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\hypersetup{
  pdftitle={X-I-ST: A supported finite-order classical BV deformation on a prescribed conical Enter spacetime},
  pdfsubject={Relative classical BV, finite-order interaction,
  complete residual-report instruments, and retained obstructions}
}

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\title{\textbf{X-I-ST: A supported finite-order classical BV deformation\\
on a prescribed conical Enter spacetime}\\[5pt]
\large A compact scalar interaction, complete residual-report instruments,\\
and preservation of the earlier diagnostic boundaries}
\author{Ngo Cheung}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We construct a controlled classical continuation of the relative
Batalin--Vilkovisky structure of X-H-ST on the unchanged
I-B/X-A-ST conical host. All analytical operations remain on the
declared Minkowski completion, and the regulator width stays
fixed and positive. The new interaction is the explicit compact
vertex
\[
 \lambda\kappa^3 V_a,\qquad
 V_a=\frac1{4!}\int a(x)\phi(x)^4\,\dd^4x,
\]
supported inside the existing relative plateau. A finite
coefficient specification of the reference Einstein--scalar
action, including its fifth geometric-order contribution,
gives a classical BV differential satisfying
\(Q_\lambda^2=0\) modulo \(\kappa^4\). The interaction changes
the scalar antifield equation by
\(\lambda\kappa^3 a\phi^3/6\), while leaving every coefficient
through order two unchanged.

Two debugging results are essential. Exact recovery of the
full free BV complex requires the centered background sector;
for a general fixed off-shell background the inherited tadpole
remains. Moreover, a complete finite instrument can report
supplied numerical coefficients of a residual, but does not
measure an arbitrary continuum or Grassmann-valued residual.
We prove the appropriate finite-report theorem, including
all selective outputs, failures, surviving systems, and
untouched references, and prove its stability under finite
adaptive composition and prescribed regular transport.

The chosen polynomial lift has an explicitly evaluated
fourth-order master residual. Its new part is
\[
 \lambda\kappa^4\mathcal B_a,\qquad
 \mathcal B_a=\frac16\int a\phi^3c^\mu\partial_\mu\phi\,\dd^4x,
\]
with a nonzero real coefficient on compact ghost data.
Thus finite-order classical closure is not silently promoted
to an exact master identity. The positive reciprocity witness
and the independent convolution gap remain unchanged in their
specified diagnostic sectors; the three-tree plateau is retained
once, not reconstructed or reinterpreted.

The contribution is a supported, order-controlled
analytical--operational assembly, not a new general BV method.
Its prospective usefulness is to make a genuine classical
interaction compatible with explicit report interfaces while
keeping the next obstruction visible. No interacting quantum
BRST differential, characteristic quantum coefficient,
loop-anomaly cancellation, ultraviolet completion, or
spacetime emergence is established.
\end{abstract}

\statementbox{Exact release and necessary corrections}{%
A relative classical BV deformation is constructed through
order \(\kappa^3\), meaning modulo \(\kappa^4\).
Its finite supplied reports have complete instruments.

X-H-ST already contains nonlinear classical interactions.
Accordingly, this paper is not advertised as the first
classical interacting construction. Its new step is an explicit
compact third-order scalar vertex and the associated
finite-order and instrument audit.

Free field BRST recovery holds at \(\kappa=0\).
Full free BV recovery additionally requires the centered
background sector specified below.
No interacting differential on the unchanged free causal
carrier is inferred.}

\tableofcontents

%=====================================================================
\section{Purpose, provenance, and the Enter discipline}
\label{sec:introduction}
%=====================================================================

\subsection{The construction actually undertaken}

The supplied plan leaves the interaction unspecified and
asks for all residuals to be instrumentable.
Neither request is a theorem without further definitions.

We make the following choices before calculating:
\begin{enumerate}
\item the interaction is the compact scalar vertex \(V_a\)
displayed in the abstract;
\item it first occurs at order \(\kappa^3\);
\item the retained coupling ring is
\(\RR[\lambda][\kappa]/(\kappa^4)\);
\item the classical reference action is specified by finite
Taylor extraction from the natural local action;
\item a report concerns a specified real coefficient of
a specified representative on supplied smooth data;
\item only finite dictionaries and finite instrument networks
are released.
\end{enumerate}

The parameter \(\lambda\) distinguishes the additional vertex
from the pre-existing gravitational interaction.
Setting \(\lambda=0\) does not turn off the reference
Einstein--scalar interactions.
Setting \(\kappa=0\) removes every positive-order interaction.

The principal predecessor records are Cheung's X-G,
X-G-ST, and X-H-ST \cite{XG,XGST,XHST}.
The free normalization, relative kinematics, and action
conventions used here are restated explicitly.
The proofs of the new deformation and its report interfaces
do not depend on a registry connection.

\subsection{Background and philosophy of Enter}

The Enter interface consumes the measured quantum names and
returns a classical record together with the unmeasured
surviving systems. Consumption is a statement about access
and output type. It is not physical annihilation of every
implementing degree of freedom.

Cheung's philosophy separates destructive readout, quantum
return, implementation, and geometric identification
\cite{EnterPhil,EnterI}. We adopt that separation:
\begin{itemize}
\item a classical BV residual is not automatically a physical
observable;
\item a finite encoding is not acquisition of a continuum
functional;
\item equality of outcome laws does not establish equality
of complete instruments;
\item reference-preserving transport is not entanglement
generated by a gravitational mediator;
\item a prescribed conical spacetime is not derived from
records.
\end{itemize}

The finite instruments below use stated Born semantics.
They do not rederive the operational characterization of
Enter from an abbreviated hypothesis list.

I-B supplies the prescribed-background interpretation:
geometry, locations, accessible algebras, and complete
control/resource data are separately declared \cite{IB}.
The spacetime remains a continuum; the experiments are finite.
No host-update rule is introduced.

\subsection{Standard mathematical context}

The local variational BV framework and its distinction
between off-shell complexes, equations, and observables
are standard \cite{FRBV}. Perturbative quantum gravity
requires additional product, renormalization, and quantum
master-equation constructions \cite{BFR}.
Those additional operations are not supplied by the
classical bracket used here.

The underlying scalar polynomial and BV construction
are not claimed as new general techniques.
The contribution is their explicit supported continuation
with a finite-order residual calculation and a complete
finite-report interface.

%=====================================================================
\section{The unchanged host and the support contract}
\label{sec:host}
%=====================================================================

\begin{assumption}[Retained supported host]
\label{ass:host}
Retain the I-B/X-A-ST instance, including the deficit,
time orientation, supported developing chart, laboratory
reserve, finite-operation permissions, and time calibration.

Write its regular conical product as
\[
 \mathcal M^\circ
 =\RR_T\times(C_\alpha\setminus\{o\})\times\RR_z,
 \qquad
 G=-c_{\mathrm h}^2\dd T^2+\dd r^2
       +\alpha^2r^2\dd\varphi^2+\dd z^2,
\]
where \(0<\alpha<1\) and \(c_{\mathrm h}>0\) retain their
inherited values.
The completion defect is the worldsheet
\[
 \mathcal W=\RR_T\times\{o\}\times\RR_z.
\]

The independent analytical completion, slab, and chart are
\[
 \widehat M=\RR^{1+3},\quad
 \eta=\operatorname{diag}(1,-1,-1,-1),\quad
 M_{T_*}=[0,T_*]\times\RR^3,\quad
 \Sigma=\{t=0\},
\]
\[
 \iota:\mathcal L\longrightarrow\mathcal M^\circ,
 \qquad
 \iota^*G=-\eta,\qquad
 t=c_{\mathrm h}(T-T_\Sigma).
\]
The chart is injective on its declared domain, and the
retained compact reserve is disjoint from \(\mathcal W\).

All global free Green operators below act on \(\widehat M\),
not on the regular conical product.
\end{assumption}

\begin{proposition}[No global cone Cauchy claim]
\label{prop:nonGH}
The regular conical product is not globally hyperbolic.
\end{proposition}

\begin{proof}
Choose a fixed radial ray, \(r_0>0\), \(z_0\), and
\(A>r_0/c_{\mathrm h}\).
Between the events with coordinates
\((-A,r_0,\varphi_0,z_0)\) and
\((A,r_0,\varphi_0,z_0)\), the causal diamond contains
\[
 (0,r_0/n,\varphi_0,z_0),\qquad n\ge1.
\]
Their only completion limit lies at the excluded apex.
They have no convergent subsequence in the regular product,
so that diamond is not compact.
\end{proof}

The signature relation is a bookkeeping identity between
the inherited analytical and host conventions.
The formal metrics used below are field-theoretic variables;
they do not replace \(G\).

\subsection{Plateau and the additional compact vertex}

Retain the smooth compact scalar switch \(g\), the observation
test \(f\), the open plateau \(U\), and the onset
\[
 \sigma_\varepsilon(t)=\chi(t/\varepsilon),
 \qquad 0<\varepsilon<\delta,
\]
where \(\chi\) is smooth, zero for arguments at most zero,
and one for arguments at least one.
Assume
\[
 \overline U\Subset\mathcal L,\qquad
 g=\sigma_\varepsilon=1
       \quad\hbox{on a neighborhood of }\overline U.
\]
The relative compact gauge parameters are
\[
 \grel=C_c^\infty(U;T\widehat M).
\]

Choose once and for all
\begin{equation}
 a\in C_c^\infty(U;\RR),\qquad a\ge0,\qquad a\not\equiv0.
 \label{eq:a}
\end{equation}
This is a new coupling profile, not a new regulator.
Its support is inside the already retained plateau and
therefore requires no enlarged host chart or smaller width.

\begin{remark}[Meaning of support inside a fixed-width prescription]
An onset is not a compact spacetime region.
Here the precise support condition is
\[
 \supp a\Subset U\Subset\{g=\sigma_\varepsilon=1\}.
\]
We do not place the new interaction in the transition layer
\(0<t<\varepsilon\), where plateau identities would require
a different audit. No \(\varepsilon\to0\) is taken.
\end{remark}

A nonzero profile satisfying \eqref{eq:a} exists in any
nonempty \(U\): choose a Euclidean ball whose closure lies
in \(U\), and a nonnegative smooth bump supported there.

For the optional finite-time response comparison in
Section~\ref{sec:response}, we additionally require
\begin{equation}
 J_\eta^-(K_{\mathrm{out}})\cap M_{T_*}
                  \Subset\mathcal L
 \label{eq:responsecertificate}
\end{equation}
for the specified output test support \(K_{\mathrm{out}}\).
This is a source-specific support certificate.
It is not inferred from the existence of a finite
reporting port.

\subsection{Density transport}

For a compact scalar density \(L(x)\dd^4x\), its expression
in host coordinates \(y\), with \(x=\iota^{-1}(y)\), is
\[
 L(\iota^{-1}(y))
       \abs{\det D\iota^{-1}(y)}\,\dd^4y.
\]
Antifields are transported as dual tensor densities.
Compact changes of variables preserve the variational
pairing and integration by parts.
The Jacobian is not omitted in a non-Cartesian presentation.

All local constructions below can therefore be represented
in the supported host chart. This is transport of an
already declared completion calculation, not a new global
conical inverse.

