
untouched references retained. The instruments report
installed evaluations; they neither acquire continuum
fields nor turn an odd BV residual into a physical field
observable.

A first-order quantum correction is formulated only as a
conditional consistency problem. Local fish ambiguities
can be encoded once specified, but the missing nonlocal
contraction and an interacting quantum BRST theorem are
not supplied. The contribution is a controlled separation
of local classical closure, free-carrier descent,
variational integrability, and complete report
implementation. This provides explicit tests against
false upgrades while preserving the earlier tree identity.
The release is local Grade S, not spacetime emergence.
\end{abstract}

\statementbox{Necessary corrections to the proposed plan}{%
A plateau classical differential can be constructed on
off-shell local fields. It is not thereby a differential
on the unchanged free causal algebra.

The classical master residual and the X-G reciprocity
pairing have different parity, ghost number, and arguments.
They are not identified.

A complete instrument for a supplied number realizes an
encoded report observable. Without a field-to-apparatus
coupling, it is not a measurement of an unknown continuum
field or a physical BRST cohomology class.

The quantum continuation remains conditional. A classical
master action and a finite report channel do not define a
missing nonlocal quantum product.}

\tableofcontents

%=====================================================================
\section{Purpose, provenance, and the Enter contract}
\label{sec:purpose}
%=====================================================================

\subsection{Exact objective and corrected ending}

The supplied X-G-ST constructs a normalized free
auxiliary--ghost multiplet and proves an exact
source-preserving variational upgrade obstruction
\cite{XGST}. Its parent is Cheung's Landmark X-G
\cite{XG}. The present manuscript continues the question
of a restricted classical interacting structure without
assuming that restriction automatically produces an
interacting differential on the free causal carrier.

The distinction is substantive. An off-shell vector field
can square to zero while failing to preserve the equations
used to define a quotient. Section~\ref{sec:descent}
gives an explicit example for the proposed nonlinear
scalar transformation. Accordingly:
\begin{enumerate}
\item a relative local classical BV complex is constructed;
\item its identification with the unchanged free equation
quotient is not made;
\item the natural proposed descent to that quotient fails
already at first order;
\item no interacting quantum differential is released.
\end{enumerate}

X-G supplies the principal normalization and reciprocity
mechanism. X-F and the supplied X-F-ST supply the separate
equation-level tree construction and its background
identity \cite{XF,XFST}. They are not reinterpreted here
as variational M{\o}ller maps.

X-H-ST denotes this proposed manuscript. No parent
X-H theorem, repository installation, registration,
submission, acceptance, or successful compilation is
assumed or asserted.

\subsection{Enter as a binding interface discipline}

Cheung's philosophy distinguishes destructive readout,
quantum return, complete implementation, and geometric
identification \cite{EnterPhilosophy,EnterI}.
We use that distinction in four precise ways:
\begin{enumerate}
\item a defined differential or residual is not acquisition
of unknown field data;
\item a complete instrument is not a formal cohomology class;
\item reference-preserving transport is not entanglement
generated by gravity;
\item the prescribed conical host is not derived from a
BRST or BV complex.
\end{enumerate}

A measured clock encoding is absent from the returned
quantum interface. The complete output retains its
classical code and the unmeasured finite system.
This is an access statement, not physical annihilation
of all implementing degrees of freedom
\cite{IB,En09}.

We call the finite object constructed below an
\emph{encoded residual-report observable}.
We do not call the master residual itself a physical
field observable merely because its supplied numerical
coefficient has been placed in a table.

No Enter characterization theorem is reconstructed from
an abbreviated list of axioms. The finite instruments
use specified Born semantics, and their channel
identities are proved directly.

\subsection{Mathematical conventions}

Both \(\kappa\) and \(\hbar\) are formal parameters.
Congruence modulo \(\kappa^3\) means equality of
coefficients through order two, not a numerical
remainder estimate.

Odd fields have coefficients in an exterior algebra.
Positivity and thresholds will be applied only to
specified real coefficient extractions, never to an
uninterpreted Grassmann-valued residual.

Local action identities are taken modulo integrated
divergences. Arbitrarily multiplying a divergence
representative by an observation function can change
its integral; that operation is not silently used to
evaluate a variational class.

%=====================================================================
\section{The retained host and the free analytical interface}
\label{sec:host}
%=====================================================================

\begin{assumption}[Unchanged supported presentation]
\label{ass:host}
Retain the instance already declared in X-A-ST and used
by X-E-ST, X-F-ST, and X-G-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 prescribed future relation;
\item an injective supported developing chart
\[
 \iota:\mathcal L\longrightarrow\mathcal M^\circ
\]
with the retained compact reserve disjoint from
\(\mathcal W\);
\item the independent 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
\[
 \iota^*G=-\eta,\qquad
 t=c_{\mathrm{host}}(T-T_\Sigma);
\]
\item a nonempty observation region
\[
 \mathcal O\Subset(0,T_*)\times\RR^3,\qquad
 J_\eta^-(\overline{\mathcal O})\cap M_{T_*}
                                                   \Subset\mathcal L.
\]
\end{enumerate}
Every global Green operator and free field below belongs
to \(\widehat M\). The regular conical product is not
used as a globally hyperbolic Cauchy spacetime.
\end{assumption}

The defect is a worldsheet, not a worldline. No
host-update map, second cone, or global conical
Green inverse is introduced \cite{IB,XAST,XGST}.

Fix the compact scalar switch \(g\), an observation
test \(f\), a plateau \(U\), and the positive onset width:
\[
 \overline U\Subset
       \mathcal L\cap\operatorname{int}\{g=1\},
 \qquad
 \gp=C_c^\infty(U;T\widehat M),
\]
\[
 \sig(t)=\chi(t/\varepsilon),\qquad
 0<\varepsilon<\delta,\qquad
 \supp f\cup\supp g\cup\overline U
                           \subset\{2\delta<t<T_*\}.
\]
Here \(\chi\) is smooth, zero on \((-\infty,0]\), one
on \([1,\infty)\), and flat at the endpoints.
Thus \(\sig=g=1\) near \(\overline U\).
There is no shrinking-layer argument.

For compact supports write
\[
 K_f=\supp f,\qquad
 S_f=\supp g\cap J_\eta^-(K_f),\qquad
 C_f=K_f\cup S_f,\qquad
 D_f=J_\eta^-(K_f)\cap M_{T_*}.
\]

\subsection{Free normalization retained from X-G-ST}

Set
\[
 \Box=\partial_t^2-\Delta,\qquad P_m=\Box+m^2,
\]
and use positive retarded/advanced inverses
\(\ERzero,\EAzero\) of \(\Box\) and
\(\ER,\EA\) of \(P_m\).
They are used only on the completion \cite{BGP}.

For covariant symmetric tensors and vector fields define
\[
 (\Ptr r)_{\mu\nu}
       =r_{\mu\nu}-\tfrac12\eta_{\mu\nu}\tr_\eta r,
 \qquad
 (\Cdiv r)^\mu=\partial_\nu(\Ptr r)^{\nu\mu},
 \qquad
 (Kc)_{\mu\nu}=\partial_\mu c_\nu+\partial_\nu c_\mu.
\]
With the tensor--density pairing,
\begin{equation}
 \Cdiv^*=-\tfrac12\Ptr K,\qquad
 \Cdiv K=\Box,\qquad
 \Akin=\tfrac12G_{\mathrm{lin}},\qquad
 \Akin+\tfrac12\Cdiv^*\Cdiv
       =\Pgr=-\tfrac14\Ptr\Box.
 \label{eq:freeoperators}
\end{equation}
In particular,
\[
 \DR=-4\Ptr\ERzero,\qquad \DA=-4\Ptr\EAzero.
\]

The fields, parities, and ghost numbers are
\[
\begin{array}{c|ccccc}
 \Phi^A&r&\phi&c&\bar c&b\\ \hline
 |\Phi^A|&0&0&1&1&0\\
 \gh(\Phi^A)&0&0&1&-1&0 .
\end{array}
\]
The independent auxiliary field is retained.
The free differential and gauge fermion are
\begin{equation}
 s_0r=Kc,\quad s_0\phi=0,\quad s_0c=0,\quad
 s_0\bar c=b,\quad s_0b=0,
 \qquad
 \Psi_0=\int\bar c_\mu((\Cdiv r)^\mu-b^\mu)\dd^4x.
 \label{eq:freeBRST}
\end{equation}
Thus
\begin{equation}
 S_0^{\mathrm{free}}
  =\tfrac12\pair r{\Akin r}
       -\tfrac12\pair\phi{P_m\phi}
       +\int\left(b\cdot\Cdiv r-b^2-\bar c\cdot\Box c\right)\dd^4x.
 \label{eq:freeaction}
\end{equation}
Eliminating \(b\) gives \(b=\Cdiv r/2\) and the
Feynman weight \((\Cdiv r)^2/4\).

For reference, the retained Green blocks are
\begin{equation}
 \mathcal G_{rb}^{\mathrm R/\mathrm A}
   =
 \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},
 \quad
 \mathcal G_{c\bar c}^{\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:Green}
\end{equation}
The scalar Hessian inverse is
\(-\mathsf E_m^{\mathrm R/\mathrm A}\), not the
positive equation inverse.

These matrices invert, respectively,
\[
 \begin{pmatrix}\Akin&\Cdiv^*\\\Cdiv&-2I\end{pmatrix},
 \qquad
 \begin{pmatrix}0&\Box\\-\Box&0\end{pmatrix}.
\]
For the first block this follows by substituting
\(b=(\Cdiv r-j)/2\) and solving
\(\Pgr r=J+\Cdiv^*j/2\).
For the ghost block it follows by matrix multiplication.
This records the normalization without constructing a
new free theory \cite{XG,XGST}.

