X-I-ST: A supported finite-order classical BV deformation on a prescribed conical Enter spacetime A compact scalar interaction, complete residual-report instruments, and preservation of the earlier diagnostic boundaries

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

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 $\lambda {\kappa }^{3}a{\varphi }^3/6$, 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

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.

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.}

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:

1. the interaction is the compact scalar vertex $V_a$ displayed in the abstract;

2. it first occurs at order ${\kappa }^3$;

3. the retained coupling ring is $\mathbb{R}[\lambda ][\kappa ]/({\kappa }^4)$;

4. the classical reference action is specified by finite Taylor extraction from the natural local action;

5. a report concerns a specified real coefficient of a specified representative on supplied smooth data;

6. only finite dictionaries and finite instrument networks are released.

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 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.

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 EnterPhil,EnterI. We adopt that separation:

• a classical BV residual is not automatically a physical observable;

• a finite encoding is not acquisition of a continuum functional;

• equality of outcome laws does not establish equality of complete instruments;

• reference-preserving transport is not entanglement generated by a gravitational mediator;

• a prescribed conical spacetime is not derived from records.

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 IB. The spacetime remains a continuum; the experiments are finite. No host-update rule is introduced.

The local variational BV framework and its distinction between off-shell complexes, equations, and observables are standard FRBV. Perturbative quantum gravity requires additional product, renormalization, and quantum master-equation constructions 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.

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

where $0<\alpha <1$ and $c_{\mathrm{h}}>0$ retain their inherited values. The completion defect is the worldsheet

The independent analytical completion, slab, and chart are

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 $\hat{M}$, not on the regular conical product.

The regular conical product is not globally hyperbolic.

Proof. Choose a fixed radial ray, $r_0>0$, $z_0$, and $A>r_0/c_{\mathrm{h}}$. Between the events with coordinates $(-A,r_0,{\phi }_0,z_0)$ and $(A,r_0,{\phi }_0,z_0)$, the causal diamond contains

Their only completion limit lies at the excluded apex. They have no convergent subsequence in the regular product, so that diamond is not compact.

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 $G$.

Retain the smooth compact scalar switch $g$, the observation test $f$, the open plateau $U$, and the onset

where $\chi$ is smooth, zero for arguments at most zero, and one for arguments at least one. Assume

The relative compact gauge parameters are

Choose once and for all

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.

An onset is not a compact spacetime region. Here the precise support condition is

We do not place the new interaction in the transition layer $0<t<\epsilon$, where plateau identities would require a different audit. No $\epsilon \to 0$ is taken.

A nonzero profile satisfying (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 sec:response, we additionally require

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.

For a compact scalar density $L(x){\mathrm{d}}^{4}x$, its expression in host coordinates $y$, with $x={\iota }^{-1}(y)$, is

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.

The independent fields are

Their parities and ghost numbers are

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

For a covariant symmetric tensor $r$, let

Compact integration by parts gives

Use the parent normalization

Here the linearized Einstein tensor is defined by

Equations (eq:A) follow by substitution. This is the normalization of X-G/X-G-ST XG,XGST.

Define the left odd derivation

It commutes with total spacetime derivatives.

The gauge fermion and free action are

These are variational actions with compact variations, not an assumption that their values are finite on every ambient smooth configuration.

The differential $s_0$ is nilpotent off shell. It preserves $I_{\mathrm{free}}$ modulo integrated divergences. Algebraic elimination of $b$ gives

and hence the metric operator ${\mathsf{P}}_{\mathrm{gr}}$.

Proof. The square vanishes on each independent generator in (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 (eq:A) identifies the reduced metric operator.

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

Let ${E_0}^{\mathrm{R}/\mathrm{A}}$ and ${E_m}^{\mathrm{R}/\mathrm{A}}$ be the positive equation inverses of $\square$ and $P_m$ on $\hat{M}$. Their existence and causal support are standard on this globally hyperbolic completion BGP.

The metric–auxiliary inverse is

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

In this source-dual convention their inverse is

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{array}{c|c@{\qquad}c|c}
 AB&\Delta_{AB}&AB&\Delta_{AB}\\ \hline
 rr&-4\mathsf P_{\mathrm{tr}}\delta_0&\phi\phi&-\delta_m\\
 rb&K\delta_0&br&-2\mathsf C\mathsf P_{\mathrm{tr}}\delta_0\\
 c\bar c&-\delta_0&\bar c c&\delta_0
\end{array}
 \label{eq:causalblocks}

and ${\Delta }_{bb}=0$. Thus the scalar sign and the odd signs are not inferred from an unsigned array.

The free equation ideal

is $s_0$-stable. The normalized causal relations

are also stable. Consequently $s_0$ induces the nilpotent free causal BRST derivation of X-G-ST.

Proof. Solving $b=(\mathsf{C}r-j)/2$ in the sourced bosonic equations reduces them to

which proves (eq:Grb). The equation variations are $AKc=0$, $\square c$, and $\square b$. The last follows from the divergence of $Ar+{\mathsf{C}}^{\ast }b=0$; 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:

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.