%=====================================================================
\section{Free normalization and the exact free field limit}
\label{sec:free}
%=====================================================================

\subsection{Fields, parities, and operators}

The independent fields are
\[
 \Phi=(r_{\mu\nu},\phi,c^\mu,\bar c_\mu,b_\mu).
\]
Their parities and ghost numbers are
\[
\begin{array}{c|ccccc}
 \Phi&r&\phi&c&\bar c&b\\ \hline
 |\Phi|&0&0&1&1&0\\
 \gh(\Phi)&0&0&1&-1&0.
\end{array}
\]
All are smooth off shell.
Odd fields are interpreted in a supercommutative coefficient
algebra; finite exterior algebras will be used for numerical
coefficient extraction.

Set
\[
 \Box=\partial_t^2-\Delta,\qquad \Pm=\Box+m^2,\qquad m\ge0.
\]
For a covariant symmetric tensor \(r\), let
\[
 (\Ptr r)_{\mu\nu}
       =r_{\mu\nu}-\frac12\eta_{\mu\nu}\tr_\eta r,
 \qquad
 (\Cop r)^\mu=\partial_\nu(\Ptr r)^{\nu\mu},
\]
\[
 (Kc)_{\mu\nu}=\partial_\mu c_\nu+\partial_\nu c_\mu.
\]
Compact integration by parts gives
\begin{equation}
 \Cop^*=-\frac12\Ptr K,\qquad
 \Cop K=\Box,\qquad \Ptr^2=I.
 \label{eq:freeidentities}
\end{equation}

Use the parent normalization
\begin{equation}
 A=\frac12G_{\mathrm{lin}},\qquad
 AK=0,\qquad
 A+\frac12\Cop^*\Cop
       =\Pgr=-\frac14\Ptr\Box.
 \label{eq:A}
\end{equation}
Here the linearized Einstein tensor is defined by
\[
\begin{split}
 (G_{\mathrm{lin}}r)_{\mu\nu}
 =\frac12\bigl(
 &\partial_\rho\partial_\mu r^\rho{}_\nu
 +\partial_\rho\partial_\nu r^\rho{}_\mu
 -\Box r_{\mu\nu}
 -\partial_\mu\partial_\nu\tr r\\
 &-\eta_{\mu\nu}
    (\partial_\rho\partial_\sigma r^{\rho\sigma}
                                      -\Box\tr r)\bigr).
\end{split}
\]
Equations \eqref{eq:A} follow by substitution.
This is the normalization of X-G/X-G-ST
\cite{XG,XGST}.

\subsection{The independent auxiliary field}

Define the left odd derivation
\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}
It commutes with total spacetime derivatives.

The gauge fermion and free action are
\begin{align}
 \Psi_0&=\int\bar c_\mu((\Cop r)^\mu-b^\mu)\dd^4x,
 \label{eq:Psi0}\\
 I_{\mathrm{free}}
 &=\frac12\pair r{Ar}
       -\frac12\pair\phi{\Pm\phi}+s_0\Psi_0
 \notag\\
 &=\frac12\pair r{Ar}
       -\frac12\pair\phi{\Pm\phi}
   +\int\bigl(b\cdot\Cop r-b^2-\bar c\cdot\Box c\bigr)\dd^4x.
 \label{eq:Ifree}
\end{align}
These are variational actions with compact variations,
not an assumption that their values are finite on every
ambient smooth configuration.

\begin{lemma}[Free off-shell identities]
\label{lem:free}
The differential \(s_0\) is nilpotent off shell.
It preserves \(I_{\mathrm{free}}\) modulo integrated
divergences. Algebraic elimination of \(b\) gives
\[
 b=\frac12\Cop r,\qquad
 b\cdot\Cop r-b^2=\frac14(\Cop r)^2,
\]
and hence the metric operator \(\Pgr\).
\end{lemma}

\begin{proof}
The square vanishes on each independent generator in
\eqref{eq:s0}; the square of an odd derivation is an even
derivation. The metric variation vanishes by \(AK=0\),
the scalar is inert, and \(s_0^2\Psi_0=0\).

Completing the square gives the auxiliary formula.
Equation~\eqref{eq:A} identifies the reduced metric operator.
\end{proof}

Eliminating \(b\) before defining the differential would give
\(s_0^2\bar c=\Box c/2\), only an on-shell identity.
We do not make that elimination.

\subsection{Completion Green operators and causal compatibility}

Let \(E_0^{\mathrm R/\mathrm A}\) and
\(E_m^{\mathrm R/\mathrm A}\) be the positive equation
inverses of \(\Box\) and \(\Pm\) on \(\widehat M\).
Their existence and causal support are standard on this
globally hyperbolic completion \cite{BGP}.

The metric--auxiliary inverse is
\begin{equation}
 \mathcal G_{rb}^{\mathrm R/\mathrm A}
 =
 \begin{pmatrix}
 -4\Ptr E_0^{\mathrm R/\mathrm A}
                    &K E_0^{\mathrm R/\mathrm A}\\
 -2\Cop\Ptr E_0^{\mathrm R/\mathrm A}&0
 \end{pmatrix}.
 \label{eq:Grb}
\end{equation}
The scalar action-Hessian inverse is
\(-E_m^{\mathrm R/\mathrm A}\).

For clarity about odd signs, place Grassmann variations
on the left. The ghost Euler derivatives are
\[
 E_c=\Box\bar c,\qquad E_{\bar c}=-\Box c.
\]
In this source-dual convention their inverse is
\[
 \begin{pmatrix}
 0&-E_0^{\mathrm R/\mathrm A}\\
 E_0^{\mathrm R/\mathrm A}&0
 \end{pmatrix}.
\]
Changing the order or parity convention for the odd source
dual changes printed inverse matrices. It does not license
changing the action or the causal brackets.

Put \(\delta_\mu=E_\mu^{\mathrm R}-E_\mu^{\mathrm A}\).
The causal brackets used here have blocks
\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\Cop\Ptr\delta_0\\
 c\bar c&-\delta_0&\bar c c&\delta_0
\end{array}
 \label{eq:causalblocks}
\end{equation}
and \(\Delta_{bb}=0\).
Thus the scalar sign and the odd signs are not inferred
from an unsigned array.

\begin{proposition}[Free causal BRST recovery]
\label{prop:freecausal}
The free equation ideal
\[
 Ar+\Cop^*b=0,\quad
 \Cop r-2b=0,\quad
 \Pm\phi=0,\quad
 \Box c=\Box\bar c=0
\]
is \(s_0\)-stable.
The normalized causal relations
\[
 [\Phi_A(u),\Phi_B(v)]_{\mathrm{gr}}
       =i\hbar\pair u{\Delta_{AB}v}\,1
\]
are also stable.
Consequently \(s_0\) induces the nilpotent free causal
BRST derivation of X-G-ST.
\end{proposition}

\begin{proof}
Solving \(b=(\Cop r-j)/2\) in the sourced bosonic equations
reduces them to
\[
 \Pgr r=J+\frac12\Cop^*j,
\]
which proves \eqref{eq:Grb}.
The equation variations are \(AKc=0\),
\(\Box c\), and \(\Box b\).
The last follows from the divergence of
\(Ar+\Cop^*b=0\); the scalar and ghost equations are
otherwise inert.

For the brackets, the only cancellation between nonzero
blocks is the variation of the initially zero
\(r\)--\(\bar c\) relation:
\[
 \{Kc,\bar c\}+[r,b]
        =-i\hbar K\delta_0+i\hbar K\delta_0=0.
\]
The remaining generator pairs use zero cross blocks or
\(\Delta_{bb}=0\).
The odd--odd blocks have the required graded symmetry
because \(\delta_0^T=-\delta_0\).
These generator checks prove compatibility.
\end{proof}

The \(\hbar\) in this comparison labels the inherited free
causal relations only. No \(\hbar\)-dependent term will
enter the new action or classical BV differential.

%=====================================================================
\section{The relative local differential and its signed BV lift}
\label{sec:kinematics}
%=====================================================================

\subsection{Domain and background split}

Retain fixed smooth classical backgrounds \(h,k\), and put
\begin{equation}
 \bar q=\eta+\kappa h+\kappa^2k,\qquad
 q=\bar q+\kappa r.
 \label{eq:split}
\end{equation}
Neither \(q\) nor \(\bar q\) is the host metric \(G\).

Let \(\mathscr P(U)[[\kappa]]\) be the algebra of local
polynomial jet expressions with supercommutative
coefficients. Formal inverse-metric and determinant
coefficients are allowed; each fixed coefficient has
finite field degree and finite jet order.

Define
\begin{align}
 sr&=Kc+\kappa\Lie_c(h+r)+\kappa^2\Lie_ck,
 \label{eq:sr}\\
 s\phi&=\kappa c^\mu\partial_\mu\phi,\qquad
 sc^\mu=\kappa c^\nu\partial_\nu c^\mu,
 \label{eq:sphi}\\
 s\bar c&=b,\qquad sb=0,\qquad sh=sk=0.
 \label{eq:snonminimal}
\end{align}
Use the left graded Leibniz rule.
These are the relative field rules of X-H-ST, restated
rather than inferred from its title \cite{XHST}.
In particular,
\[
 s=s_0+\kappa s_1+\kappa^2s_2,\qquad
 sq=\kappa\Lie_cq.
\]

\begin{theorem}[Off-shell field nilpotence]
\label{thm:kinematic}
The derivation \(s\) satisfies \(s^2=0\) on the local
formal field algebra.
Its leading field operator is exactly \eqref{eq:s0}.
\end{theorem}

\begin{proof}
For the odd vector field \(c\), set
\(w^\mu=c^\nu\partial_\nu c^\mu\).
Expanding the Lie derivative with the graded Leibniz
rule gives
\[
 \Lie_c^2=\Lie_w.
\]
The second-derivative terms cancel by antisymmetry
of the ghost coefficients.
Therefore
\[
 s^2q
    =\kappa\Lie_{sc}q-\kappa\Lie_c(sq)
    =\kappa^2(\Lie_w-\Lie_c^2)q=0.
\]
Since \(\bar q\) is fixed, \(s^2r=0\) coefficientwise.

Similarly,
\[
 s^2\phi
 =\kappa(sc^\mu)\partial_\mu\phi
        -\kappa c^\mu\partial_\mu(s\phi)=0.
\]
Expansion proves \(s(c^\nu\partial_\nu c^\mu)=0\)
after substituting \(sc=\kappa c\partial c\).
The nonminimal pair is nilpotent by definition.
The generator calculation extends to the algebra.
\end{proof}

The theorem concerns the classical field algebra.
It asserts neither interacting equation descent nor
interacting quantum BRST.

For compact evaluation configurations, a common support
restriction on all fields is stable under these field
rules. In particular, a compact antighost restriction
must include the same restriction on \(b\).
The principal algebra here instead uses unrestricted
smooth germs and compact tests.

\subsection{Multilocal functionals and antifields}

Let \(\Fml(U)\) be the supercommutative algebra generated
by compactly supported local functionals under ordinary
multiplication. A product of two single integrals is
generally multilocal, not another single local integral.

Local actions are horizontal four-forms modulo horizontal
divergences. Their Euler derivatives act on
\(\Fml(U)\). This is the local variational interpretation
of the classical BV bracket \cite{FRBV}.