The free causal algebra \(\Afree\) is the graded
polynomial algebra of compactly smeared linear fields
with these causal relations and the equations
\begin{equation}
 \Akin r+\Cdiv^*b=0,\quad
 \Cdiv r-2b=0,\quad
 P_m\phi=0,\quad \Box c=\Box\bar c=0.
 \label{eq:freeequations}
\end{equation}
Its equation ideal is denoted \(\Jfree\).
X-G-ST's nilpotent causal derivation \(s_0\) is
retained. No interacting action on \(\Afree\) follows
from that free result.

%=====================================================================
\section{Stage A: a relative off-shell nonlinear differential}
\label{sec:stageA}
%=====================================================================

\subsection{Domain and field/background split}

All constructions in this section are local on \(U\).
Fields are unrestricted smooth local sections before
equations are imposed. Backgrounds \(h,k\) are fixed
smooth covariant symmetric tensors, not dynamical
fields and not the host metric.

Declare the formal local reference split
\begin{equation}
 \bar q=\eta+\kappa h+\kappa^2k,\qquad
 q=\bar q+\kappa r.
 \label{eq:split}
\end{equation}
The use of \(\bar q\) through second order fixes a
local reference expansion. It does not assert a
new retarded prescription for X-F.

Let \(\mathcal P(U)[[\kappa]]\) be the graded
differential algebra of local polynomial jet
expressions, allowing the formal inverse metric
coefficients determined by \eqref{eq:split}.
At each formal order the expressions have finite
jet order and finite field degree.

Define \(s\) by fixing \(\bar q,h,k\), commuting
with total spacetime derivatives, and setting
\begin{align}
 sr&=Kc+\kappa\Lie_c(h+r)+\kappa^2\Lie_c k,
 \nonumber\\
 s\phi&=\kappa c^\mu\partial_\mu\phi,\qquad
 sc^\mu=\kappa c^\nu\partial_\nu c^\mu,
 \nonumber\\
 s\bar c&=b,\qquad sb=0.
 \label{eq:nonlinear}
\end{align}
The Leibniz rule is left graded:
\(s(FG)=(sF)G+(-1)^{|F|}F(sG)\).
In particular,
\begin{equation}
 sq=\kappa\Lie_cq.
 \label{eq:sq}
\end{equation}

The coefficient operators are explicit:
\[
\begin{array}{c|ccc}
 &s_0&s_1&s_2\\ \hline
 r&Kc&\Lie_c(h+r)&\Lie_ck\\
 \phi&0&c^\mu\partial_\mu\phi&0\\
 c&0&c^\nu\partial_\nu c&0\\
 \bar c&b&0&0\\
 b&0&0&0 .
\end{array}
\]
The leading field differential is exactly
\eqref{eq:freeBRST}.

\begin{theorem}[Relative off-shell nilpotence]
\label{thm:kinematic}
The derivation \eqref{eq:nonlinear} satisfies
\(s^2=0\) on \(\mathcal P(U)[[\kappa]]\).
Consequently its truncation through second order is
nilpotent modulo \(\kappa^3\), without imposing free
or interacting equations.
\end{theorem}

\begin{proof}
For the odd vector field \(c\), put
\(w^\mu=c^\nu\partial_\nu c^\mu\).
Expansion of the tensor Lie derivative with the odd
Leibniz rule gives
\[
 \Lie_c^2=\Lie_w.
\]
Terms containing symmetric second derivatives
cancel against antisymmetric ghost coefficients.
It follows from \(sc=\kappa w\) and \eqref{eq:sq}
that
\[
 s^2q=\kappa\Lie_{sc}q-\kappa\Lie_c(sq)
        =\kappa^2(\Lie_w-\Lie_c^2)q=0.
\]
Since \(q=\bar q+\kappa r\) and \(s\bar q=0\),
this gives \(s^2r=0\) coefficientwise.

For the scalar,
\[
 s^2\phi
  =\kappa(sc^\mu)\partial_\mu\phi
     -\kappa c^\mu\partial_\mu(s\phi)=0
\]
by the same cancellation. Direct expansion of
\(s(c^\nu\partial_\nu c^\mu)\) gives zero after
substitution of \(sc=\kappa c\partial c\).
The nonminimal pair is nilpotent by definition.
The square is an even derivation, so the generator
checks prove the assertion on the algebra.
\end{proof}

This is the natural diffeomorphism kinematics already
distinguished from interacting quantum BRST in X-G-ST
\cite{XGST}. The theorem constructs a classical
local differential. It does not yet claim an
action symmetry or quotient descent.

\subsection{Support restrictions that are actually stable}

For any compact \(A\Subset U\), simultaneous compact
support of all dynamic fields in \(A\) is stable
under \eqref{eq:nonlinear}. In particular, restricting
\(\bar c\) requires the same support restriction on
\(b\), because \(s\bar c=b\).

One may instead work with arbitrary smooth germs and
compact smearing tests. That is the main local
algebraic interpretation here. Compact configurations
will be used only as an evaluation sector for finite
reports.

A space with compact antighost but unrestricted
auxiliary field is not used: \(s\) need not be tangent
to such a space.

%=====================================================================
\section{Stage B: an explicit relative classical master action}
\label{sec:stageB}
%=====================================================================

\subsection{Local variational meaning}

Let a local Lagrangian mean a horizontal four-form
modulo a horizontal divergence:
\[
 [\mathcal L]\in
 \Omega_{\mathrm H}^4(U)/\dh\Omega_{\mathrm H}^3(U).
\]
Euler differentiation annihilates divergences.
Generalized actions act on compactly supported
local functionals by their local Euler derivatives.

We use the algebra generated by compact local
functionals under ordinary multiplication.
It is a \emph{multilocal} algebra, denoted
\(\Fml(U)\); products of two single integrals are
not falsely declared to be single local integrals.
The classical variational bracket is defined on
this algebra with its usual integration-by-parts
identifications \cite{FRBV}.

No time-ordered product, Wick extension, or BV
Laplacian is required for this classical operation.

\subsection{The natural reference action}

Fix the curvature convention by
\begin{align*}
 \Gamma^\rho_{\mu\nu}[q]
 &=\tfrac12q^{\rho\lambda}
      (\partial_\mu q_{\lambda\nu}
        +\partial_\nu q_{\lambda\mu}
        -\partial_\lambda q_{\mu\nu}),\\
 R^\rho{}_{\sigma\mu\nu}
 &=\partial_\mu\Gamma^\rho_{\nu\sigma}
      -\partial_\nu\Gamma^\rho_{\mu\sigma}
      +\Gamma^\rho_{\mu\lambda}\Gamma^\lambda_{\nu\sigma}
      -\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\sigma},\\
 R_{\sigma\nu}&=R^\rho{}_{\sigma\rho\nu}.
\end{align*}
Write \(\mathcal I[q]=J_qR[q]\), \(J_q=\sqrt{-\det q}\),
and
\[
 \mathcal L_q[\phi]
   =\tfrac12J_q(q^{\mu\nu}\partial_\mu\phi\partial_\nu\phi
                                                    -m^2\phi^2).
\]

The local reference invariant action is the class
\begin{equation}
 I_{\mathrm{inv}}
  =-\frac1{2\kappa^2}\int
              \bigl(\mathcal I[q]-\mathcal I[\bar q]\bigr)\dd^4x
       +\int\mathcal L_q[\phi]\dd^4x.
 \label{eq:Iinv}
\end{equation}
These integrals denote variational classes, not an
assumption of globally finite action.

The coefficient of \(\kappa^{-1}\) is
\[
 -\tfrac12\bigl(
   \partial_\mu\partial_\nu r^{\mu\nu}
                       -\Box\tr_\eta r\bigr),
\]
a divergence. It is removed before the action is
viewed as a nonnegative formal series.
Thus no manipulation of \(\kappa^{-1}\) in a
nilpotent coupling ring is involved.

The background de Donder vector and fermion are
\begin{align}
 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),
 \nonumber\\
 \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,
 \nonumber\\
 I&=I_{\mathrm{inv}}+s\Psi_{\bar q}.
 \label{eq:Igf}
\end{align}
In particular,
\begin{equation}
 s\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\Lie_cr]\right)\dd^4x.
 \label{eq:gfexplicit}
\end{equation}
The independent \(b\) has not been eliminated.

\begin{lemma}[Relative action symmetry]
\label{lem:action}
On \(U\), \(sI=0\) as a local variational class.
\end{lemma}

\begin{proof}
The curved densities are natural under
\(sq=\kappa\Lie_cq\) and
\(s\phi=\kappa c\partial\phi\):
\[
 s\mathcal I[q]=\kappa\partial_\mu(c^\mu\mathcal I[q]),
 \qquad
 s\mathcal L_q=\kappa\partial_\mu(c^\mu\mathcal L_q).
\]
The background-only subtraction is \(s\)-inert.
The resulting variation is a divergence.
Theorem~\ref{thm:kinematic} gives
\(s(s\Psi_{\bar q})=0\).
\end{proof}

On the plateau the original switched scalar density
equals \(\mathcal L_q\), so this is its relative
local action identity. It is not a new claim of
unrestricted symmetry of the globally switched
action \cite{XG,XGST}.