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

Retain fixed smooth classical backgrounds $h,k$, and put

Neither $q$ nor $\bar{q}$ is the host metric $G$.

Let $\mathcal{P}(U)[[\kappa ]]$ 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

Use the left graded Leibniz rule. These are the relative field rules of X-H-ST, restated rather than inferred from its title XHST. In particular,

The derivation $s$ satisfies $s^2=0$ on the local formal field algebra. Its leading field operator is exactly (eq:s0).

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

The second-derivative terms cancel by antisymmetry of the ghost coefficients. Therefore

Since $\bar{q}$ is fixed, $s^{2}r=0$ coefficientwise.

Similarly,

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.

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.

Let ${\mathcal{F}}_{\mathrm{ml}}(U)$ 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 ${\mathcal{F}}_{\mathrm{ml}}(U)$. This is the local variational interpretation of the classical BV bracket FRBV.

Adjoin antifields with

They are dual tensor densities. Fix the canonical odd bracket by

with the odd biderivation and integration-by-parts rules.

For the even Hamiltonian acting on the left, $QF=(\mathcal{S},F)$, define

Explicitly,

For every field generator,

Moreover, $({\mathcal{H}}_s,{\mathcal{H}}_s)=0$ modulo divergences.

Proof. If $p=|{\Phi }^A|$ and $R^A=s{\Phi }^A$, the odd biderivation rule gives

The factor in (eq:H) therefore gives $R^A$ for both even and odd generators. In particular,

so an unsigned common antifield coupling would be wrong in this convention.

The cotangent-lift calculation gives $\frac{1}{2}({\mathcal{H}}_s,{\mathcal{H}}_s)$ linear in antifields, with coefficients $s^2{\Phi }^A$, after integration by parts. They vanish by Theorem thm:kinematic. There is no antifield-independent term.

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

For any formal metric $p$, put

Write

The generalized invariant action is

The sole negative-power coefficient is a divergence:

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

The auxiliary field remains independent.

For $j=0,1,2,3$, define the local variational coefficients

Coefficient extraction means differentiation at zero divided by the relevant factorial. Thus (eq:Ij) is a finite prescription, not an unnamed cubic or quartic vertex.

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

Let ${\mathcal{H}}_s={{\sum }_{j=0}}^2{\kappa }^j{H}_j$, and put

The natural reference action satisfies $sI=0$ as a local variational class. Consequently

Proof. Naturality of the curvature and scalar densities gives

The background subtraction is $s$-inert. Also $s(s{\Psi }_{\bar{q}})=0$. Thus $sI=0$ modulo divergences.

For the formal reference Hamiltonian $I+{\mathcal{H}}_s$,

Extracting the first four nonnegative coefficients gives the asserted finite-order identity.

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.

At order zero,

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.

Proof. The quadratic gravitational coefficient is $\frac{1}{2}⟨h+r,A(h+r)⟩$. Subtracting its background-only term gives

The scalar and gauge-fixing coefficients at zero are those of (eq:Ifree). The extra term is linear in $r$, so it changes no Hessian and no field rule. Its bracket with $r^{\ast }$ is $Ah$.

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 XHST.

Thus the phrase “free-limit recovery” has two distinct, proved meanings here:

1. on fields and the homogeneous free causal carrier, the limit is exactly $s_0$;

2. on the complete BV algebra, exact free recovery holds in the centered background sector.

No unconditional full-BV free-limit claim is made for arbitrary fixed off-shell backgrounds.

With $a$ fixed by (eq:a), define

over

The full classical BV differential is

The field differential is denoted $s$, not $Q$. The latter also contains the antifield equations.

The Hamiltonian (eq:newaction) satisfies

It is a classical deformation with no quantum term. Its field rules remain (eq:sr)–(eq:snonminimal), while its scalar antifield equation changes by

Every other fundamental generator has zero change from the additional vertex.

Proof. The vertex has no antifields, so $(V_a,V_a)=0$. Since $s_0\varphi =0$,

All other cross brackets with the added ${\kappa }^3{V}_a$ have order at least four. Lemma lem:referenceCME therefore gives the first identity in (eq:newCME).

The odd Jacobi identity yields

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 $\varphi$, namely $a{\varphi }^3/6$.

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.

The additional action, its first variations, and the new local Euler term have support in $\mathrm{supp}a$. 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 ${\mathcal{F}}_{\mathrm{ml}}(U)$.

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.

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 $\mathrm{supp}a$.

Let two action or operator data sets differ by ${\kappa }^{N}V$, with $N\ge 1$. 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.

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.

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 prop:freecausal and Proposition prop:tadpole give the precise free-limit statements.

Proof. The additional vertex starts at order three, and (eq:Ij) reproduces the predecessor coefficients for $j=0,1,2$. The remaining assertions are direct substitutions.

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

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

Then its new scalar contribution is uniquely

The added contribution to the metric–auxiliary response at this order is zero. The new scalar dependence is contained in $J_{\eta }^{+}(\mathrm{supp}a)$.