Adjoin antifields with
\[
 |\Phi_A^*|=|\Phi^A|+1\pmod2,\qquad
 \gh(\Phi_A^*)=-1-\gh(\Phi^A).
\]
They are dual tensor densities.
Fix the canonical odd bracket by
\begin{equation}
 (\Phi^A(x),\Phi_B^*(y))
      =\delta^A_B\delta(x-y),\qquad
 (F,G)=-(-1)^{(|F|+1)(|G|+1)}(G,F),
 \label{eq:bracket}
\end{equation}
with the odd biderivation and integration-by-parts rules.

For the even Hamiltonian acting on the left,
\(QF=(\mathcal S,F)\), define
\begin{equation}
 \Hkin=\sum_A(-1)^{|\Phi^A|}
                    \int\Phi_A^*s\Phi^A\,\dd^4x.
 \label{eq:H}
\end{equation}
Explicitly,
\[
\begin{split}
 \Hkin=\int\Bigl[
 &r^{*\mu\nu}
      \bigl((Kc)_{\mu\nu}
       +\kappa(\Lie_c(h+r))_{\mu\nu}
       +\kappa^2(\Lie_ck)_{\mu\nu}\bigr)\\
 &+\kappa\phi^*c^\mu\partial_\mu\phi
       -\kappa c_\mu^*c^\nu\partial_\nu c^\mu
       -\bar c^{*\mu}b_\mu
                   \Bigr]\dd^4x.
\end{split}
\]

\begin{lemma}[Generator-level sign audit]
\label{lem:sign}
For every field generator,
\[
 (\Hkin,\Phi^A)=s\Phi^A.
\]
Moreover, \((\Hkin,\Hkin)=0\) modulo divergences.
\end{lemma}

\begin{proof}
If \(p=|\Phi^A|\) and \(R^A=s\Phi^A\), the odd
biderivation rule gives
\[
 (\Phi_A^*R^A,\Phi^A)=(-1)^pR^A.
\]
The factor in \eqref{eq:H} therefore gives \(R^A\)
for both even and odd generators.
In particular,
\[
 (-\bar c^{*\mu}b_\mu,\bar c_\nu)=b_\nu,
\]
so an unsigned common antifield coupling would be wrong
in this convention.

The cotangent-lift calculation gives
\(\tfrac12(\Hkin,\Hkin)\) linear in antifields, with
coefficients \(s^2\Phi^A\), after integration by parts.
They vanish by Theorem~\ref{thm:kinematic}.
There is no antifield-independent term.
\end{proof}

The signed lift agrees with X-H-ST's convention.
No BV Laplacian or quantum product is used.

%=====================================================================
\section{A finite reference action through third order}
\label{sec:referenceaction}
%=====================================================================

\subsection{Explicit coefficient specification}

For any formal metric \(p\), put
\[
 J_p=\sqrt{-\det p},\qquad
 \Gamma^\rho_{\mu\nu}[p]
  =\frac12p^{\rho\lambda}
       (\partial_\mu p_{\lambda\nu}
       +\partial_\nu p_{\lambda\mu}
       -\partial_\lambda p_{\mu\nu}),
\]
\[
 R_{\mu\nu}[p]
 =\partial_\rho\Gamma^\rho_{\mu\nu}
  -\partial_\nu\Gamma^\rho_{\mu\rho}
  +\Gamma^\rho_{\rho\lambda}\Gamma^\lambda_{\mu\nu}
  -\Gamma^\rho_{\nu\lambda}\Gamma^\lambda_{\mu\rho}.
\]
Write
\[
 \mathscr E[p]=J_p\,p^{\mu\nu}R_{\mu\nu}[p],
 \qquad
 L_p[\phi]=\frac12J_p
       (p^{\mu\nu}\partial_\mu\phi\partial_\nu\phi-m^2\phi^2).
\]

The generalized invariant action is
\begin{equation}
 I_{\mathrm{inv}}
 =-\frac1{2\kappa^2}
     \int(\mathscr E[q]-\mathscr E[\bar q])\dd^4x
                +\int L_q[\phi]\dd^4x.
 \label{eq:Iinv}
\end{equation}
The sole negative-power coefficient is a divergence:
\[
 -\frac1{2\kappa}
  (\partial_\mu\partial_\nu r^{\mu\nu}-\Box\tr r).
\]
It is removed as a variational class before passing to
a nonnegative formal coefficient ring.

Let \(\bar\nabla\) be the connection of \(\bar q\), and set
\[
 C_{\bar q}^{\mu}[r]
 =\bar\nabla_\nu\left(
    \bar q^{\nu\alpha}\bar q^{\mu\beta}r_{\alpha\beta}
       -\frac12\bar q^{\nu\mu}
                        \bar q^{\alpha\beta}r_{\alpha\beta}\right),
\]
\[
 \Psi_{\bar q}
 =\int J_{\bar q}\bar c_\mu
             (C_{\bar q}^{\mu}[r]-\bar q^{\mu\nu}b_\nu)\dd^4x,
 \qquad
 I=I_{\mathrm{inv}}+s\Psi_{\bar q}.
 \label{eq:referencegauge}
\]
The auxiliary field remains independent.

For \(j=0,1,2,3\), define the local variational coefficients
\begin{equation}
 I_j
 =\int\left[
 -\frac12[\kappa^{j+2}]
                 (\mathscr E[q]-\mathscr E[\bar q])
 +[\kappa^j]L_q[\phi]
 +[\kappa^j](s\Psi_{\bar q})_{\mathrm{dens}}
                           \right]\dd^4x.
 \label{eq:Ij}
\end{equation}
Coefficient extraction means differentiation at zero
divided by the relevant factorial.
Thus \eqref{eq:Ij} is a finite prescription, not an
unnamed cubic or quartic vertex.

\begin{remark}[Why the fifth geometric coefficient is retained]
The order-\(\kappa^3\) gravitational action coefficient
requires \([\kappa^5]\mathscr E[q]\).
Truncating the geometry at fourth order would generally
invalidate the third-order master identity.
The division by \(\kappa^2\) in \eqref{eq:Iinv} is handled
before quotienting by \(\kappa^4\), not by inverting
a nilpotent element.
\end{remark}

Let \(\Hkin=\sum_{j=0}^2\kappa^jH_j\), and put
\[
 S_j=I_j+H_j\quad(j=0,1,2),\qquad S_3=I_3,
\]
\begin{equation}
 \Sref^{[3]}=\sum_{j=0}^3\kappa^jS_j.
 \label{eq:Sref}
\end{equation}

\subsection{Reference master identity}

\begin{lemma}[Relative reference symmetry]
\label{lem:referenceCME}
The natural reference action satisfies \(sI=0\)
as a local variational class.
Consequently
\[
 \frac12(\Sref^{[3]},\Sref^{[3]})
                         =0\pmod{\kappa^4}.
\]
\end{lemma}

\begin{proof}
Naturality of the curvature and scalar densities gives
\[
 s\mathscr E[q]
       =\kappa\partial_\mu(c^\mu\mathscr E[q]),
 \qquad
 sL_q=\kappa\partial_\mu(c^\mu L_q).
\]
The background subtraction is \(s\)-inert.
Also \(s(s\Psi_{\bar q})=0\).
Thus \(sI=0\) modulo divergences.

For the formal reference Hamiltonian \(I+\Hkin\),
\[
 \frac12(I+\Hkin,I+\Hkin)
       =sI+\frac12(\Hkin,\Hkin)=0.
\]
Extracting the first four nonnegative coefficients gives
the asserted finite-order identity.
\end{proof}

The formal reference series is used only to prove
identities between the explicitly defined finite
coefficients. No convergence or numerical all-orders
solution is claimed.

\subsection{The tadpole correction to the free-limit assertion}

\begin{proposition}[Precise zeroth-order comparison]
\label{prop:tadpole}
At order zero,
\begin{equation}
 I_0=I_{\mathrm{free}}+\pair r{Ah}
                                      \quad\bmod\text{ divergences}.
 \label{eq:tadpole}
\end{equation}
The free Hessian and the field differential \(s_0\)
are unchanged for all fixed \(h,k\).
The full zeroth-order BV Hamiltonian equals the free
one in the centered sector \(h=k=0\).
For a general off-shell \(h\), it instead retains the
affine source \(Ah\) in the metric antifield equation.
\end{proposition}

\begin{proof}
The quadratic gravitational coefficient is
\(\frac12\pair{h+r}{A(h+r)}\).
Subtracting its background-only term gives
\[
 \frac12\pair r{Ar}+\pair r{Ah}.
\]
The scalar and gauge-fixing coefficients at zero are
those of \eqref{eq:Ifree}.
The extra term is linear in \(r\), so it changes no
Hessian and no field rule.
Its bracket with \(r^*\) is \(Ah\).
\end{proof}

This is not a removable bookkeeping term.
Erasing it for arbitrary \(h\) would change the reference
action and confuse it with a different source prescription
\cite{XHST}.

Thus the phrase ``free-limit recovery'' has two distinct,
proved meanings here:
\begin{enumerate}
\item on fields and the homogeneous free causal carrier,
the limit is exactly \(s_0\);
\item on the complete BV algebra, exact free recovery
holds in the centered background sector.
\end{enumerate}
No unconditional full-BV free-limit claim is made for
arbitrary fixed off-shell backgrounds.

%=====================================================================
\section{Section A: the compact classical interaction}
\label{sec:interaction}
%=====================================================================

\begin{definition}[The selected deformation]
\label{def:deformation}
With \(a\) fixed by \eqref{eq:a}, define
\begin{equation}
 V_a=\frac1{4!}\int a\phi^4\dd^4x,\qquad
 \Snew=\Sref^{[3]}+\lambda\kappa^3V_a,
 \label{eq:newaction}
\end{equation}
over
\[
 R_3=\RR[\lambda][\kappa]/(\kappa^4).
\]
The full classical BV differential is
\[
 \Qnew F=(\Snew,F).
\]
The field differential is denoted \(s\), not \(Q\).
The latter also contains the antifield equations.
\end{definition}

\begin{theorem}[Finite-order interacting classical BV]
\label{thm:deformation}
The Hamiltonian \eqref{eq:newaction} satisfies
\begin{equation}
 \frac12(\Snew,\Snew)=0\quad\text{in }R_3,
 \qquad
 (\Qnew)^2=0\quad\text{in }R_3.
 \label{eq:newCME}
\end{equation}
It is a classical deformation with no quantum term.
Its field rules remain \eqref{eq:sr}--\eqref{eq:snonminimal},
while its scalar antifield equation changes by
\begin{equation}
 (\Qnew-Q_{\mathrm{ref}}^{[3]})\phi^*(x)
                =\frac{\lambda\kappa^3}{6}a(x)\phi(x)^3.
 \label{eq:antifieldchange}
\end{equation}
Every other fundamental generator has zero change from
the additional vertex.
\end{theorem}

\begin{proof}
The vertex has no antifields, so \((V_a,V_a)=0\).
Since \(s_0\phi=0\),
\[
 (S_0,V_a)=s_0V_a=0.
\]
All other cross brackets with the added
\(\kappa^3V_a\) have order at least four.
Lemma~\ref{lem:referenceCME} therefore gives the first
identity in \eqref{eq:newCME}.

The odd Jacobi identity yields
\[
 (\Qnew)^2F
 =\left(\frac12(\Snew,\Snew),F\right),
\]
giving the second identity.