\subsection{A finite coefficient specification through second order}

For \(j=0,1,2\), define
\begin{equation}
 I_j=\int\left[
  -\tfrac12[\kappa^{j+2}]
                    \bigl(\mathcal I[q]-\mathcal I[\bar q]\bigr)
  +[\kappa^j]\mathcal L_q
  +[\kappa^j](s\Psi_{\bar q})_{\mathrm{dens}}
                         \right]\dd^4x.
 \label{eq:Ij}
\end{equation}
Here \([\kappa^n]\) is Taylor coefficient extraction,
equivalently the \(n\)-th derivative at zero divided
by \(n!\). The curvature, split, and fermion have
all been specified. Thus \eqref{eq:Ij} is a finite
definition of every vertex required at this order,
not an unspecified choice of a cubic or quartic action.

In particular, \(I_2\) retains the Einstein--Hilbert
coefficient through fourth geometric order.
Omitting that coefficient would generally invalidate
the second-order master identity.

At order zero,
\begin{equation}
 I_0
     =S_0^{\mathrm{free}}+\pair r{\Akin h}
                         \quad\text{modulo divergences}.
 \label{eq:tadpole}
\end{equation}
Indeed, the quadratic Einstein--Hilbert coefficient
is \(\frac12\pair l{\Akin l}\), and the
background-only part has been subtracted.

\begin{remark}[The background tadpole is not erased]
The Hessian of \(I_0\) is the retained free Hessian,
but for general off-shell \(h\) its metric Euler
equation has the additional source \(\Akin h\).
At \(h=0\), \(I_0\) is exactly the free action
\eqref{eq:freeaction}.

Equation \eqref{eq:tadpole} is not silently replaced
by the fluctuation-independent-source subtraction
used in X-F's response prescription.
Such a replacement would change the action being
used in this section. Therefore this reference BV
action is not asserted to be an exact variational
realization of that prescription.
\end{remark}

\subsection{Antifields and a generator-level sign check}

Adjoin antifields \(\Phi_A^*\) with
\[
 |\Phi_A^*|=|\Phi^A|+1\pmod2,\qquad
 \gh(\Phi_A^*)=-1-\gh(\Phi^A).
\]
They are dual tensor densities of the appropriate
bundle type.

Use the canonical odd bracket with
\[
 (\Phi^A(x),\Phi_B^*(y))
       =\delta^A_B\delta(x-y),
 \qquad
 (F,G)=-(-1)^{(|F|+1)(|G|+1)}(G,F),
\]
extended as an odd biderivation and by integration
by parts on local functionals.
For an even Hamiltonian \(\Sbv\), its left action
is \(\Qcl F=(\Sbv,F)\).

Let \(R^A=s\Phi^A\). Define
\begin{equation}
 H_s=\sum_A(-1)^{|\Phi^A|}
                         \int\Phi_A^*R^A\,\dd^4x.
 \label{eq:Hs}
\end{equation}
In components this is
\begin{align}
 H_s=\int\Bigl\{&
 r^{*\mu\nu}\bigl(
       (Kc)_{\mu\nu}
       +\kappa(\Lie_c(h+r))_{\mu\nu}
       +\kappa^2(\Lie_ck)_{\mu\nu}\bigr)
       +\kappa\phi^*c^\mu\partial_\mu\phi
 \nonumber\\
 &-\kappa c_\mu^*c^\nu\partial_\nu c^\mu
       -\bar c^{*\mu}b_\mu\Bigr\}\dd^4x.
 \label{eq:Hsexplicit}
\end{align}

\begin{lemma}[Signed cotangent lift]
\label{lem:sign}
One has \((H_s,\Phi^A)=R^A\) for every field.
Moreover,
\[
 \tfrac12(H_s,H_s)=0
\]
as a local variational class.
\end{lemma}

\begin{proof}
Put \(\epsilon_A=|\Phi^A|\). The biderivation rule
and the canonical coordinate bracket give
\[
 (\Phi_A^*R^A,\Phi^B)
                =(-1)^{\epsilon_A}\delta_A^BR^A.
\]
The prefactor in \eqref{eq:Hs} therefore gives
\((H_s,\Phi^A)=R^A\), for even and odd species.
In particular the minus signs in the ghost and
antighost terms of \eqref{eq:Hsexplicit} are required.

The bracket square of an antifield-linear
Hamiltonian is antifield linear. Its action on
every field is the square of the induced
evolutionary vector field, by the graded Jacobi
identity. That square is zero by
Theorem~\ref{thm:kinematic}.
Consequently the bracket square is zero modulo
a divergence: a variationally trivial
antifield-linear functional has no nonzero
antifield-independent constant.
\end{proof}

\begin{theorem}[Relative classical BV construction]
\label{thm:BV}
Let
\[
 \Sbv=I+H_s,\qquad
 \Sbv^{[2]}=\Sbv_0+\kappa\Sbv_1+\kappa^2\Sbv_2
\]
be its nonnegative formal expansion through second
order. Then on \(U\),
\begin{equation}
 \frac12(\Sbv^{[2]},\Sbv^{[2]})
                         =0\pmod{\kappa^3}
 \label{eq:CME}
\end{equation}
as a local variational class.

The Hamiltonian operator
\[
 \Qcl^{[2]}F=(\Sbv^{[2]},F)
\]
is an odd differential on the classical multilocal
BV algebra modulo \(\kappa^3\).
Its restriction to fields is
\eqref{eq:nonlinear} through that order.
\end{theorem}

\begin{proof}
The antifield-independent functional \(I\)
satisfies \((I,I)=0\).
By Lemma~\ref{lem:sign}, \(H_s\) acts on
antifield-independent functionals as \(s\).
Since the bracket of two even arguments is
symmetric,
\[
 \tfrac12(\Sbv,\Sbv)
       =(H_s,I)+\tfrac12(H_s,H_s)=sI=0.
\]
The last equality is Lemma~\ref{lem:action}.
The action has no negative formal coefficients
after removal of the specified divergence.
Thus truncation preserves this equality through
order two. Explicitly,
\[
 \tfrac12(\Sbv_0,\Sbv_0)=0,\qquad
 (\Sbv_0,\Sbv_1)=0,\qquad
 (\Sbv_0,\Sbv_2)+\tfrac12(\Sbv_1,\Sbv_1)=0.
\]
The graded Jacobi identity yields
\[
 (\Qcl^{[2]})^2F
   =\left(\tfrac12(\Sbv^{[2]},\Sbv^{[2]}),F\right)
   =0\pmod{\kappa^3}.
\]
The generator action follows from
Lemma~\ref{lem:sign}.
\end{proof}

The construction uses the standard classical
cotangent-lift mechanism \cite{FRBV}. It provides
a differential, not a theorem of Koszul--Tate
acyclicity, physical cohomology, or a physical state.
For off-shell \(h\), its order-zero antifield
equations include \eqref{eq:tadpole}; they are not
identified with the unchanged free equations.

%=====================================================================
\section{The essential limitation: no automatic free-carrier descent}
\label{sec:descent}
%=====================================================================

\begin{proposition}[A necessary quotient condition]
\label{prop:quotient}
An odd derivation \(D\) of an off-shell algebra
induces a derivation on its quotient by an ideal
\(\mathscr I\) only if
\(D\mathscr I\subseteq\mathscr I\).
The equation \(D^2=0\), even off shell, does not
imply this condition.
\end{proposition}

\begin{proof}
If \(F\) and \(F+J\) represent the same quotient
element, their proposed images differ by \(DJ\).
Well-definedness requires \(DJ\in\mathscr I\).
Nilpotence tests a different composition and
imposes no such requirement.
\end{proof}

\begin{theorem}[Natural first-order scalar non-descent]
\label{thm:nondescent}
The differential \eqref{eq:nonlinear} does not
preserve the unchanged free equation ideal, even
locally on the plateau. In particular its natural
first-order action cannot induce an interacting
derivation on \(\Afree\) merely by using the
unchanged free equations.

This failure occurs at order \(\kappa\), so a
second-order correction leaving \(s_1\phi\)
unchanged cannot remove it.
\end{theorem}

\begin{proof}
Work in coordinates centred at a point \(p\in U\),
with \(\tau=t-t_p\) and \(y^1=x^1-x_p^1\).
Take a nonzero odd generator \(\theta\), a number
\(\lambda>0\), and
\[
 c=\theta\tau\partial_t,\qquad
 \phi=\cos(\omega\tau)\cos(\lambda y^1),
 \qquad \omega^2=\lambda^2+m^2.
\]
Set \(r=b=\bar c=0\).
These are smooth local free configurations:
\(\Box c=0\), \(P_m\phi=0\), and
\eqref{eq:freeequations} holds.

Since \(s\) commutes with spacetime differentiation,
\[
 s(P_m\phi)=\kappa P_m(c^\mu\partial_\mu\phi).
\]
On free inputs the right side is
\[
 \kappa\left(
       (\Box c^\mu)\partial_\mu\phi
        +2(\partial^\alpha c^\mu)
                            \partial_\alpha\partial_\mu\phi\right).
\]
At \(p\) it equals
\[
 -2\kappa\omega^2\theta\ne0.
\]
Choose a compact real scalar test supported in a
small neighbourhood of \(p\) on which the
coefficient has one nonzero sign.
The corresponding free equation functional
vanishes on all free configurations, but its
\(s\)-image does not.