Proof. Coefficients below order three agree by Lemma lem:orderprotection. At order three, subtraction of the two scalar Euler equations leaves

The source is smooth and compact. The completion retarded inverse proves (eq:response), uniqueness, and causal support. The vertex has no metric or auxiliary derivative.

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 (eq:responsecertificate), its specified compact output pairing has a supported laboratory presentation.

A statement in $R_3$ does not assign canonical values to fourth-order coefficients. To discuss them, fix the ordinary polynomial lift

with no additional terms. Every higher residual below refers to this named lift.

Define

Also define the compact odd functional

As ordinary finite polynomial variational functionals,

In particular,

Coefficients through order three vanish. The exact lift obeys

Proof. All $I_j$ and $V_a$ are antifield independent. Only ${\mathcal{H}}_s$ can bracket nontrivially with $V_a$. Its scalar field rule gives

There is no $s_2$-action on $\varphi$. This proves (eq:Rdifference) exactly. Expanding the finite polynomial proves (eq:R4)–(eq:R5). The third coefficient $S_3$ and $V_a$ have no antifields, proving (eq:R6). The lower coefficients vanish by Theorem thm:deformation.

Equation (eq:consistency) is the odd Jacobi identity applied three times to the even Hamiltonian.

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

Fix a nonzero exterior generator $\vartheta$. For this diagnostic only, use the centered background and the smooth evaluation

The ghost coefficient is compactly supported in $U$.

On this evaluation,

Also $[{\kappa }^4]{\mathcal{R}}_{\mathrm{ref}}=0$ on this evaluation. Thus for $\lambda >0$ the displayed fourth-order coefficient of the chosen polynomial lift is nonzero.

Proof. Substitution in (eq:B) gives (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 (eq:R4) at $\lambda =0$ proves the claim.

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.

Let

This is the first volume-density correction to the compact potential.

Modulo compact divergences,

Proof. Since $s_0(\mathrm{tr}r/2)={\partial }_{\mu }{c}^{\mu }$,

Also ${\mathcal{B}}_a=\int a{c}^{\mu }{\partial }_{\mu }({\varphi }^4/4!)$. One integration by parts proves (eq:cutoffresidual).

Thus a future fourth-order continuation must audit both density completion and the fixed profile $a$. The plateau of $g,{\sigma }_{\epsilon }$ does not make $a$ constant. No spurion, compensator, changed ghost algebra, or fourth-order repair is introduced here.

The natural positive-order scalar field rule $s\varphi =\kappa c^{\mu }{\partial }_{\mu }\varphi$ does not preserve the unchanged free scalar equation ideal. The additional vertex $\lambda {\kappa }^3{V}_a$ does not repair this failure.

Proof. For arbitrary smooth coefficient fields,

Choose $p>0$, $\omega =\sqrt{p^2+m^2}$, and

Both $P_m{\varphi }_0=0$ and $\square c=0$, but

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.

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 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 off-shell classical relative BV complex. It is not an interacting derivation of the unchanged free causal algebra.

Let ${\mathsf{Q}}_w$ denote the first background coefficient of the natural gauge-fixed metric Hessian, with

The two Lie derivatives have different tensor types. Define the frozen source-switched operator

For free metric solutions $u,v$, its reciprocity defect is

These are the X-G/X-G-ST diagnostic objects XG,XGST, not the Euler operator of (eq:newaction).

Choose a nonnegative compact spatial bump ${\chi }_0$ with $M_{\chi }=\int {\chi }_0{\mathrm{d}}^{3}x>0$, and choose $\zeta \in {C_c}^{\infty }((0,\epsilon ))$, nonnegative, such that

Set

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 ${\mathfrak{g}}_U$.

For these data,

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.

Proof. Since ${\mathsf{P}}_{\mathrm{gr}}u={\mathsf{P}}_{\mathrm{gr}}v=0$, equation (eq:typedcommutator) gives ${\mathsf{Q}}_{w}u=-{\mathsf{P}}_{\mathrm{gr}}{\mathcal{L}}_{{\xi }_{\mathrm{on}}}u$ and likewise for $v$. The $00$ Lie components are

Their $0i$ components do not change the $00$ trace-reversed output used in the pairing. For such a tensor $z$, $({\mathsf{P}}_{\mathrm{gr}}z{)}_{00}=-\square z_{00}/8$. The spatial Laplacians integrate to zero, leaving

Two compact integrations by parts prove (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.

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 (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 (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 $\mathrm{supp}a⋐U$.

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

The matter normalization $-2$ belongs to that already counted source coefficient.

The additional vertex in (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.

Proof. The new data begin at order three. Apply Lemma lem:orderprotection.

The parent identity and its kernel-admission hypotheses remain those of X-F/X-F-ST 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 (eq:trees). No equality of different quantum ordering presentations is assumed.

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

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.

Proof. The added vertex has no quadratic part at the zero configuration. The definitions of the free operators in Section sec:free are therefore unchanged. Applying the same operations to the same free distributions gives the same diagnostic objects.

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 BF; it is not a definition of a missing nonlocal characteristic product.

Retain the fixed convolution data of X-A-conv and X-A-conv-ST XConv,XConvST. Their witness coefficient has the form