An antifield-independent vertex changes no field rule.
Its only nonzero fundamental antifield bracket is its
Euler derivative with respect to \(\phi\), namely
\(a\phi^3/6\).
\end{proof}

The interaction is not merely a new status word: it
changes a nonlinear Euler equation.
No assertion is made that this deformation represents
a new nontrivial BV cohomology class modulo all possible
canonical transformations.

\begin{proposition}[Support and continuity of the new vertex]
\label{prop:support}
The additional action, its first variations, and the
new local Euler term have support in \(\supp a\).
Their coefficient pairings are continuous polynomial
maps in the compact-open smooth topology.

Bracketing a compact local functional with the new
vertex produces a compact local functional.
Finite multilocal products remain in \(\Fml(U)\).
\end{proposition}

\begin{proof}
Every new variation contains a factor \(a\) or one of
its derivatives. Differentiation does not enlarge its
support.
On a compact set, the integral and its derivatives are
bounded by finite products of smooth seminorms.

The variational bracket uses a finite number of Euler
derivatives and integrations by parts.
Smooth classical configurations have no undefined
pointwise field products.
The super-Leibniz rule proves the multilocal statement.
\end{proof}

If an insertion is multiplied by an independent spectator
functional, the support of the entire multilocal product
also includes that spectator's support.
Only the additional interaction vertex is claimed to
be supported in \(\supp a\).

\subsection{Order protection}

\begin{lemma}[Protection of lower coefficients]
\label{lem:orderprotection}
Let two action or operator data sets differ by
\(\kappa^NV\), with \(N\ge1\).
An expression obtained by finitely many additions,
multiplications, coefficientwise linear maps, derivatives,
and formal inversions with invertible order-zero part
has the same coefficients below order \(N\), provided
the construction uses no negative coupling powers.
\end{lemma}

\begin{proof}
All listed operations preserve the filtration
\(\kappa^j\).
For a formal inverse, the coefficient recursion starts
with the common order-zero inverse and determines its
\(j\)-th coefficient from data of orders at most \(j\).
Induction proves the assertion.
\end{proof}

\begin{corollary}[Exact predecessor comparison]
\label{cor:recoverH}
The action and full BV differential agree with X-H-ST's
stated relative construction through order two.
At \(\lambda=0\), the present action is its explicitly
specified reference continuation through order three.
At \(\kappa=0\), Theorem~\ref{prop:freecausal} and
Proposition~\ref{prop:tadpole} give the precise free-limit
statements.
\end{corollary}

\begin{proof}
The additional vertex starts at order three, and
\eqref{eq:Ij} reproduces the predecessor coefficients
for \(j=0,1,2\).
The remaining assertions are direct substitutions.
\end{proof}

\subsection{A compact-source response comparison}
\label{sec:response}

A differential is not itself a Cauchy evolution theorem.
Nevertheless, the new source has a simple conditional
response interface.

\begin{proposition}[Incremental scalar response]
\label{prop:response}
In the centered sector, suppose a smooth formal reference
Euler solution and a deformed Euler solution have the
same free input and zero difference Cauchy data.
Write their scalar difference as
\[
 \phi_\lambda-\phi_{\mathrm{ref}}
                 =\kappa^3\delta\phi_3\pmod{\kappa^4}.
\]
Then its new scalar contribution is uniquely
\begin{equation}
 \delta\phi_3
        =\frac{\lambda}{6}
                    E_m^{\mathrm R}(a\phi_0^3).
 \label{eq:response}
\end{equation}
The added contribution to the metric--auxiliary response
at this order is zero.
The new scalar dependence is contained in
\(J_\eta^+(\supp a)\).
\end{proposition}

\begin{proof}
Coefficients below order three agree by
Lemma~\ref{lem:orderprotection}.
At order three, subtraction of the two scalar Euler
equations leaves
\[
 -\Pm\delta\phi_3+\frac{\lambda}{6}a\phi_0^3=0.
\]
The source is smooth and compact.
The completion retarded inverse proves
\eqref{eq:response}, uniqueness, and causal support.
The vertex has no metric or auxiliary derivative.
\end{proof}

The proposition constructs the incremental response once
the two formal Euler problems are compared.
It does not claim global nonlinear existence,
constraint propagation, a physical gravitational state,
or transport of the complete relative BV complex onto
a free causal algebra.
Under \eqref{eq:responsecertificate}, its specified
compact output pairing has a supported laboratory
presentation.

%=====================================================================
\section{The finite lift and its explicit next residual}
\label{sec:residual}
%=====================================================================

A statement in \(R_3\) does not assign canonical values
to fourth-order coefficients.
To discuss them, fix the ordinary polynomial lift
\[
 \widetilde{\mathcal S}_\lambda
     =S_0+\kappa S_1+\kappa^2S_2
                          +\kappa^3(S_3+\lambda V_a)
\]
with no additional terms.
Every higher residual below refers to this named lift.

\begin{definition}[Lift residual]
Define
\[
 \Rnew=\frac12
   (\widetilde{\mathcal S}_\lambda,
                         \widetilde{\mathcal S}_\lambda),
 \qquad
 \Rref=\frac12
   (\widetilde{\mathcal S}_0,
                         \widetilde{\mathcal S}_0).
\]
Also define the compact odd functional
\begin{equation}
 \mathcal B_a
   =\frac16\int a\phi^3c^\mu\partial_\mu\phi\,\dd^4x.
 \label{eq:B}
\end{equation}
\end{definition}

\begin{theorem}[Exact new residual of the chosen lift]
\label{thm:residual}
As ordinary finite polynomial variational functionals,
\begin{equation}
 \Rnew-\Rref=\lambda\kappa^4\mathcal B_a.
 \label{eq:Rdifference}
\end{equation}
In particular,
\begin{align}
 [\kappa^4]\Rnew
     &=(S_1,S_3)+\frac12(S_2,S_2)+\lambda\mathcal B_a,
 \label{eq:R4}\\
 [\kappa^5]\Rnew&=(S_2,S_3),
 \label{eq:R5}\\
 [\kappa^6]\Rnew&=0.
 \label{eq:R6}
\end{align}
Coefficients through order three vanish.
The exact lift obeys
\begin{equation}
 (\widetilde{\mathcal S}_\lambda,\Rnew)=0.
 \label{eq:consistency}
\end{equation}
\end{theorem}

\begin{proof}
All \(I_j\) and \(V_a\) are antifield independent.
Only \(\Hkin\) can bracket nontrivially with \(V_a\).
Its scalar field rule gives
\[
 (\Hkin,V_a)=sV_a
                  =\kappa\mathcal B_a.
\]
There is no \(s_2\)-action on \(\phi\).
This proves \eqref{eq:Rdifference} exactly.
Expanding the finite polynomial proves
\eqref{eq:R4}--\eqref{eq:R5}.
The third coefficient \(S_3\) and \(V_a\) have no
antifields, proving \eqref{eq:R6}.
The lower coefficients vanish by
Theorem~\ref{thm:deformation}.

Equation~\eqref{eq:consistency} is the odd Jacobi identity
applied three times to the even Hamiltonian.
\end{proof}

This is a classical consistency identity.
It is not a quantum anomaly equation.

\subsection{A nonzero evaluated coefficient}

Fix a nonzero exterior generator \(\vartheta\).
For this diagnostic only, use the centered background and
the smooth evaluation
\[
 r=h=k=b=\bar c=\Phi^*=0,\qquad
 \phi=x^1,\qquad
 c=\vartheta\,a(x)(x^1)^3\,\partial_1.
\]
The ghost coefficient is compactly supported in \(U\).

\begin{proposition}[A visible fourth-order boundary]
\label{prop:positiveB}
On this evaluation,
\begin{equation}
 [\vartheta]\mathcal B_a
      =\frac16\int a(x)^2(x^1)^6\,\dd^4x>0.
 \label{eq:positiveB}
\end{equation}
Also \([\kappa^4]\Rref=0\) on this evaluation.
Thus for \(\lambda>0\) the displayed fourth-order
coefficient of the chosen polynomial lift is nonzero.
\end{proposition}

\begin{proof}
Substitution in \eqref{eq:B} gives
\eqref{eq:positiveB}.
A nonzero smooth \(a\) is nonzero on an open set,
which cannot be contained in the hyperplane \(x^1=0\).
The integral is therefore strictly positive.

When \(h=k=0\), the order-three reference action contains
a gravitational term of degree five in \(r\) and a scalar
term with at least three metric factors.
The gauge-fixing action has no order-three term in this
centered split.
Consequently \(I_3\) and all its first derivatives vanish
at \(r=0\).
Moreover \(H_2=0\) in that sector, so
\((S_2,S_2)=0\).
Equation~\eqref{eq:R4} at \(\lambda=0\) proves the claim.
\end{proof}

Positivity here means positivity of the specified real
coefficient of \(\vartheta\), not of an odd functional.

The result disproves exact master closure of this
specified polynomial lift.
It does not exclude every higher-order completion:
adding a fourth-order term changes the fourth-order
residual.

\subsection{Why a covariant density does not erase all switching terms}

Let
\[
 V_1=\frac1{4!}\int
          \frac12\tr_\eta(h+r)\,a\phi^4\,\dd^4x.
\]
This is the first volume-density correction to the
compact potential.

\begin{proposition}[Next-order density diagnostic]
\label{prop:density}
Modulo compact divergences,
\begin{equation}
 \mathcal B_a+s_0V_1
       =-\frac1{4!}\int
                 (\partial_\mu a)c^\mu\phi^4\,\dd^4x.
 \label{eq:cutoffresidual}
\end{equation}
\end{proposition}

\begin{proof}
Since
\(s_0(\tr r/2)=\partial_\mu c^\mu\),
\[
 s_0V_1=\frac1{4!}\int
                      a(\partial_\mu c^\mu)\phi^4\,\dd^4x.
\]
Also
\(\mathcal B_a=\int a c^\mu\partial_\mu(\phi^4/4!)\).
One integration by parts proves
\eqref{eq:cutoffresidual}.
\end{proof}

Thus a future fourth-order continuation must audit both
density completion and the fixed profile \(a\).
The plateau of \(g,\sigma_\varepsilon\) does not make
\(a\) constant.
No spurion, compensator, changed ghost algebra, or
fourth-order repair is introduced here.

%=====================================================================
\section{No automatic descent to the unchanged free carrier}
\label{sec:descent}
%=====================================================================

\begin{theorem}[Persistence of the free-equation non-descent]
\label{thm:nondescent}
The natural positive-order scalar field rule
\(s\phi=\kappa c^\mu\partial_\mu\phi\) does not preserve
the unchanged free scalar equation ideal.
The additional vertex \(\lambda\kappa^3V_a\) does not
repair this failure.
\end{theorem}

\begin{proof}
For arbitrary smooth coefficient fields,
\[
 [\Pm,c^\mu\partial_\mu]\phi
  =(\Box c^\mu)\partial_\mu\phi
        +2(\partial^\nu c^\mu)
                              \partial_\nu\partial_\mu\phi.
\]
Choose \(p>0\), \(\omega=\sqrt{p^2+m^2}\), and
\[
 \phi_0=\cos(\omega t-px^1),\qquad
 c=\vartheta t\,\partial_1.
\]
Both \(\Pm\phi_0=0\) and \(\Box c=0\), but
\[
 \Pm(c^\mu\partial_\mu\phi_0)
      =2\vartheta\,\partial_t\partial_1\phi_0
      =2\vartheta p\omega\cos(\omega t-px^1),
\]
which has a nonzero compact pairing on a sufficiently
small test region.