Every element of the free dynamical equation
ideal has zero normal symbol on smooth free
configurations: equation factors vanish there,
and their contractions vanish by the free
bisolution identities. Hence the displayed image
cannot be an equation-ideal element.
No term beginning at order \(\kappa^2\) changes
its nonzero order-\(\kappa\) coefficient.
\end{proof}

Thus Stage A is constructed on off-shell fields,
and Stage B is constructed on a relative
classical BV algebra. The stronger target of an
interacting differential on the unchanged free
causal algebra has not been supplied.

\begin{proposition}[Compact relative ghosts do not solve this problem]
\label{prop:compactghost}
A smooth ghost coefficient compactly supported
in spacetime and satisfying \(\Box c=0\) on the
completion is zero. Consequently restricting
ghosts to \(C_c^\infty(U)\) does not produce a
nontrivial interacting action on the unrestricted
free ghost carrier.
\end{proposition}

\begin{proof}
A Cauchy surface earlier than the compact support
has zero position and velocity data. Wave
uniqueness gives the zero solution.
\end{proof}

The theorem in Section~\ref{sec:stageA} and this
support observation are compatible. The former
uses off-shell local ghosts; the latter concerns
homogeneous free solutions.
Compact gauge parameters are not nonzero compact
free ghost configurations.

A future retarded transport to an interacting
equation space would require a separate map,
a compatible interaction prescription, and its
domain theorem. The X-G obstruction prevents
assuming that the unchanged post-variation source
map already supplies that variational transport.

%=====================================================================
\section{Unrestricted breaking and variational reciprocity are different}
\label{sec:obstructions}
%=====================================================================

\subsection{The natural switched master residual}

On the completion, retain
\[
 S_{\mathrm{sc},g}
  =\int\left((1-g)\mathcal L_\eta[\phi]
                           +g\mathcal L_q[q,\phi]\right)\dd^4x.
\]
Use the same kinematical transformation and the
natural gauge fermion, but no post-variation
source switching.

For compact ghost coefficients, direct variation
gives
\begin{align}
 sS_{\mathrm{sc},g}
   &=\kappa\int c^\nu\mathfrak B_\nu\,\dd^4x,
 \nonumber\\
 \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:breaking}
\end{align}
Here
\[
 T_0^{\mu\nu}
   =\partial^\mu\phi\partial^\nu\phi
                           -\eta^{\mu\nu}\mathcal L_\eta.
\]
To verify the formula, use
\[
 s\mathcal L_q=\kappa\partial_\mu(c^\mu\mathcal L_q),
 \qquad
 s\mathcal L_\eta
  =\kappa\left(
       \partial_\mu(c^\mu\mathcal L_\eta)
                    +T_0^{\mu\nu}\partial_\mu c_\nu\right),
\]
and
\(\partial_\mu T_0^\mu{}_\nu=(P_m\phi)\partial_\nu\phi\).
All switch derivatives remain \cite{XG,XGST}.

Let \(\Sbv_{\mathrm{sw}}\) be this natural reference
action plus its signed antifield lift. The argument
of Theorem~\ref{thm:BV} now gives
\begin{equation}
 \frac12(\Sbv_{\mathrm{sw}},\Sbv_{\mathrm{sw}})
          =\kappa\int c^\nu\mathfrak B_\nu\,\dd^4x.
 \label{eq:swCME}
\end{equation}
The result has ghost number one and odd parity.

For example, with
\[
 \phi_m(t)=
 \begin{cases}\sin(mt)/m,&m>0,\\t,&m=0,\end{cases}
\]
one has \(P_m\phi_m=0\) and \(T_0^{00}=1/2\).
For a nonzero compact switch with a plateau,
choose \(\beta\ge0\), nonzero and compactly
supported where \(\partial_tg>0\), and put
\(c=\theta\beta\partial_t\).
The leading coefficient of \eqref{eq:swCME} is
\[
 \frac{\theta}{2}\int\beta\,\partial_tg\,\dd^4x\ne0.
 \label{eq:matterwitness}
\]
Such a transition exists on the completion along
a vertical line reaching the plateau.
Calling this detector a retained laboratory test
additionally requires its support to lie in
\(\mathcal L\), as qualified in X-G-ST.

The full off-shell breaking is not confined to
\(\supp\dd g\), because of the term containing
\(1-g\). On \(U\), all terms in
\eqref{eq:breaking} vanish.

\subsection{The exact source-preserving upgrade obstruction}

Let \(Q_w\) be the first background coefficient
of the natural metric Hessian including background
de Donder gauge:
\[
 Q[\eta+\kappa w]=\Pgr+\kappa Q_w+\ord(\kappa^2).
\]
Its tensor types give the identity
\begin{equation}
 Q_{K\xi}=\Lied_\xi\Pgr-\Pgr\Liec_\xi.
 \label{eq:typed}
\end{equation}
It follows by differentiating naturality under
simultaneous background and fluctuation pullback
\cite{XF,XG}.

X-F's mixed equation-level source map is
\[
 N_w=\sig Q_w,\qquad k_w[u]=-\DR N_wu.
\]
For compact \(w\), \(Q_w\) is formally symmetric,
but \(N_w^{\mathrm T}=Q_w\sig\).
A symmetric action Hessian \(L_w\) reproducing
\(k_w[u]\) on every smooth free metric input
would necessarily satisfy
\begin{equation}
 \mathcal D_w(u,v)
   :=\pair u{N_wv}-\pair v{N_wu}=0
             \qquad(\Pgr u=\Pgr v=0).
 \label{eq:reciprocity}
\end{equation}
Indeed, equality of retarded fields and application
of \(\Pgr\) imply \(L_wu=N_wu\); symmetry then
gives \eqref{eq:reciprocity}.
Compact background coefficients make these
pairings well defined.

Choose nonnegative
\(\zeta\in C_c^\infty((0,\varepsilon))\), nonzero
where \(\sig''\ne0\), and
\[
 \psi=\zeta\sig'',\qquad
 \chi_0\in C_c^\infty(\RR^3),\quad
 \chi_0\ge0,\quad M_\chi=\int\chi_0\,\dd^3x>0.
\]
Put
\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:recipdata}
\end{equation}

\begin{proposition}[Positive upgrade incompatibility]
\label{prop:recip}
The data \eqref{eq:recipdata} give
\begin{equation}
 W:=\mathcal D_w(u,v)
    =\frac{5M_\chi}{8}
           \int_0^\varepsilon\zeta(t)(\sig''(t))^2\,\dd t>0.
 \label{eq:W}
\end{equation}
The background can be supported in the retained regular
reserve, and the detecting free inputs can be replaced
by free inputs with compact Cauchy data.
\end{proposition}

\begin{proof}
On a free input, \eqref{eq:typed} gives
\(Q_wu=-\Pgr\Liec_{\xi_{\mathrm{on}}}u\).
The relevant Lie components are
\[
 (\Liec_{\xi_{\mathrm{on}}}u)_{00}=2\psi'\chi_0,\qquad
 (\Liec_{\xi_{\mathrm{on}}}v)_{00}
                              =(\psi+2t\psi')\chi_0.
\]
There are also \(0i\) components but no spatial--spatial
components. For tensors of this form,
\((\Pgr a)^{00}=-\Box a_{00}/8\).
Spatial Laplacians integrate to zero, and
\[
 (\psi+2t\psi')''-t(2\psi')''=5\psi''.
\]
Thus \(W=(5M_\chi/8)\int\sig\psi''\,\dd t\).
Two integrations by parts yield \eqref{eq:W}.

Choose an observation event
\((t_o,\mathbf x_o)\in\mathcal O\).
Since \(t_o>2\varepsilon\), a small spatial ball
\(B\) satisfies
\[
 [0,\varepsilon]\times\overline B
        \subset I_\eta^-((t_o,\mathbf x_o))\cap M_{T_*}
        \subset\mathcal L.
\]
Choose \(\chi_0\) supported in \(B\).
For compact-Cauchy replacements, choose initial
data equal to those of \(u,v\) on the initial
causal shadow of a neighbourhood of \(\supp w\),
and compactly cut off the data outside a larger
ball. Finite propagation preserves \(u,v\) near
\(\supp w\), so the local pairing is unchanged
\cite{BGP}.
\end{proof}

This is Cheung's X-G witness, with its normalization
retained \cite{XG,XGST}.
It excludes the exact source-preserving variational
upgrade class, not every possible changed classical
action. Algebraic auxiliary elimination leaves
a symmetric reduced Hessian and does not remove
the contradiction. Corrections beginning at
positive powers of \(\hbar\) cannot cancel a
nonzero classical formal coefficient.

\subsection{The correction to Stage E}

\begin{proposition}[The two residuals cannot be identified]
\label{prop:distinct}
The pairing \(W\) is not the classical
master-equation residual in \eqref{eq:swCME}.
In particular, the statement that the unrestricted
master equation fails ``exactly by \(W\)'' is not
valid for the frozen data.
\end{proposition}

\begin{proof}
The master residual is odd with ghost number one.
The pairing \(W\) is an even real number of ghost
number zero, bilinear in two free metric probes.
No coefficient-extraction or comparison map
identifying those arguments has been specified.

There is also a direct separation. In the natural
reference action, set the scalar and ghosts to
zero and all antifields to zero.
The master residual is then zero, including in
the compact onset background of
\eqref{eq:recipdata}. The independent reciprocity
experiment in that same background still has
\(W>0\). The pairing detects the absence of a
symmetric Hessian reproducing a different,
source-switched map; it is not evaluation of a
master action that has not been constructed.
\end{proof}

Nonconstant onset backgrounds do not all carry the
same evaluated defect. The theorem gives a specified
witness excluding an all-background upgrade.
It does not say that every nonconstant background
has nonzero \(W\).