Thus the variation of a free equation is nonzero on a
free configuration and cannot belong to that unchanged
equation ideal.
The new vertex changes no field rule and begins only
at order three, whereas this failure occurs at order one.
\end{proof}

This is the same type of domain obstruction distinguished
in X-H-ST, now exhibited without identifying compact
relative ghosts with nonzero compact free ghost solutions
\cite{XHST}.

For the relative local comparison, one can choose a compact
ghost coefficient equal to \(t\partial_1\) near the compact
observation test. It gives the same local non-descent
diagnostic, but it is not a homogeneous free ghost on
the full completion.

A compactly supported smooth solution of the homogeneous
wave equation on spacetime is zero: take a Cauchy surface
before its support and use uniqueness.
Consequently the compact relative ghost sector and the
unrestricted free ghost carrier cannot be identified.

The construction here is therefore an interacting
\emph{off-shell classical relative BV complex}.
It is not an interacting derivation of the unchanged
free causal algebra.

%=====================================================================
\section{Section C: diagnostics preserved in their exact scopes}
\label{sec:diagnostics}
%=====================================================================

\subsection{Reciprocity is a separate classical obstruction}

Let \(\mathsf Q_w\) denote the first background coefficient
of the natural gauge-fixed metric Hessian, with
\[
 \mathsf Q_{K\xi}
       =\Lie_\xi^{\mathrm{dens}}\Pgr
                         -\Pgr\Lie_\xi^{\mathrm{cov}}.
 \label{eq:typedcommutator}
\]
The two Lie derivatives have different tensor types.
Define the frozen source-switched operator
\[
 \mathsf N_w=\sigma_\varepsilon\mathsf Q_w.
\]
For free metric solutions \(u,v\), its reciprocity defect is
\[
 D_w(u,v)
    =\pair v{\mathsf N_wu}-\pair u{\mathsf N_wv}.
\]
These are the X-G/X-G-ST diagnostic objects
\cite{XG,XGST}, not the Euler operator of
\eqref{eq:newaction}.

Choose a nonnegative compact spatial bump \(\chi_0\)
with \(M_\chi=\int\chi_0\,\dd^3x>0\), and choose
\(\zeta\in C_c^\infty((0,\varepsilon))\), nonnegative,
such that
\[
 \int_0^\varepsilon\zeta(\sigma_\varepsilon'')^2\dd t>0.
\]
Set
\[
 \psi=\zeta\sigma_\varepsilon'',\qquad
 \xi_{\mathrm{on}}=\psi(t)\chi_0(\mathbf x)\partial_t,\qquad
 w=K\xi_{\mathrm{on}},
\]
\[
 u_{00}=1,\qquad v_{00}=t,
\]
with all other components zero.
The background is compact in the onset layer.
The vector \(\xi_{\mathrm{on}}\) is not being declared a
member of the relative plateau algebra \(\grel\).

\begin{proposition}[Retained positive reciprocity witness]
\label{prop:reciprocity}
For these data,
\begin{equation}
 D_w(u,v)
   =\frac{5M_\chi}{8}
           \int_0^\varepsilon
                \zeta(\sigma_\varepsilon'')^2\dd t>0.
 \label{eq:reciprocity}
\end{equation}
The additional scalar interaction contributes zero to
this metric diagnostic.
In particular, it supplies no repair of the X-G exact
source-preserving variational-upgrade obstruction.
\end{proposition}

\begin{proof}
Since \(\Pgr u=\Pgr v=0\), equation
\eqref{eq:typedcommutator} gives
\(\mathsf Q_wu=-\Pgr\Lie_{\xi_{\mathrm{on}}}u\)
and likewise for \(v\).
The \(00\) Lie components are
\[
 (\Lie_{\xi_{\mathrm{on}}}u)_{00}=2\psi'\chi_0,
 \qquad
 (\Lie_{\xi_{\mathrm{on}}}v)_{00}
                              =(\psi+2t\psi')\chi_0.
\]
Their \(0i\) components do not change the \(00\)
trace-reversed output used in the pairing.
For such a tensor \(z\),
\((\Pgr z)_{00}=-\Box z_{00}/8\).
The spatial Laplacians integrate to zero, leaving
\[
 D_w(u,v)=\frac{M_\chi}{8}\int
   \sigma_\varepsilon
        \bigl((\psi+2t\psi')''-t(2\psi')''\bigr)\dd t
       =\frac{5M_\chi}{8}\int\sigma_\varepsilon\psi''\dd t.
\]
Two compact integrations by parts prove
\eqref{eq:reciprocity}.

The added potential is independent of \(r\), so its
metric first and second derivatives vanish identically.
It changes neither \(\mathsf N_w\) nor this pairing.
\end{proof}

A variational Hessian is formally symmetric.
If it agreed with the frozen source-switched metric map
on every free input, its antisymmetric pairing on the
displayed free pair would vanish, contradicting
\eqref{eq:reciprocity}.
This is the specified source-preserving exclusion,
not a theorem against every changed action prescription.

The master residual has ghost number one; \(D_w(u,v)\)
has ghost number zero and different arguments.
Equation~\eqref{eq:reciprocity} is not a loop anomaly and
is not identified with \(\mathcal B_a\).

For laboratory interpretation of this onset witness,
its compact source support and the required Cauchy shadows
must fit the inherited regular reserve.
That is the existing X-G-ST support certificate, not a
consequence of \(\supp a\Subset U\).

\subsection{The three-tree plateau is retained once}

Let \(\mathcal T_{\mathrm{mat}},
\mathcal T_{\mathrm{mix}},\mathcal T_{\mathrm{self}}\)
denote the already specified X-F/X-F-ST coefficients.
Their observable contribution is
\begin{equation}
 \kappa^2\bigl(
     \mathcal T_{\mathrm{mat}}
        +\mathcal T_{\mathrm{mix}}
        +\mathcal T_{\mathrm{self}}\bigr).
 \label{eq:trees}
\end{equation}
The matter normalization \(-2\) belongs to that
already counted source coefficient.

\begin{proposition}[Order-level tree non-interference]
\label{prop:trees}
The additional vertex in \eqref{eq:newaction} cannot
alter a coefficient through order two in any
coupling-filtration-preserving calculation with the
same lower-order data.
Thus the retained three-tree plateau identity remains
a valid comparison at its original order.
\end{proposition}

\begin{proof}
The new data begin at order three.
Apply Lemma~\ref{lem:orderprotection}.
\end{proof}

The parent identity and its kernel-admission hypotheses
remain those of X-F/X-F-ST \cite{XF,XFST}.
The proposition does not derive those kernels again,
add their coefficients a second time, or identify their
source response with a variational BV realization.

In particular, no observable BV-closure theorem follows
from \eqref{eq:trees}.
No equality of different quantum ordering presentations
is assumed.

\subsection{Free fish and wavefront conditionals}

At the zero dynamical configuration, \(V_a\) has vanishing
first, second, and third field derivatives.
In particular its Hessian vanishes there.

\begin{proposition}[Unchanged free analytical tests]
\label{prop:freeWF}
The free Hessian, propagators, causal brackets, and the
specific free numerical distributions entering the
X-G-ST fish and null-wavefront diagnostics are
independent of \(\lambda\).
Every conditional statement about those unchanged objects
retains its original hypotheses and conclusion.
\end{proposition}

\begin{proof}
The added vertex has no quadratic part at the zero
configuration. The definitions of the free operators
in Section~\ref{sec:free} are therefore unchanged.
Applying the same operations to the same free
distributions gives the same diagnostic objects.
\end{proof}

This does not treat graphs containing the new quartic
vertex. No such quantum graph is formed.
Finite-scaling-degree extension theory applies only
after its own hypotheses have been established
\cite{BF}; it is not a definition of a missing
nonlocal characteristic product.

\subsection{The convolution gap remains independent}

Retain the fixed convolution data of X-A-conv and
X-A-conv-ST \cite{XConv,XConvST}.
Their witness coefficient has the form
\[
 c_\varepsilon
   =\frac1{2N}\int\varrho_\varepsilon(z)
       \int(\psi(x)-\psi(x-z))^2\dd^4x\,\dd^4z>0.
\]
The new interaction changes none of
\(\varrho_\varepsilon,\psi,N\), the convolution operator,
its coordinate Cauchy problem, or its frozen gauge algebra.

Thus this number is unchanged, and its first-order
obstruction is not repaired.
No statement about a different interacting convolution
observable is inferred.

%=====================================================================
\section{Section B: what counts as a released residual report}
\label{sec:reports}
%=====================================================================

\subsection{Finite coefficient extraction, not Grassmann measurement}

\begin{definition}[Admissible finite residual task]
\label{def:task}
An admissible task specifies:
\begin{enumerate}[label=\textup{(\roman*)}]
\item one of the local coefficient expressions generated
by the finite Hamiltonian, its master residual, or a
specified tested generator identity;
\item an actual density or Euler-expression representative,
or a complete compact integral modulo divergences;
\item a finite exterior algebra
\(\Lambda(\vartheta_1,\ldots,\vartheta_d)\), with an ordered
monomial basis;
\item supplied smooth field and antifield coefficients,
and any compact observation tests;
\item one real coefficient extraction in that ordered basis;
\item a finite report rule and a supplied value, rational
enclosure, or analytically proved status.
\end{enumerate}
A released dictionary contains finitely many such tasks.
\end{definition}

A local variational class is not evaluated by multiplying
an arbitrary divergence representative by an unrelated
test. If a density representative is used, it is part of
the task specification.
Euler expressions, which annihilate divergences before
testing, are an alternative.

\begin{proposition}[Existence of the declared evaluations]
\label{prop:evaluations}
Every admissible task yields a finite real number.
The dependence on its supplied smooth coefficient fields
is continuous on compact supports.
For any finite collection of tasks, the resulting report
dictionary is finite.

This is not an algorithm for evaluating arbitrary smooth
descriptions or a finite certificate of every continuum
identity.
\end{proposition}

\begin{proof}
Each retained action coefficient has finite jet order
and polynomial field degree. Its brackets and specified
Euler derivatives have the same finiteness property.
The compact tests or compact interaction profile make
the integral finite.
Expanding in the finite exterior basis produces finitely
many ordinary smooth coefficient integrals.
Finite seminorm estimates give continuity.
The last assertion follows from choosing finitely many
tasks and finite report alphabets.
\end{proof}

\subsection{A nonempty dictionary requiring no numerical execution}

For example, an eight-entry dictionary may report:
\[
\begin{array}{c|p{.70\linewidth}}
000 & \loc{ZERO}: a specified component of \(s^2\Phi\).\\[2pt]
001 & \loc{ZERO}: the zeroth master coefficient.\\[2pt]
010 & \loc{ZERO}: the first master coefficient.\\[2pt]
011 & \loc{ZERO}: the second master coefficient.\\[2pt]
100 & \loc{ZERO}: the third master coefficient.\\[2pt]
101 & \loc{POSITIVE}: the real coefficient
      \eqref{eq:positiveB} of the specified fourth-order lift.\\[2pt]
110 & \loc{POSITIVE}: the separate reciprocity value
      \eqref{eq:reciprocity}.\\[2pt]
111 & \loc{NOT_ACQUIRED}: no continuum acquisition map
      has been supplied.
\end{array}
\]
These are theorem-based finite status words.
No decimal approximation of an unevaluated integral
is invented.

The two positive entries describe different objects.
Their juxtaposition does not identify them.

\statementbox{Corrected residual-instrument assertion}{%
For every \emph{finite supplied dictionary of admissible residual
tasks}, a complete finite instrument is constructed below.

We do not claim a single finite detector that acquires every
continuum residual, resolves arbitrary real values exactly,
or measures a Grassmann-valued BV expression.}

%=====================================================================
\section{Complete destructive instruments and their composition}
\label{sec:instruments}
%=====================================================================

\subsection{The complete task-clock instrument}

Let \(C\cong\CC^{2^q}\), \(q\ge1\), have a calibrated
orthonormal basis \(\{\ket{\tau_j}\}\).
Let \(A_{\mathrm{fin}}\) contain every surviving finite
quantum register.
An untouched reference is denoted \(E\).

For installed finite words \(w_j\), put
\[
 D_j=\bra{\tau_j}_C\otimes I_{A_{\mathrm{fin}}},
\]
\begin{equation}
 \mathcal M(Z)
    =\sum_j\proj{j,w_j}_R\otimes D_jZD_j^\dagger.
 \label{eq:instrument}
\end{equation}
If fewer than \(2^q\) task entries are used, the remaining
codes report a retained invalid-task status.

\begin{theorem}[Complete residual-report instrument]
\label{thm:instrument}
Equation~\eqref{eq:instrument} is CPTP and satisfies
\begin{equation}
 \Tr_R[(\mathcal M\otimes\id_E)(Z)]
                       =\Tr_CZ
 \label{eq:drop}
\end{equation}
for every operator \(Z\), including all reference
extensions.

On a positive branch, the surviving state is the
corresponding subnormalized block divided by its trace.
No normalized state is assigned to a zero branch.
The returned quantum names are those of
\(A_{\mathrm{fin}}\), not those of \(C\).
\end{theorem}

\begin{proof}
Use the Kraus maps
\[
 L_j=\ket{j,w_j}_R\otimes D_j.
\]
Orthonormality gives
\[
 \sum_jL_j^\dagger L_j
      =\sum_j\proj{\tau_j}\otimes I=I.
\]
Thus the map is completely positive and trace preserving.

Expand
\(Z=\sum_{j,l}\kb{\tau_j}{\tau_l}\otimes Z_{jl}\).
Its \(j\)-th output block is \(Z_{jj}\); summing gives
\(\Tr_CZ\).
The calculation is unchanged on adjoining \(E\).
The branch and consumption statements follow from the
actual domain and codomain.
\end{proof}

This is the complete I-B/En09-A pattern, also used by
X-H-ST and X-Enter-ST \cite{IB,En09,XHST,XEnterST}.
It does not include conditional unitary time evolution
without the separate stationarity and clock-covariance
hypotheses.

The calibrated labels are history labels unless a
separate event-time support condition is imposed.
Several distinct random readings cannot all equal
the coordinate time of one fixed event.

\subsection{Numerical report encoding and an exact error formula}

For a finite list \(v_j\in[-1,1]\), define
\(p_y(v)=(1+yv)/2\), \(y=\pm1\), and
\begin{equation}
 \mathcal I_v(Z)
   =\sum_{j,y}\proj{j,y,w_j}_R
            \otimes p_y(v_j)D_jZD_j^\dagger.
 \label{eq:numericinstrument}
\end{equation}
A supplied real coefficient can be scaled by
\(z\mapsto z/(1+|z|)\); any approximation or enclosure
used for installation remains part of the input contract.

\begin{proposition}[Complete numerical-report comparison]
\label{prop:reporterror}
The maps \(\mathcal I_v\) are CPTP.
On the same report interface,
\begin{equation}
 \ddia(\mathcal I_v,\mathcal I_{\tilde v})
       =\frac12\max_j|v_j-\tilde v_j|,
 \qquad
 \ddia(\Lambda,\Gamma)=\frac12\norm{\Lambda-\Gamma}_\diamond.
 \label{eq:reporterror}
\end{equation}
The comparison includes every untouched reference.
\end{proposition}

\begin{proof}
The Kraus maps are
\(\sqrt{p_y(v_j)}\ket{j,y,w_j}\otimes D_j\).
Their completeness follows from
\(\sum_yp_y(v_j)=1\).

After the clock pinching, the difference has orthogonal
\((j,y)\)-blocks
\(y(v_j-\tilde v_j)Z_{jj}/2\).
Pinching is trace-norm contractive, giving the diamond
upper bound \(\max_j|v_j-\tilde v_j|\).
A basis input at a maximizing \(j\) attains it.
Divide by two.
\end{proof}

This continuity applies to this probabilistic encoding.
Two distinct deterministic orthogonal status words may
instead have channel distance one.

\subsection{Why complete reports are not acquisition}

\begin{proposition}[No acquisition conclusion]
\label{prop:noacquisition}
The report instruments implement observables of the
installed finite input, not measurements of an unknown
continuum BV residual.
\end{proposition}

\begin{proof}
For a diagonal report observable
\(B_R=\sum_jb_j\proj{j,w_j}\), the pullback is
\[
 \mathcal M^*(B_R\otimes I)
             =\sum_jb_j\proj{\tau_j}_C\otimes I.
\]
It is an observable of the supplied clock encoding.
No continuum-field input space or field-to-apparatus
coupling appears in \eqref{eq:instrument}.
The same observation applies to
\eqref{eq:numericinstrument}.
\end{proof}

The finite exterior-coefficient extraction in a task is
classical mathematical processing.
It is not a physical ghost detector.

\subsection{Successive instruments}

\begin{theorem}[Finite adaptive composition]
\label{thm:composition}
A finite sequence or finite adaptive tree of the declared
instruments, finite allocations, allowed unitaries,
and compatible report transports defines a complete
instrument.

If \(h\) ranges over its complete histories, there are
path maps \(K_h\) such that
\begin{equation}
 \Phi_y(Z)=\sum_{h:\,\mathrm{out}(h)=y}K_hZK_h^\dagger,
 \qquad
 \sum_hK_h^\dagger K_h=I.
 \label{eq:pathmaps}
\end{equation}
These formulas hold with arbitrary untouched references.
Every failure history is retained.
Each new clock use requires a fresh or separately returned
clock encoding; a consumed name is not reused as a quantum
input.
\end{theorem}

\begin{proof}
Compose the isometries, gates, transport maps, and
destructive measurement bras along each history.
They give \(K_h\).

Backward induction on the finite instruction tree proves
completeness. A unitary or allocation is isometric.
At a readout, summing the complete outcome family gives
the identity; the same is true of every complete
continuation by induction.
Thus \eqref{eq:pathmaps} holds.
Tensoring each path map with \(I_E\) proves the reference
statement.
The declared input/output names enforce consumption.
\end{proof}

This is a complete finite-interface theorem, consistent
with the resource-certified programme patterns in
Cheung's finite operational papers \cite{FinitePrograms}.
It does not assert an infinite-time instrument limit.

%=====================================================================
\section{Scheduled locality and reference-preserving report routes}
\label{sec:locality}
%=====================================================================

\subsection{Complete backwards support}

Let \(X_{\mathrm{site}}\) be a finite site set.
At tick \(s\), the complete channel factors over a
partition \(\mathcal B_s\):
\[
 \Gamma_s=\bigotimes_{B\in\mathcal B_s}\Gamma_{s,B}.
\]
Every control, task selector, record, reset, allocation,
and retained workspace is included in its block interface.

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

\begin{theorem}[Scheduled locality]
\label{thm:locality}
The complete network satisfies
\[
 (\Gamma_{s+1}^*\circ\cdots\circ\Gamma_t^*)(\Alg(Y))
                                     \subseteq\Alg(N_{s,t}(Y)).
\]
Equal input reductions on \(N_{s,t}(Y)\) give equal output
reductions on \(Y\).
For a joint reference conclusion, the equalities must
include the reference.
A deterministic intervention outside the complete
backwards region does not change the output.

If each dependency edge has a supplied future-causal
host embedding, its ancestors lie in the prescribed
metric past of the output.
\end{theorem}

\begin{proof}
An excluded block sees an identity factor in the
pulled-back observable. Its adjoint is unital.
An included block enlarges support only inside itself.
Backward induction proves the inclusion.
Testing all output observables proves the reduced-state
statements and the intervention conclusion.
Adjoining the reference algebra gives the joint version.
Concatenation of the separately certified causal edges
proves the geometric containment.
\end{proof}

This is the En03-A/VII-E mechanism
\cite{En03,VIIE}.
Scheduled ancestry is an upper bound, not necessarily
minimal operational influence or the complete physical
causal past.

Scheduler consistency does not by itself determine a
Lorentz cone \cite{En08A}.
The geometric embedding in
Theorem~\ref{thm:locality} is additional data.

\subsection{A nonempty regular route}

Within the retained developed regular tube choose a
nonzero straight spatial segment of length \(\ell\),
with enough existing time reserve for the chosen finite
protocol and its declared buffers.

For \(n\ge1\), place equal payload registers
\(A_0,\ldots,A_n\) at equally spaced stations.
The payload contains the finite report and every
surviving quantum factor that must be returned.
Initialize \(A_1,\ldots,A_n\) in independent blanks
\(\omega\), and apply successive SWAPs over developed
durations \(\ell/n\).

\begin{theorem}[Reference-preserving finite report route]
\label{thm:route}
For every input \(\rho_{A_0E}\), the complete state
after hop \(j\) is
\[
 \rho_{A_jE}\otimes\bigotimes_{i\ne j}\omega_{A_i}.
\]
The logical payload channel is the identity including
every reference.
The arrival time is \(\ell\) in developed units, or
\(\ell/c_{\mathrm h}\) in the calibrated host time.
The route stays in the regular tube.
\end{theorem}

\begin{proof}
A SWAP exchanges the entire current payload, including
its reference correlations, with the next independent
blank. Induction proves the tensor identity.

Each developed spatial step has length \(\ell/n\) and
is assigned the same developed duration. Its straight
lift is null. The retained time calibration gives the
host duration.
\end{proof}

This is the target-adapted I-B/En08-B construction
\cite{IB,En08B}.
It uses the declared finite-operation permissions.
No microscopic relativistic SWAP Hamiltonian is derived.
No exact null-attainment theorem on an arbitrary previously
fixed lattice is asserted.

The route preserves correlations already supplied.
It neither creates a gravitationally mediated
entanglement witness nor transports a continuum field.

\subsection{Resources and complete errors}

A \(p\)-qubit payload route uses \(n\) register SWAPs and
\((n+1)p\) reserved route qubits.
Every fresh clock, finite signal, supplier holding,
workspace, and returned register is separately charged.
External untouched references are listed in their own
storage ledger.

For any finite compatible implementation,
\begin{equation}
 \ddia(\widetilde\Gamma,\Gamma)
 \le\min\left\{1,\,
        e_{\mathrm{prep}}+
         \sum_{\text{readouts}}e_{\mathrm{read}}
        +\sum_{\text{gates/routes}}e_j
        +e_{\mathrm{decode}}\right\}.
 \label{eq:errorledger}
\end{equation}
Replace one channel factor at a time, use contraction
under channel composition, and apply the triangle
inequality \cite{Watrous,En08B}.
All comparisons retain their classical and quantum outputs.