The instrument in Stage E will therefore report
\emph{reciprocity incompatibility}, not a quantum
anomaly or an identified master residual.

%=====================================================================
\section{Compatibility with the retained three-tree identity}
\label{sec:trees}
%=====================================================================

The X-F-ST direct identity acts on the classical
background coefficients while holding the free
fields fixed. For \(\xi,\zeta_p\in\gp\),
\[
 h'=h+K\xi,\qquad
 k'=k+\Liec_\xi h+K\zeta_p+\tfrac12\Liec_\xi K\xi.
\]
It is a different operation from \(s\), which
acts on the field variables.

For the mixed response,
\[
 -\DR\sig Q_{K\xi}r
    =\Liec_\xi r-\DR\Lied_\xi\Pgr r.
\]
The proof uses \eqref{eq:typed},
\(\sig=1\) on \(\supp\xi\), and the retarded
left inverse on the compact tensor
\(\Liec_\xi r\).
After compact smearing, the last term is a
graviton-slot equation image.

This remains the cancellation supplied by the
mixed tree. The matter and self trees are inert
under that background transformation
\cite{XF,XFST}.
No additional copy of the scalar second-order
pair, matter tree, or mixed cancellation is added.

The relative BV construction does not establish
that the relational observable is \(\Qcl\)-closed.
Action symmetry, observable closure, and the
fixed-free-input background identity are different
claims. The supplied tree identity is not
reinterpreted as a variational M{\o}ller theorem
or an interacting quantum Ward identity.

%=====================================================================
\section{Stage C: finite evaluations of the classical residuals}
\label{sec:reports}
%=====================================================================

\subsection{What is evaluated}

Fix a finite exterior algebra
\(\Lambda_N\) with a specified ordered basis.
A coefficient extraction means a specified real
linear map
\[
 \lambda:\Lambda_N\otimes\CC\longrightarrow\RR,
\]
for example the real or imaginary part of one
basis coefficient. It is not a state, positivity
functional, or ghost measurement.

There are two classes of classical report tasks:
\begin{enumerate}[label=\textup{(R\arabic*)}]
\item a coefficient of the integrated classical master
residual
\[
 \Rme^{[2]}
     =\tfrac12(\Sbv^{[2]},\Sbv^{[2]})
\]
evaluated on supplied compact field and antifield
profiles supported in a common \(A\Subset U\);
\item a compactly smeared coefficient of
\[
 s^2\Phi^A
 \quad\text{or}\quad
 (\Qcl^{[2]})^2F
\]
through order two, evaluated on supplied local
profiles, optionally satisfying the free
equations near the smearing support.
\end{enumerate}

For (R1), one integrates the complete variational
bracket, not an arbitrarily weighted representative
of its divergence class.
On compact evaluation profiles its divergence
current is compact, so its integral is zero.
Equivalently, one may insert a cutoff equal to
one near the support of that current; no derivative
of the cutoff then contributes.

For (R2), the phrase ``free-shell projection'' means
evaluation of the already defined off-shell square
on supplied free configurations. It does \emph{not}
mean that \(s\) descends to the free quotient.
Theorem~\ref{thm:nondescent} excludes that inference.

\begin{proposition}[Defined classical report values]
\label{prop:residualvalues}
Every task (R1) or (R2), at the released orders,
has real extracted value zero.
The evaluations are continuous in their smooth
data on common compact support stages.
A nonempty finite dictionary of these tasks exists.
\end{proposition}

\begin{proof}
Theorems~\ref{thm:kinematic} and \ref{thm:BV}
give the zero identities before evaluation.
The compact evaluation rule respects the
divergence identification.

At each formal order the expressions are finite
local differential polynomials.
Compact integration and a fixed coefficient
extraction are continuous in the smooth
seminorms of their finitely many input jets.

For nonemptiness choose a nonzero real bump
\(\rho\in C_c^\infty(U)\), any compact tensor
profile for \(r\), and, for example,
\[
 \phi=\rho,\quad
 c=\theta_1\rho\,\partial_t+\theta_2\rho\,\partial_1,\quad
 \bar c=\theta_3\rho\,\dd t,\quad
 b=\rho\,\dd t,
\]
with compact antifield profiles and a finite
exterior algebra containing the indicated
generators. These are off-shell test profiles.
At least one such task can be installed without
requiring a nonzero compact free ghost.
\end{proof}

The theorem proving zero is analytical.
A later table readout does not independently prove
that universal identity. It reports a task whose
value has been supplied and justified.

\subsection{Reports and sound threshold certificates}

A dictionary entry contains:
\begin{enumerate}
\item a task identifier, quantity type, formal order,
and coefficient-extraction rule;
\item the support and input-description contract;
\item a supplied real value \(a_j\) together with a
rational enclosure
\[
 I_j=[L_j,U_j],\qquad L_j\le a_j\le U_j;
\]
\item a declared rational threshold \(\tau_j>0\).
\end{enumerate}
For the proved zero tasks one can use
\(I_j=[0,0]\).

Define the status
\[
 \mathrm{status}(I,\tau)=
 \begin{cases}
 \mathrm{PASS},& I\subset[-\tau,\tau],\\
 \mathrm{FAIL},& I\cap[-\tau,\tau]=\varnothing,\\
 \mathrm{UNDETERMINED},&\text{otherwise}.
 \end{cases}
\]

\begin{proposition}[Certificate soundness]
\label{prop:threshold}
If the installed enclosure is valid, PASS implies
\(|a_j|\le\tau_j\), and FAIL implies
\(|a_j|>\tau_j\).
No conclusion is drawn from UNDETERMINED.
\end{proposition}

\begin{proof}
The value belongs to its enclosure. The three
cases follow from the stated set inclusions.
\end{proof}

No algorithm for obtaining enclosures of arbitrary
smooth field descriptions is assumed. The finite
dictionary is supplied data.
Distinct deterministic numerical words need not
define nearby channels when the numbers are close.
No false continuity claim for thresholding or
orthogonal report words is made.

%=====================================================================
\section{Complete destructive instruments for the two dictionaries}
\label{sec:instrument}
%=====================================================================

\subsection{The classical residual-report instrument}

Let the finite dictionary have \(J\) entries.
Choose \(d_C=2^{n_C}\ge J\), \(n_C\ge1\), and
a calibrated orthonormal clock basis
\(\{\ket{\tau_j}\}_{j=1}^{d_C}\).
Unused labels receive an explicit
\(\mathrm{UNUSED}\) report.

Let \(A\) contain all unmeasured finite quantum
registers. Let \(q_j\) be a finite classical word
containing the entry identifier, enclosure,
status, and order tags. Put
\[
 D_j=\bra{\tau_j}_C\otimes I_A.
\]
Distinct output labels retain \(j\), even when
several entries have the same zero residual.

Define
\begin{equation}
 \mathcal M_{\mathrm{cl}}(X)
     =\sum_{j=1}^{d_C}\proj{j,q_j}_R
                                \otimes D_jXD_j^\dagger.
 \label{eq:instrument}
\end{equation}

\begin{theorem}[Complete supplied-residual instrument]
\label{thm:instrument}
The map \eqref{eq:instrument} is CPTP and
\begin{equation}
 \Tr_R\mathcal M_{\mathrm{cl}}(X)=\Tr_CX.
 \label{eq:drop}
\end{equation}
The identities hold on all input operators and
after adjoining any untouched reference \(E\).

For clock preparation
\(\proj{\tau_j}_C\otimes\rho_{AE}\), the output is
\[
 \proj{j,q_j}_R\otimes\rho_{AE}.
\]
For an arbitrary correlated input, the returned
quantum block is the subnormalized conditional
block \(D_jXD_j^\dagger\), not an assumed
unchanged conditional state.

The clock factor \(C\) is consumed from the
returned quantum interface.
\end{theorem}

\begin{proof}
A Kraus family is
\[
 K_j=\ket{j,q_j}_R\otimes D_j,\qquad
 \sum_jK_j^\dagger K_j
          =\sum_j\proj{\tau_j}_C\otimes I_A=I.
\]
This proves complete positivity and trace
preservation.

Expand
\(X=\sum_{j,l}\kb{\tau_j}{\tau_l}\otimes X_{jl}\).
The \(j\)-block in \eqref{eq:instrument} is
\(X_{jj}\); their sum is \(\Tr_CX\).
The same expansion with \(A\otimes E\) proves
the reference identities and the basis-input
formula.

The quantum codomain is \(A\), not \(C\otimes A\).
The measured clock names are therefore unavailable
as quantum inputs to a continuation through this
interface.
\end{proof}

This is the I-B/En09-A complete instrument pattern,
with the present report words explicitly attached
\cite{IB,En09}. Conditional reference steering
is compatible with the unchanged unconditional
reference marginal.
Only positive-probability branches are normalized.

\subsection{The encoded report is not a field observable}

For a chosen rational displayed value \(\widehat a_j\)
of each report, the bounded register observable is
\[
 B_{\mathrm{rep}}
       =\sum_j\widehat a_j\proj{j,q_j}_R.
\]
The Heisenberg pullback is
\begin{equation}
 \mathcal M_{\mathrm{cl}}^*
                  (B_{\mathrm{rep}}\otimes I_A)
       =\sum_j\widehat a_j
                         \proj{\tau_j}_C\otimes I_A.
 \label{eq:reportobservable}
\end{equation}
It is an observable of the finite task encoding.

\begin{proposition}[No acquisition or physical-cohomology inference]
\label{prop:noacquisition}
The instrument above does not measure an unknown
continuum master residual, select a physical BRST
state, or turn a ghost-number-one expression into
a self-adjoint field operator.
\end{proposition}

\begin{proof}
Its input has only the declared finite factors.
Its Kraus maps contain the installed table but
no map from continuum configurations to those
factors. Two hypothetical continuum inputs
attached to the same finite preparation therefore
give the same output.