A small complete-channel error does not prove exact
causal containment or validity of an incorrect
supplied numerical enclosure.

%=====================================================================
\section{Release, significance, and what remains unconstructed}
\label{sec:release}
%=====================================================================

\begin{definition}[Local Grade S]
\label{def:grade}
For this manuscript, local Grade S consists of:
\begin{enumerate}
\item the retained host, chart, completion, fixed width,
and declared finite permissions;
\item the explicit relative classical action and
differential through order three;
\item the exact polynomial-lift residual and its scoped
diagnostics;
\item a nonempty finite residual-report dictionary;
\item a complete instrument for every released report,
including all surviving systems and references;
\item finite adaptive composition, scheduled locality,
a nonempty regular report route, and the resource/error
ledger.
\end{enumerate}
No field-acquisition identification is included in the grade.
\end{definition}

\begin{theorem}[X-I-ST mathematical release]
\label{thm:release}
Under the retained supported-host contract:
\begin{enumerate}[label=\textup{(\roman*)}]
\item \(\Snew\) is a specified classical relative BV
Hamiltonian modulo \(\kappa^4\), and \(\Qnew^2=0\)
in that ring;
\item its new scalar Euler term is
\(\lambda\kappa^3a\phi^3/6\), with compact support;
\item coefficients through order two agree with X-H-ST;
the field free limit is exactly X-G-ST's \(s_0\), and
the full free BV limit is exact in the centered sector;
\item the off-shell background tadpole remains visible
outside that centered sector;
\item the chosen polynomial lift has the exact new
residual \(\lambda\kappa^4\mathcal B_a\), with the
nonzero real evaluation \eqref{eq:positiveB};
\item the natural interacting field rule still fails
descent to the unchanged free equation quotient;
\item the earlier three-tree coefficient packet is not
changed through its retained order, and the specified
positive reciprocity diagnostic remains positive;
\item the independent convolution obstruction and
the free fish/null-wavefront conditionals retain their
separate scopes;
\item every released finite supplied residual report
has the complete instrument and finite-composition
theorems proved above;
\item scheduled locality and regular reference-preserving
transport hold under the stated finite permissions.
\end{enumerate}
These results form local Grade S.

They are not interacting quantum BRST, not a characteristic
coefficient, not loop-anomaly cancellation, not an
ultraviolet completion, and not a derived metric or
spacetime emergence.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:deformation} proves (i)--(ii).
Corollary~\ref{cor:recoverH},
Proposition~\ref{prop:freecausal}, and
Proposition~\ref{prop:tadpole} prove (iii)--(iv).
Theorem~\ref{thm:residual} and
Proposition~\ref{prop:positiveB} prove (v).
Theorem~\ref{thm:nondescent} proves (vi).
Propositions~\ref{prop:trees} and
\ref{prop:reciprocity} prove (vii).
Section~\ref{sec:diagnostics} proves precisely the
non-interference statements in (viii).
Theorems~\ref{thm:instrument} and
\ref{thm:composition}, with
Proposition~\ref{prop:noacquisition}, prove (ix).
Theorems~\ref{thm:locality} and \ref{thm:route} prove (x).
\end{proof}

\begin{center}
\small
\renewcommand{\arraystretch}{1.12}
\begin{tabularx}{\linewidth}{@{}p{.31\linewidth}p{.23\linewidth}X@{}}
\toprule
Object & Verdict & Boundary\\
\midrule
Compact scalar interaction &
Constructed &
Starts at order three\\
Relative classical BV differential &
Constructed &
Modulo \(\kappa^4\), off shell\\
Full free BV limit &
Exact when centered &
General off-shell tadpole retained\\
Polynomial-lift master identity &
Not exact &
Displayed nonzero fourth-order evaluation\\
Unchanged free-carrier descent &
Natural rule fails &
No retarded BV transport constructed\\
Three-tree plateau &
Retained once &
No new variational identification\\
Reciprocity witness &
Positive diagnostic retained &
Not a master residual or loop anomaly\\
Finite residual instruments &
Constructed &
Supplied finite evaluations only\\
Continuum residual acquisition &
Not constructed &
No field-to-apparatus coupling\\
Interacting quantum BRST &
Not constructed &
No quantum product or master operation\\
Characteristic coefficient &
Not admitted &
Separate analytical chain remains open\\
Spacetime emergence &
Not constructed &
Geometry and speed are inputs\\
\bottomrule
\end{tabularx}
\end{center}

\subsection{Novelty and prospective impact}

The predecessor free multiplet, relative nonlinear
kinematics, and reciprocity mechanism retain their
attribution to Cheung's X-G/X-G-ST/X-H-ST.
The three-tree identity retains its separate X-F/X-F-ST
attribution.

The new content is the explicitly chosen compact
third-order interaction, its finite coefficient
completion, the exact next residual of its polynomial
lift, and the proof that its released finite reports
compose as complete instruments.

Its prospective use is methodological and local:
a successor can distinguish a genuine new classical
Euler term from an instrument that merely reports it,
and can identify the next uncancelled residual before
making a quantum or geometric inference.
No general priority claim for scalar polynomial
interactions or classical BV is made.

\subsection{Successor obligations}

A successor seeking more than this release must separately:
\begin{enumerate}
\item construct an admissible fourth-order completion
for the fixed profile and ghost domain, or state its
remaining residual;
\item construct interacting equation-space transport,
rather than identify the result with the unchanged free
carrier;
\item supply any proposed continuum-field encoding,
apparatus coupling, and complete instrument comparison;
\item construct the required quantum products before
writing an interacting quantum master equation;
\item discharge the separate characteristic-map,
remainder, transverse-estimate, contact, and output
wavefront obligations before claiming the inherited
quantum coefficient.
\end{enumerate}
None follows from the finite report channel.

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

\begin{longtable}{@{}p{.36\linewidth}p{.59\linewidth}@{}}
\toprule
Potential overstatement & Correction proved or imposed here\\
\midrule
\endfirsthead
\toprule
Potential overstatement & Correction proved or imposed here\\
\midrule
\endhead

X-I-ST is the first classical interacting sheet &
X-H-ST already has nonlinear classical kinematics
and a relative interacting reference action.
The new step is the compact order-three vertex.\\[4pt]

An unspecified interaction is controlled &
The profile, monomial, coupling order, ring,
and finite reference coefficients are fixed explicitly.\\[4pt]

The new interaction must alter the field BRST rules &
A gauge-invariant leading scalar vertex can change
Euler/antifield equations without changing field
kinematics at the retained order.\\[4pt]

Nilpotent field kinematics implies BV closure &
The action identity and signed antifield lift are
separately proved.\\[4pt]

A third-order action needs only fourth-order geometry &
The Einstein--Hilbert normalization requires the fifth
geometric coefficient for the third action coefficient.\\[4pt]

\(\kappa=0\) removes an off-shell background tadpole &
It does not. Equation~\eqref{eq:tadpole} is retained.
Full free BV equality is stated in the centered sector.\\[4pt]

A positive-order truncation is an exact master action &
The lift residual is explicitly nonzero at order four
on the displayed evaluation.\\[4pt]

Adding the volume density cancels every residual &
The fixed-profile gradient remains in
\eqref{eq:cutoffresidual}.\\[4pt]

The new residual is a universal completion obstruction &
It is the residual of one named polynomial lift.
A higher-order candidate changes it.\\[4pt]

All residuals are one finite observable &
Only a specified finite dictionary of real coefficient
tasks is instrumented.\\[4pt]

Odd residuals possess an intrinsic positivity order &
Positivity is applied only after a specified real
exterior-coefficient extraction.\\[4pt]

A local action class can be arbitrarily weighted &
A task names a representative or first passes to
Euler expressions. Divergences are not silently
reweighted.\\[4pt]

Local functionals form an ordinary product algebra &
Their ordinary products are included in the
multilocal algebra.\\[4pt]

Compact antighost, unrestricted auxiliary field &
That domain need not be tangent to \(s\bar c=b\).
We use unrestricted germs, or compatible common
support restrictions.\\[4pt]

A compact ghost test is a nonzero free ghost solution &
The two domains are distinct; compact spacetime
homogeneous wave solutions vanish.\\[4pt]

The reciprocity witness must vanish in any BV theory &
The witness excludes an exact source-preserving
identification, not every different variational action.\\[4pt]

The witness is a loop anomaly &
It is a classical ghost-number-zero pairing.\\[4pt]

Preserving the three trees requires counting them again &
The added interaction begins beyond their order.
They are retained exactly once.\\[4pt]

Free fish statements control the new interaction graphs &
Only the unchanged free numerical distributions are
covered by the non-interference statement.\\[4pt]

The ramp and convolution share an obstruction &
Their operators, algebras, and witnesses remain distinct.\\[4pt]

A complete residual instrument acquires a field &
Its pullback is an observable of the installed finite
encoding.\\[4pt]

Postselection gives a deterministic instrument &
All branches, including failures, are retained
subnormalized until conditioning.\\[4pt]

A consumed clock can be reused in the next round &
A new quantum input requires a separately supplied
clock factor and its resource charge.\\[4pt]

Clock consumption removes time &
Only finite quantum names are consumed;
the geometric time axis remains.\\[4pt]

Finite network locality derives the cone &
Causal embedding is separately supplied.
The host is unchanged.\\[4pt]

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

A new source file is an executed registration &
The integration appendix remains proposed and false
for all execution fields.\\

\bottomrule
\end{longtable}

%=====================================================================
\section{Standing boundaries and documentary comparisons}
\label{app:boundaries}
%=====================================================================

\paragraph{The independent characteristic programme.}
The supplied XIV-H, XIV-I, and XIV-J records distinguish
the historical H2 stop, the narrower internal-amplitude
construction at I2, and the missing full characteristic
maps at I3/J2 \cite{CharacteristicRecords}.
This paper constructs none of those maps.
A finite classical coupling coefficient is not
\(R_1^{(1)}\), and no operation is applied to an absent
quantum coefficient.

The downstream XV-B quantum-admission exit is not opened.
This does not deny the existence of technical manuscripts
with that identifier.
Their reconstruction and extension interfaces are not
a substitute for admission of the complete coefficient.

\paragraph{Prepared-source and prescription boundaries.}
XVI-N is not discharged.
No result here changes the XVI-O prescription or closes
its \loc{NOT_CLOSED} O.1 record.
Later XVI-P/XVI-Q descriptions concern their own
version-qualified subsectors and counterterm domains;
their scoped assertions are neither denied nor
transferred by this manuscript \cite{PreparedRecords}.
No blanket all-prescription closure or nonclosure claim
is inferred.

\paragraph{Other carriers and products.}
The VII--IX sharp-product and topology questions,
XI's distinct free multiplet and equation-space questions,
and the present off-shell relative BV algebra are not
identified.
A free causal derivation does not construct a renormalized
interacting product.

\paragraph{OpenAI comparisons.}
The supplied descriptions of Families 282, 362, and 376
and related finite-interface records are documentary
comparisons only \cite{OpenAICatalogue}.
No catalogue theorem proves the present interaction,
master identity, instrument theorem, or free-limit
comparison. No catalogue proof or reported formalization
is independently verified here.

The finite-certificate style is a presentation discipline,
not a mathematical transfer from an unrelated family.
Finite coefficient extraction does not mean that the
continuum normalization space or the full equation
problem is finite dimensional.

\paragraph{Physical scope.}
No interacting quantum BRST differential, loop anomaly,
anomaly cancellation, quantum characteristic coefficient,
physical-state selection, Kugo--Ojima construction,
ultraviolet completion, Einstein dynamics from records,
gravity-generated entanglement, Page curve,
holographic identification, or spacetime emergence is
constructed.

The sentence that no inherited characteristic coefficient
is admitted does not deny the separately admitted free
carrier or the earlier tree coefficients.

%=====================================================================
\section{Proposed companion integration record}
\label{app:integration}
%=====================================================================

This appendix is documentary metadata, not an executed
repository transaction.
The file has not been registered, hashed, compiled,
or tested by this response.

\paragraph{Proposed routing note.}
X-I-ST constructs a classical interacting relative BV
deformation through order three on the unchanged
I-B/X-A-ST host and declared completion.
The free limit recovers X-G-ST and X-H-ST in the
precise field, Hessian, and centered-BV senses proved
above; arbitrary off-shell background tadpoles remain.
It is not quantum BRST, not a characteristic coefficient,
and not a premise of any earlier sheet.
Residual instruments report supplied finite coefficients,
not continuum acquisition.
The reciprocity and convolution obstructions retain
their separate scopes.
No spacetime emergence is constructed.

\begin{verbatim}
identifier: X-I-ST
sourceStatus: proposed
grade: local Grade S
track: ramp
problemConnections:
  - relational
  - products
nearUv: true

host:
  retained: I-B / X-A-ST
  analyticalCompletion: Minkowski
  signatureLedger: iota-star G = -eta
  defect: worldsheet
  width: fixed positive

mathematicalScope:
  relative classical BV
  retainedOrder: kappa^3 modulo kappa^4
  addedInteraction: lambda kappa^3 integral(a phi^4 / 4!)
  fullFreeBVRecovery: centered background sector
  unrestrictedTadpole: retained
  freeCarrierInteractingDescent: not constructed
  characteristicQuantumCoefficient: not admitted

reportScope:
  finite supplied real coefficient dictionaries
  complete destructive instruments
  failures and quantum residuals retained
  untouched references included
  continuumAcquisition: not constructed

obligations:
  interacting-classical-not-quantum:
    Third-order classical CME is not quantum BRST.
  residual-instrument-not-acquisition:
    Installed coefficients are not acquired fields.
  reciprocity-still-obstruction:
    Exact source-preserving upgrade remains excluded.
  free-limit-recovery-not-premise:
    The precise limit theorem creates no backward dependency.
  fourth-order-profile-completion:
    No fourth-order repair is selected.
  unchanged-free-carrier-descent:
    Natural positive-order scalar rule fails the old ideal.

documentaryConnections:
  X-G-ST: separation and version-qualified comparison
  X-H-ST: explicit coefficient comparison
  X-Enter-ST: finite complete-interface comparison
  X-A-conv-ST: independent unrepaired gap
  X-F-ST: retained tree packet counted once
  I-B: prescribed host and finite permissions
  En-series: scheduled complete-interface comparisons
  OpenAI: documentary only; not a premise

timeLandmark:
  proposed: true
  reason: complete finite clock instrument is explicitly constructed

emergenceStatus: not constructed
includeInEmergenceSources: false

registryEditsExecuted: false
repositoryTestsExecuted: false
compilationAsserted: false
numericalExecutionAsserted: false
machineVerificationAsserted: false
\end{verbatim}

Any actual integration must use the repository's current
schema and real identifiers, add one appropriately typed
connection per intended relation, and run its actual
source, face, connection, time-index, problem, and proposal
tests. Expected array lengths, passing results, or a
no-edit-after-registration condition are not asserted
before that transaction exists.

A citation does not automatically create a formal premise.
Conversely, the explicitly stated definitions and
hypotheses used in a mathematical proof are not erased
by labeling a registry edge ``separation.''

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

\paragraph{Authorship.}
The author field is intentionally empty for anonymous
review. Citing N.~Cheung does not assign authorship of
this adaptation. Authorized submission metadata must be
supplied for a signed submission.

\paragraph{Attribution and novelty.}
Cheung's X-G/X-G-ST supply the normalized free multiplet
and reciprocity diagnostic. X-H-ST supplies the relative
classical action and signed-lift convention through the
predecessor order. X-F/X-F-ST supply the distinct
three-tree comparison.
The additional compact third-order vertex, its exact
finite-lift residual, and the complete finite-report
assembly are the scoped continuation developed here.

\paragraph{Analytical status.}
The construction is a finite-order local variational
one on smooth off-shell fields. No resolution acyclicity,
physical cohomology theorem, interacting carrier transport,
global nonlinear existence, or quantum master operation
is asserted.

\paragraph{Operational status.}
The instruments act on finite supplied encodings.
They do not prepare such encodings from unknown continuum
fields or provide physical representations of ghosts
and antifields. Every unmeasured finite output and
untouched reference belongs to the complete comparison.

\paragraph{Source status.}
The source manuscripts are identified by manuscript and
version. Supplied passages and inspected documentary
records are not independent proof certificates.
Internal publication dates, DOIs, or catalogue-wide
verification are not invented.

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

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

\bibitem{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}.
Free normalization, typed background operator,
positive reciprocity witness, and conditional
fish/nonlocal-product distinctions.

\bibitem{XGST}
N.~Cheung,
\emph{X-G-ST: Free causal BRST compatibility and
a variational upgrade obstruction}.
\loc{X-G-ST.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-G-ST}.
Especially \loc{lem:s0nil}, \loc{prop:gfnorm},
\loc{prop:Ghb}, \loc{thm:freeBRST},
\loc{prop:Dsigned}, \loc{thm:obstruction},
\loc{prop:fish}, and \loc{prop:nullWF}.

\bibitem{XHST}
N.~Cheung,
\emph{X-H-ST: Relative classical BV structure and
complete instruments for supplied residual reports}.
\loc{X-H-ST.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-H-ST}.
Relative field rules, natural local reference action,
background tadpole, signed antifield lift,
free-equation non-descent, and complete report interfaces.
The order-three continuation in the present paper is
specified explicitly rather than attributed to that source.

\bibitem{XEnterST}
N.~Cheung,
\emph{X-Enter-ST: Readout closure of an obstructed
convolution coefficient on a prescribed conical host}.
\loc{X-Enter-ST.tex}, version 1.
Supplied manuscript record, Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-Enter-ST}.
Complete finite instruments, untouched references,
scheduled locality, and finite transport.
Its orbit theorem is not an informational-completeness
or field-acquisition premise here.

\bibitem{IB}
N.~Cheung,
\emph{I-B: A Finite Enter Spacetime:
Calibrated Destructive Time Readout on a Prescribed
Conical Product}.
\loc{I-B-v1.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/I-B}.
Prescribed geometry and worldsheet, finite clock,
calibration, complete instrument, and regular transport.

\bibitem{XAST}
N.~Cheung,
\emph{X-A-ST: Supported classical covariance of a
fixed-ramp relational scalar}.
Version 1 predecessor source record supplied with
the request.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-A-ST}.
Supported chart, declared completion, fixed width,
signature ledger, and plateau data.

\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}.
Three-tree coefficient and direct background identity.
Its stronger variational terminology is separately
qualified by X-G and the ST continuation.

\bibitem{XFST}
N.~Cheung,
\emph{X-F-ST: Three fixed-width retarded trees and
the background identity}.
\loc{X-F-ST.tex}, version 1.
Supplied manuscript record, Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-F-ST}.
Equation-level source response, typed commutator,
support distinctions, and one-time tree counting.

\bibitem{XConv}
N.~Cheung,
\emph{X-A-conv}.
\loc{X-A-conv.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/X-A-conv}.
Frozen spacetime convolution and its strictly positive
first classical covariance defect.

\bibitem{XConvST}
\emph{X-A-conv-ST: A fixed-width convolution obstruction
on a prescribed conical Enter spacetime}.
Version 1 source supplied in the preceding development
and cited in the request.
\url{https://enter.grok.me/browse/X-A-conv-ST}.
Source-specific support certificates, density transport,
retained obstruction, and supplied-report separation.
Citation does not assign authorship of a new adaptation
or independently certify registration.

\bibitem{EnterPhil}
N.~Cheung,
\emph{The Philosophy of Enter:
Destructive Readout, Quantum Recovery, and the
Refusal of Identification}.
\loc{EntPhil-v2.tex}, version 2.
Introductory essay, Enter Calculus Companion.
\url{https://enter.grok.me/enter}.
Interpretive background, not a classical BV 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}.
Characterization within its full structural class;
not a field-measurement construction.

\bibitem{En09}
N.~Cheung,
\emph{Realization of Relational Time by Destructive
Enter on a Clock Register}.
\loc{En09-A.tex}, version 1.
Enter Calculus Companion.
\url{https://enter.grok.me/browse/En09-A}.
Complete clock instrument, reference extension,
consumption, and separate covariance 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}.
Complete scheduled interfaces and backwards-support
theorem.

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

\bibitem{En08A}
N.~Cheung,
\emph{En08-A}.
Version 1, supplied manuscript record,
Enter Calculus Companion.
\url{https://enter.grok.me/browse/En08-A}.
Scheduler consistency does not reconstruct a
Lorentz cone without additional geometric data.

\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,
resources, and complete-channel error bounds
relative to prescribed geometry.

\bibitem{FinitePrograms}
N.~Cheung,
\emph{En02-B}, version 1, and
\emph{En02-C}, version 1.
Version-qualified excerpts supplied with the request.
Enter Calculus Companion.
Finite complete-history soundness and explicit
live-name/storage ledgers.
Comparison of finite programme certification only;
their recovery tasks are not imported.

\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 local variational BV framework and its
configuration-space hypotheses.

\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 quantum completion of the
finite classical construction here.

\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,
used on the declared Minkowski completion.

\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 theory under its
stated hypotheses; no missing characteristic
composition is imported.

\bibitem{Watrous}
J.~Watrous,
\emph{The Theory of Quantum Information}.
Cambridge University Press, 2018.
Author manuscript and errata:
\url{https://cs.uwaterloo.ca/~watrous/TQI/}.
Finite channels, complete instruments, trace norms,
and diamond-norm comparison.

\bibitem{CharacteristicRecords}
N.~Cheung,
\emph{XIV-H}, \emph{XIV-I}, and \emph{XIV-J},
version 1, and technical \emph{XV-B}, version 2.
Version-qualified supplied manuscript records,
Enter Calculus Companion.
Historical internal-amplitude boundaries, reconstruction
descent, partial references, the missing full I3/J2
maps, and the downstream quantum-admission distinction.

\bibitem{PreparedRecords}
N.~Cheung,
\emph{XVI-N}, revision r12;
\emph{XVI-O}, revision o70-sync;
\emph{XVI-P}, version p23;
and \emph{XVI-Q}, version Q11.
Version-qualified excerpts and status descriptions
supplied with the request.
Prepared-insertion and distinct subsector-prescription
boundaries only. No calculation or verification of
those records is performed here.

\bibitem{OpenAICatalogue}
OpenAI,
\emph{Mathematics catalogue}.
Supplied documentary descriptions, including
Families 282, 362, and 376, recorded separately
in the Enter Calculus Companion.
No catalogue theorem is a premise of this manuscript,
and no catalogue proof or reported formalization is
independently verified here.

\end{thebibliography}

\end{document}