The only self-adjoint observable constructed
for the report is the finite diagonal operator
in \eqref{eq:reportobservable}.
Coefficient extraction from an exterior algebra
is not a positive field representation.
No physical cohomology or such representation
has been specified.
\end{proof}

Thus the instrument licenses the phrase
``observed encoded residual report'' on its
declared interface. It does not supply the
stronger physical identification suggested by
calling the unsmeared BV residual an observable
of an interacting quantum theory.

The finite residual \(A\) returned by the
instrument and the mathematical master residual
\(\Rme\) are different objects despite the shared
word ``residual''.

\subsection{Stage E: the separate obstruction instrument}

Choose a separate finite dictionary of the
reciprocity tasks \eqref{eq:W}, with its own clock
\(C_{\mathrm{rec}}\).
A task can contain the fixed analytic expression
for \(W\), a supplied rational enclosure
\([L,U]\subset(0,\infty)\), and the tag
\(\mathrm{RECIPROCITY\_INCOMPATIBLE}\).

Such an enclosure exists for each fixed witness:
the proof gives \(W>0\), so rational numbers
\(0<L\le W\le U\) can be chosen and supplied.
This existence is not a general evaluation
algorithm. For example, if
\(|\sig''|\ge a>0\) on \(\supp\zeta\), then
\[
 W\ge\frac{5M_\chi a^2}{8}\int\zeta\,\dd t>0
\]
is an explicit lower-bound interface.

Define \(\mathcal M_{\mathrm{rec}}\) by the same
Kraus formula as \eqref{eq:instrument}, using
these separate report words and clock names.

\begin{corollary}[Complete obstruction-report instrument]
\label{cor:obstructioninstrument}
The map \(\mathcal M_{\mathrm{rec}}\) is CPTP,
consumes its clock encoding, retains the
unmeasured finite residual, and satisfies
\eqref{eq:drop} including every reference.
On a task-basis input it returns its certified
positive reciprocity report with probability one.

It does not cancel \(W\), evaluate a master action
that does not exist, or identify the report with
a quantum anomaly.
\end{corollary}

\begin{proof}
Theorem~\ref{thm:instrument} applies to any finite
installed report table.
The last assertions follow from
Proposition~\ref{prop:distinct} and from the
fact that readout changes no source equation.
\end{proof}

An exact arbitrary real value is not asserted to
have a finite numerical code. The returned word
is its installed identifier and certified
enclosure, or an exact number when one has
actually been supplied.

\subsection{Clock calibration and continuation}

Nontrivial dictionary labels may be calibrated as
history or task labels and read at one later event.
They are not all declared to equal the time of that
same event. If distinct calibrated labels must
equal the coordinate of one fixed reading event
on every positive branch, only one label can
occur \cite{IB}.

Neither clock is returned. The geometric coordinate
\(t\), the host time \(T\), and the prescribed
future relation remain.
No relational unitary evolution follows from
clock extraction alone; En09-A's additional
stationarity and clock-covariance hypotheses
are not assumed here.

%=====================================================================
\section{Stage D: the first quantum correction remains conditional}
\label{sec:quantum}
%=====================================================================

\subsection{The local fish does not supply the full residual}

Under the specified flat Feynman conventions, the
local scalar--metric fish components are
\[
 t_{\alpha\beta}(z)
    =D_0^{\mathrm F}(z)
          \partial^\alpha\partial^\beta D_m^{\mathrm F}(z),
 \qquad z\ne0,\qquad |\alpha|+|\beta|\le2.
\]
They are defined off zero and have scaling degree
at most six. Finite-scaling-degree extensions
exist, and their internal ambiguity has the form
\[
 \sum_{|\gamma|\le2}a_\gamma\partial^\gamma\delta_0.
\]
This is the local extension interface retained
from X-G and the standard extension theorem
\cite{XG,XGST,BF}.

For a separately chosen extension and finite
coefficient list, the resulting local counterterm
functional and its classical BV variation can be
specified and evaluated on compact smooth inputs.
All external differentiations and derivatives of
test coefficients must be retained.
The internal derivative bound is not automatically
a bound of two on the total derivative order of
the resulting local density.

None of these local data defines the missing
nonlocal coordinate contraction or the entire
one-loop master residual.

\subsection{A conditional first-order consistency equation}

The following statement fixes the logic without
introducing an undefined BV Laplacian.

\begin{definition}[A supplied first-order quantum residual interface]
\label{def:quantuminterface}
Assume a later construction supplies, on the same
relative domain and through the same formal order:
\begin{enumerate}[label=\textup{(Q\arabic*)}]
\item all quantum products and insertion maps needed
to define the first-order residual;
\item an admitted odd ghost-number-one functional
\(\mathfrak A_1\), including every normalization
and transport contribution;
\item a convention in which adding an even
ghost-number-zero counterterm \(C_1\) changes
that residual by \(\Qcl^{[2]}C_1\);
\item the consistency identity
\(\Qcl^{[2]}\mathfrak A_1=0\) modulo \(\kappa^3\).
\end{enumerate}
These are hypotheses, not conclusions of the
local fish extension.
\end{definition}

\begin{proposition}[Conditional quantum cancellation criterion]
\label{prop:quantumcriterion}
Under Definition~\ref{def:quantuminterface}, a
permitted counterterm cancels the first-order
residual exactly when
\begin{equation}
 \Qcl^{[2]}C_1-\mathfrak A_1=0
                                      \pmod{\kappa^3}.
 \label{eq:quantumcriterion}
\end{equation}
Existence is an exact-image question in the
specified permitted counterterm space. It is not
implied by the closedness of \(\mathfrak A_1\).
\end{proposition}

\begin{proof}
This is the change law (Q3).
Condition (Q4) is necessary for
\eqref{eq:quantumcriterion} because
\((\Qcl^{[2]})^2=0\), but a closed element need
not be in the image of the differential.
No vanishing theorem for the relevant
ghost-number-one cohomology or permitted
counterterm image has been proved here.
\end{proof}

The framework of quantum BV and perturbative
gravity requires precisely such domain and
normalization constructions \cite{RejznerBV,BFR}.
The references do not automatically instantiate
them for the unchanged source-switched nonlocal
observable.

\subsection{Instrumenting only what exists}

A finite chosen local counterterm contribution
\(\Qcl^{[2]}C_1\), once defined and supplied with
a real coefficient extraction and enclosure, is
routed through the same instrument architecture
\eqref{eq:instrument}. Its quantity tag says
\(\mathrm{LOCAL\_COUNTERTERM\_CONTRIBUTION}\).

If the full \(\mathfrak A_1\) and the corrected
residual in \eqref{eq:quantumcriterion} are later
constructed, their finite supplied evaluations
can be added as separately tagged entries.

Until then, an entry for the total corrected
residual must say
\(\mathrm{NOT\_CONSTRUCTED}\), not zero, PASS,
or ``quantum symmetry restored''.
This is metadata about a missing construction,
not a numerical value for an undefined expression.

A complete channel can report that status. Its
existence does not construct the missing quantity.
Consequently Stage D has a conditional criterion
and an instrument interface, not a quantum lift.

Positive powers of \(\hbar\) also cannot remove
the classical reciprocity witness or the leading
switched-scalar breaking. Equality of formal
series is coefficientwise.

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

\subsection{The complete network}

Let a finite set of sites carry finite-dimensional
matrix algebras. At tick \(s+1\), let the complete
channel factor over a supplied partition
\(\mathcal B_{s+1}\).
Every table selector, report register, control,
reset, encoder, and resource wire is included
in its actual block interface.

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

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

Deterministic interventions outside the complete
backwards region cannot change the corresponding
joint output law. If every dependency edge has
an independently certified regular future-causal
host route, scheduled ancestry is contained in
the prescribed metric causal past of the output.
\end{theorem}

\begin{proof}
A trace-preserving channel has a unital adjoint.
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.
The reference algebra is retained unchanged in
this argument.

Concatenating the separately certified host
routes proves causal containment.
\end{proof}

This is the prescribed-network mechanism of
I-B, En03-A, and VII-E \cite{IB,En03,VIIE}.
Ancestry is an upper bound; it need not equal
actual influence or an entire physical causal
past. Postselection is not a deterministic
intervention. Scheduler consistency does not
derive a Lorentz cone \cite{En08A}.

\subsection{A nonempty regular transport route}

Choose a straight developed spatial segment of
length \(\ell>0\) in the inherited regular convex
tube, with a time reserve after readout, lookup,
and a positive processing buffer.
At \(n+1\) target-adapted stations put \(p\)-qubit
payload registers \(A_0,\ldots,A_n\).
Registers \(A_1,\ldots,A_n\) start in independent
blank states \(\omega\).
At hop \(j\), SWAP \(A_{j-1}\) with \(A_j\)
across developed duration \(\ell/n\).

These are declared tick-boundary finite permissions,
not instantaneous equal-time interactions or a
derived gravitational Hamiltonian.

\begin{proposition}[Reference-preserving finite route]
\label{prop:route}
For every \(\rho_{A_0E}\), after hop \(j\) the
state is
\[
 \rho_{A_jE}\otimes
   \bigotimes_{\substack{0\le i\le n\\i\ne j}}\omega_{A_i}.
\]
The logical payload channel is the identity with
every reference. Arrival is at
\[
 t_{\mathrm d}+\ell,\qquad
 T_{\mathrm d}+\ell/c_{\mathrm{host}}.
\]
All routes remain in the regular tube.
\end{proposition}

\begin{proof}
Each SWAP moves the whole payload factor,
including its reference correlations, to the
next blank and leaves the previous factor blank.
Induction proves the state formula.
Station spacing \(\ell/n\) and tick duration
\(\ell/n\) give null cross-port dependencies;
same-station dependencies are timelike.
\end{proof}

The finite residual in the instrument can be this
payload, and classical report words can be
forwarded on separately certified wires.
This is the I-B/En08-B transport pattern
\cite{IB,En08B}. It transports a finite register,
not an asserted physical ghost multiplet, and
preserves existing correlations rather than
generating entanglement by gravity.

\subsection{Resource and error ledger}

For two simultaneously reserved clocks, payloads,
report buffers, and lookup workspace, a conservative
preallocated reserve is
\[
 n_C^{\mathrm{cl}}+n_C^{\mathrm{rec}}
       +(n+1)p+(n+1)b_{\mathrm{rep}}+w_{\mathrm{lookup}}.
\]
Sequential reuse can lower the peak only if its
actual schedule and reset permissions are supplied.
Descriptions, tables, supplier holdings, external
references, and returned factors are separately
charged when present.

The route uses \(n\) register SWAPs.
Clock basis changes, readouts, lookup, record
forwarding, buffers, and durations are also
charged. Table generation is not an unpriced
field-acquisition step.

For compatible ideal and implemented complete
channels,
\[
 \ddia(\widetilde\Gamma,\Gamma)
  \le\min\left\{1,\,
        \epsilon_{\mathrm{enc}}
       +\epsilon_{\mathrm{clock}}
       +\epsilon_{\mathrm{lookup}}
       +\sum_j\epsilon_j
       +\epsilon_{\mathrm{dec}}\right\},
 \qquad
 \ddia(\Lambda,\Gamma)
           =\tfrac12\|\Lambda-\Gamma\|_\diamond.
\]
Replace one channel factor at a time and use
diamond-norm contraction and the triangle inequality
\cite{Watrous}. The comparison includes references.
A probability of receiving a particular status
changes by at most this complete-channel allowance.

The error bound does not verify an invalid enclosure,
construct a missing analytic value, or prove exact
causal containment of a faulty implementation.

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

\begin{definition}[Local Grade S for this manuscript]
\label{def:grade}
The release consists of the inherited supported
host; the explicit relative classical differential
and master action; the non-descent and obstruction
accounting; finite supplied-evaluation dictionaries;
complete destructive instruments including
unmeasured residuals and references; and scheduled
locality with a nonempty regular route under the
declared finite permissions.

No physical field-measurement identification is
included merely by the term Grade S.
\end{definition}

\begin{theorem}[X-H-ST mathematical release]
\label{thm:release}
At fixed positive width:
\begin{enumerate}[label=\textup{(\roman*)}]
\item the field differential of
Section~\ref{sec:stageA} is explicitly constructed
and nilpotent off shell;
\item the relative classical BV master action is
constructed through \(\kappa^2\), modulo
divergences on the plateau, with independent
auxiliary field and the signed antifield lift;
\item the unchanged free causal algebra is not
identified with this interacting classical
complex; the natural first-order scalar rule
fails free-equation descent;
\item the unrestricted switched master residual and
the positive reciprocity pairing are separately
defined and are not identified;
\item the X-G exact source-preserving variational
upgrade obstruction remains in force;
\item finite supplied classical residual reports and
the separate reciprocity reports have the
complete destructive instruments proved above;
\item Stage D remains a conditional consistency and
encoding interface, not a constructed quantum lift;
\item scheduled locality, regular reference-preserving
transport, and the stated resource/error ledger
hold for the complete finite interface.
\end{enumerate}
These statements form local Grade S.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:kinematic} proves (i).
Lemmas~\ref{lem:action} and \ref{lem:sign}, and
Theorem~\ref{thm:BV}, prove (ii).
Theorem~\ref{thm:nondescent} and
Proposition~\ref{prop:compactghost} prove (iii).
Equations \eqref{eq:swCME} and \eqref{eq:W},
with Proposition~\ref{prop:distinct}, prove (iv).
The reciprocity argument and
Proposition~\ref{prop:recip} give (v).
Propositions~\ref{prop:residualvalues} and
\ref{prop:threshold}, Theorem~\ref{thm:instrument},
and Corollary~\ref{cor:obstructioninstrument}
prove (vi), with the interpretation limited by
Proposition~\ref{prop:noacquisition}.
Definition~\ref{def:quantuminterface} and
Proposition~\ref{prop:quantumcriterion} give
exactly the conditional conclusion in (vii).
Section~\ref{sec:locality} proves (viii).
\end{proof}

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.31\linewidth}p{.23\linewidth}X@{}}
\toprule
Target & Verdict & Exact boundary\\
\midrule
Nonlinear field nilpotence &
Constructed &
Off-shell local fields, not quotient descent\\
Plateau classical BV action &
Constructed through \(\kappa^2\) &
Natural reference action modulo divergences\\
Interacting action on unchanged free carrier &
Natural proposal fails descent &
No retarded interacting transport supplied\\
Classical residual report instrument &
Constructed &
Supplied coefficient evaluations only\\
Reciprocity report instrument &
Constructed &
Separate upgrade-incompatibility quantity\\
Exact X-F source-preserving variational upgrade &
Obstructed in the named class &
Not every changed classical prescription\\
X-F direct tree identity &
Retained &
Not reinterpreted as observable BV closure\\
First quantum lift &
Not constructed &
Conditional anomaly/counterterm interface only\\
Physical residual field measurement &
Not constructed &
No field-to-apparatus acquisition map\\
Physical cohomology and positivity &
Not opened &
No Kugo--Ojima analysis or physical state\\
Spacetime emergence &
Not constructed &
Host and causal dictionary remain inputs\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Potential use and novelty boundary}

The classical diffeomorphism algebra and canonical
BV mechanism are standard, and the X-G normalization
and reciprocity obstruction are Cheung's results
\cite{FRBV,XG}. No priority claim is made for them.

The useful contribution is the explicit same-model
audit of four different operations:
\[
 \text{nilpotent local vector field},\qquad
 \text{free-shell descent},\qquad
 \text{variational response},\qquad
 \text{complete report channel}.
\]
The construction shows exactly which can coexist
and which implication fails.
The signed antifield lift, the free-solution
non-descent witness, the background-tadpole
accounting, and the typed separation of the two
obstructions prevent a local classical success
from being advertised as an interacting quantum
completion.

The complete instruments add a genuine finite
implementation interface for supplied reports.
Their value is not inflated into a continuum
detector, a cohomology measurement, or an independent
experimental verification of a universal identity.

\subsection{Successor package}

A successor receives:
\begin{enumerate}
\item the unchanged host, completion, width, and free
normalization;
\item the explicit off-shell nonlinear field rules;
\item the relative master action, its coefficient
specification, and its sign convention;
\item the free-equation non-descent witness;
\item separate switched-action and reciprocity
obstruction data;
\item complete supplied-report instruments and
threshold semantics;
\item the conditional first-order quantum consistency
equation;
\item the complete finite locality and transport contract.
\end{enumerate}

It does not receive an interacting differential on
the unchanged free causal algebra, a nonlocal
one-loop observable, a BV Laplacian, a quantum
master theorem, a physical-state selection, a
variational identification of the frozen X-F source
law, or a field-acquisition instrument.

\subsection{Standing boundaries}

O.1 remains \loc{NOT_CLOSED} in the inherited scope.
XV-B is not opened as the downstream
quantum-admission exit. The existence of technical
manuscripts with that identifier is not denied.
XVI-N's separate prepared-insertion obstruction is
not discharged \cite{XVIN,BoundaryRecords}.

Historical H2 and the missing full I3/J2 characteristic
maps are not discharged. XIV-I's narrower internal
amplitude is not withdrawn or identified with the
missing full map \cite{BoundaryRecords}.
The present local BV calculation contains no new
characteristic composition.

No quantum coefficient of the inherited
characteristic graph is admitted.
The free carrier and the previously admitted
tree coefficients are not denied by that sentence.

No X-D-ST sharp scalar limit is inherited as metric
admission. The host metric, worldsheet, geometric
time, and prescribed future relation remain fixed.
This is not spacetime emergence, Einstein dynamics
derived from records, gravity-generated entanglement,
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 overstatement & Correction\\
\midrule
\endfirsthead
\toprule
Potential overstatement & Correction\\
\midrule
\endhead

Nilpotence constructs a differential on the free quotient &
Ideal preservation is separately required and fails
for the natural scalar rule.\\[4pt]

Compact plateau ghosts are nonzero homogeneous free ghosts &
Spacetime-compact free wave solutions vanish.\\[4pt]

A compact antighost restriction is automatically tangent &
The auxiliary field must obey the matching restriction
because \(s\bar c=b\).\\[4pt]

A local action density is a finite global action &
Only variational classes and compact evaluation
sectors are used.\\[4pt]

An arbitrary smearing evaluates a divergence class &
It can introduce boundary terms. The complete compact
variational integral is evaluated instead.\\[4pt]

The quadratic and cubic gravitational terms suffice
for the second-order master action &
The \(\kappa^2\) action coefficient contains the
fourth geometric Einstein--Hilbert coefficient.\\[4pt]

One unweighted antifield coupling works with the stated bracket &
The factor \((-1)^{|\Phi^A|}\) is checked on every
field parity.\\[4pt]

Off-shell backgrounds can have their leading tadpole deleted &
The reference action retains \(\pair r{\Akin h}\);
deleting it changes the model.\\[4pt]

The reference BV action realizes the unchanged X-F source law &
No such identification is made; its exact variational
upgrade is obstructed.\\[4pt]

The positive reciprocity pairing is the master residual &
They have different parity, ghost number, and
arguments.\\[4pt]

Every nonconstant onset background has the same defect &
A specified witness excludes an all-background upgrade.\\[4pt]

An instrument measures a Grassmann-odd quantity directly &
It reads a supplied real coefficient extraction with
a finite encoding.\\[4pt]

A dictionary of zero reports proves the universal identity &
The identity is proved analytically; the dictionary
reports selected supplied evaluations.\\[4pt]

The word ``residual'' identifies the master residual
with the unmeasured quantum output &
One is an analytical functional; the other is a
finite returned register state.\\[4pt]

A finite word returns an arbitrary real number exactly &
It returns an installed identifier, rational report,
or certified enclosure.\\[4pt]

Nearby numerical words imply nearby deterministic channels &
Orthogonal encodings can have unit channel distance.
No such continuity is claimed.\\[4pt]

A local fish counterterm cancels the total quantum residual &
The full nonlocal residual and its permitted
normalizations must first be constructed.\\[4pt]

An absent quantum residual can be encoded as zero &
Its status is \loc{NOT_CONSTRUCTED}, not PASS.\\[4pt]

An action master identity proves relational observable closure &
The observable's BV variation is a separate problem.\\[4pt]

The table instrument consumes geometric time &
Only finite measured clock names are consumed.\\[4pt]

Reference preservation is gravity-generated entanglement &
The route preserves correlations already present.\\[4pt]

Scheduled ancestry is the entire metric causal past &
Only containment follows from separate route
certificates.\\[4pt]

The source file proves registration or successful compilation &
Neither is asserted.\\

\bottomrule
\end{longtable}

%=====================================================================
\section{Documentary comparisons and integration status}
\label{app:status}
%=====================================================================

XVI-N concerns a separately specified prepared
insertion and antifield-linear completion problem.
Theorem~\ref{thm:BV} changes neither its seed,
equation space, nor its obstruction hypotheses
\cite{XVIN}. No repair is inferred from sharing
antifield notation.

The OpenAI catalogue is documentary only.
No catalogue family is used to prove the classical
BV action, free-shell descent, an instrument, or a
quantum correction \cite{OpenAICatalogue}.
No claim of independent catalogue verification is
made.

A proposed installation may use
\loc{content/manuscripts/X-H-ST.tex}.
Actual source hashes, metadata schemas, imports,
registry status, and repository tests must be
obtained from the repository. They are not
generated by the mathematical argument.

\begin{verbatim}
identifier: X-H-ST
sourceStatus: proposed
grade: local Grade S
host: unchanged I-B / X-A-ST instance
widthStatus: fixed-positive-width
classicalDomain: relative off-shell local BV algebra
classicalOrder: kappa^2
freeCarrierInteractingDescent:
  naturalProposal: fails at order kappa
constructed:
  - nonlinear off-shell field differential
  - signed relative classical BV master action
  - compactly tested free-shell non-descent witness
  - classical supplied-residual report instrument
  - separate reciprocity report instrument
  - scheduled locality and regular reference transport
retained:
  - X-G exact source-preserving upgrade obstruction
  - X-F direct tree plateau identity
quantumLift: not constructed
nonlocalOneLoopObservable: unresolved
fieldAcquisitionInstrument: not constructed
physicalCohomology: not opened
spacetimeEmergence: not constructed
registryEditsExecuted: false
repositoryTestsExecuted: false
compilationAsserted: false
\end{verbatim}

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

\paragraph{Authorship and attribution.}
The author name follows the supplied predecessor
metadata. Cheung's X-G supplies the normalized free
multiplet and the reciprocity obstruction. X-F
supplies the separate direct tree identity.
The classical BV mechanism is also placed in its
standard mathematical context. No priority claim
is made for those parent constructions.

\paragraph{Analytical scope.}
The positive classical result is relative and
off shell. It is not an interacting derivation
of the unchanged free causal algebra.
No acyclicity, physical cohomology, positivity,
interacting quantum BRST, or Kugo--Ojima theorem
is claimed.

\paragraph{Operational scope.}
The constructed instruments read finite supplied
evaluations or status words. They do not acquire
unknown continuum fields or provide a physical
representation of ghost-valued residuals.
All unmeasured finite output and reference factors
are retained in the complete channel.

\paragraph{Source status.}
Supplied manuscripts and retrieved documentary
records are citations, not independent proof
certificates. No registration or publication
status is inferred for the ST source from its
identifier.

\paragraph{Verification.}
No successful \loc{pdflatex} execution,
proof-assistant verification, symbolic-program
execution, numerical evaluation, hardware
realization, repository installation, independent
execution, referee approval, experiment, or
journal acceptance is asserted.
The mathematical release is the explicit
definitions and written proofs above.

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

\bibitem{XGST}
N.~Cheung,
\emph{X-G-ST: Free causal BRST compatibility and
a variational upgrade obstruction}.
Supplied \loc{X-G-ST.tex}.
Normalized free multiplet, Green matrix, switched
breaking, supported reciprocity witness, and
the distinction between free causal compatibility
and an interacting quantum differential.

\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}.
Principal free normalization, source-versus-variation
distinction, reciprocity obstruction, and local
fish interface.

\bibitem{XFST}
N.~Cheung,
\emph{X-F-ST: Three fixed-width retarded trees and
the background identity}.
Supplied \loc{X-F-ST.tex}.
Equation-level response, typed commutator,
three-tree identity, support and variational audit.
No registration is inferred.

\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 direct tree mechanism is retained; the stronger
variational terminology is qualified by X-G.

\bibitem{XAST}
N.~Cheung,
\emph{X-A-ST: Supported classical covariance of a
fixed-ramp relational scalar}.
Version-specific predecessor source record.
Inherited supported host, chart, completion,
signature ledger, fixed-width prescription,
and plateau algebra.

\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 conical host and worldsheet, complete
clock instrument, calibration, scheduled locality,
and regular finite transport.

\bibitem{EnterPhilosophy}
N.~Cheung,
\emph{The Philosophy of Enter: Destructive Readout,
Quantum Recovery, and the Refusal of Identification}.
\loc{EntPhil-v2.tex}, version 2.
Introductory essay, Enter Calculus Companion.
\url{https://enter.grok.me/enter}.
Interface orientation, not a gravitational or BV
construction premise.

\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}.
Contextual attribution within its full stated
structural class; no abbreviated characterization
is used here.

\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 destructive instrument, untouched
references, conditioning, consumption, and separate
dynamical hypotheses.

\bibitem{En03}
N.~Cheung,
\emph{Observation Boundaries for Destructive Quantum
Readout and External Geometry}.
\loc{En03-A.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/En03-A}.
Prescribed scheduled blocks, complete interfaces,
and locality bounds.

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

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

\bibitem{En08B}
N.~Cheung,
\emph{Resource-controlled operational approximation
of null boundary transport}.
\loc{En08-B.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/En08-B}.
Reference-preserving finite SWAP transport relative
to supplied geometry and operational permissions.

\bibitem{FRBV}
K.~Fredenhagen and K.~Rejzner,
``Batalin--Vilkovisky formalism in the functional
approach to classical field theory,''
\emph{Communications in Mathematical Physics}
\textbf{314} (2012), 93--127.
\url{https://arxiv.org/abs/1101.5112}.
Classical functional BV framework and its domain
distinctions.

\bibitem{RejznerBV}
K.~Rejzner,
\emph{BV quantization in perturbative algebraic QFT:
Fundamental concepts and perspectives}.
arXiv:2004.14272.
\url{https://arxiv.org/abs/2004.14272}.
Classical and quantum BV distinctions, M{\o}ller
maps, and anomalous master-Ward interfaces.

\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 quantum-gravity framework;
not an automatic construction of the frozen
source-switched nonlocal observable.

\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 extensions, used for the
stated local fish interface only.

\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 causal
Green operators on the declared completion.

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

\bibitem{XVIN}
N.~Cheung,
\emph{XVI-N}.
Revision r12, version-specific supplied record.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/XVI-N}.
Separate prepared-insertion and antifield-linear
completion obstruction; not discharged here.

\bibitem{BoundaryRecords}
N.~Cheung,
\emph{XIV-H}, \emph{XIV-I}, and \emph{XIV-J},
version 1; technical \emph{XV-B}, version 2;
and the supplied O.1 standing-boundary records.
Enter Calculus Companion.
Version-specific excerpts supplied with the request.
Historical H2, the narrower internal-amplitude
advance, missing full I3/J2 maps, and downstream
admission requirements remain distinct.
No characteristic coefficient is admitted by
the present local classical construction.

\bibitem{OpenAICatalogue}
OpenAI,
\emph{Mathematics catalogue}.
Supplied documentary catalogue descriptions,
recorded separately in the Enter Calculus Companion.
Comparison only. No catalogue theorem is used
or independently verified here.

\end{thebibliography}

\end{document}