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

We examine the proposed one-loop continuation of the fixed-regulator relational scalar of Landmarks X-E and X-F. The graviton normalization is retained: $P_{\mathrm{gr}}=-\frac{1}{4}\mathcal{P}\square$ and ${{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}=-4\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$. The fermion $\int \bar{c}(C-b)$ constructs the corresponding free auxiliary–ghost multiplet. It gives $b=C[h]/2$, the required Feynman-weight gauge fixing, and the ghost term $-\bar{c}\square c$. The independent auxiliary field makes the free differential nilpotent off shell. Its Green matrix and compatibility with the causal field algebra are proved.

Two obstacles must be distinguished. The switched scalar action has an explicitly evaluated gauge-breaking coefficient. Separately, X-F multiplies gravitational sources by the ramp after variation. An exact variational upgrade retaining that field response on all allowed backgrounds is impossible: a compact pure-gauge background in the ramp yields a nonzero reciprocity pairing on genuine free metric solutions. This obstruction is classical and cannot be cancelled by a positive-quantum-order counterterm.

Under a stated Wick graph convention, two non-tadpole contraction graphs exhaust the ${\kappa }^2\hslash$ sector. Their coefficients contain no classical background. This proves a conditional background-identity statement, not existence of the complete observable. The local scalar–graviton fish admits finite-scaling-degree extensions. A scalar model of the nonlocal coordinate contraction has a null-collinear exceptional set for the sufficient wavefront product criterion. Failure of that criterion is not called nonexistence. The required full retarded product is not supplied by local covariance alone.

The free BRST construction and the exact-upgrade obstruction are released. An unconditional renormalized one-loop background identity for the unchanged X-F prescription is not claimed.

Figure fig:split-X-G records one view of the same separation. Figure fig:chain-X-G records one view of the same separation. Figure fig:pair-X-G records one view of the same separation. Figure fig:gate-X-G records one view of the same separation. The drawings add no hypothesis.

Figure fig:scope-X-G records that separation. The drawing adds no hypothesis.

and quantifiers{ Constructed: the free graded multiplet, one gauge fermion, its Green operators, the free differential, the natural background ghost operator, the switched-matter breaking, and the conditional graph inventory.

Proved obstructed: a variational/Bogoliubov upgrade whose classical metric field map agrees exactly with X-F's source-switched mixed map for every allowed background and free metric input.

Unresolved: the complete nonlocal one-loop retarded product and an unconditional one-loop background-identity theorem for the requested prescription.

These statements have different quantifiers. The obstruction is not a universal no-go for perturbative gravity, nor a disproof of X-F's direct polynomial plateau identity.}

X-F constructs three fixed-width tree symbols and proves their direct background identity through ${\kappa }^2$ at classical order XF. It also states that its gravitational switch acts on sources, not on action densities. The latter distinction matters when a quantum retarded map is requested.

The proposed continuation contains three logically separate tasks:

A tree kernel's admission proves none of the later tasks by itself.

Local covariance of renormalized products is also different from comparison of two interacting splittings with the free input fields held fixed. The latter comparison requires an explicit quantum transport or perturbative-agreement statement. The additional normalization conditions in HW, and the separate background transport analysis in TTZ, illustrate this distinction. Neither paper is used as an automatic completion theorem for the present switched nonlocal observable.

We give complete proofs of the finite algebraic and classical compatibility statements below. Conditional quantum statements are labelled as such.

An exact source-preserving upgrade of X-F consists of:

1. its frozen free metric operator and causal relations;

2. an even interaction functional with symmetric bosonic second variation, after elimination of the algebraic auxiliary field;

3. a classical retarded metric field map agreeing with X-F's mixed response on every smooth free metric solution and every allowed smooth classical background;

4. quantum corrections containing only nonnegative powers of $\hslash$.

Agreement in (c) is equality of the field map, equivalently equality after every compact smooth linear metric smearing. It is stronger than agreement of one selected relational observable.

This definition makes precise one natural meaning of a “Bogoliubov completion of X-F”. Its failure does not exclude a weaker prescription assigning selected polynomial symbols to an algebra, or a new density-switched theory.

Retain

Fix the X-E/X-F observation and scalar switch

Let $U$ be relatively compact with

Write $K_f=\mathrm{supp}f$. The separation condition is

The ramp ${\sigma }_{\epsilon }(t)$ is smooth, zero for $t\le 0$, and one for $t\ge \epsilon$. Monotonicity is not needed for the obstruction.

The backgrounds $h_{\mathrm{cl}},k_{\mathrm{cl}}$ remain arbitrary smooth off-shell tensors. In particular, (eq:separation) does not require them to vanish in $0<t<\epsilon$.

Metrics and metric fluctuations are covariant symmetric tensors. Their Euler derivatives are contravariant tensor densities. The Lie derivative on either side of an operator acts in the appropriate type.

For a vector field $\xi$ define

The plateau transformation is the inherited two-jet rule

Put

Then

The formal adjoint of $C$ is

Parentheses include the factor $1/2$.

Let $A=\frac{1}{2}{G}_{\mathrm{lin}}$. The identities copied from X-E are

The metric Green operators are

Thus the metric equation is equivalent, through a constant invertible fibre map, to a normally hyperbolic equation. We do not change its coefficient to $+4$.

The scalar action Hessian is $-P_m$, with retarded inverse $-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$. Its equation inverse is ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$. These two uses must not be conflated.

For the background de Donder gauge,

The unswitched $Q_w$ is the second variation of a quadratic background action and is formally self-adjoint.

Naturality supplies

This local identity holds for all smooth $\xi$, not only $\xi \in {\mathfrak{g}}_U$.

X-F's mixed source prescription is

It is (eq:mixed), rather than a silently density-switched replacement, that is tested in Section sec:obstruction.

The support assertion for $D_1$ in X-F requires correction. Its scalar completion contains $-{\mathsf{L}}_{1,g}[u]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f$, whose support need not be contained in $K_f$.

Define the compact sets

The first completed test

has a compact kernel $D_1(x,z)$ satisfying

Its local completion terms are supported on $x=z$. Its coordinate term has $x\in K_f$ and $z\in C_f$.

Proof. The coordinate term has the factor $f(x)$ and a retarded coordinate kernel with $z\in J^{-}(x)$ and $t_z\ge 0$. The local terms have factors $g(x)$, derivatives of the smooth ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f(x)$, and derivatives of $\delta (x-z)$. Since $\mathrm{supp}{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f\subset J^{-}(K_f)$, their first slot lies in $\mathrm{supp}g\cap J^{-}(K_f)$ and their second slot equals it. All these sets are compact.

Consequently, the first scalar slot of a completed tree is bounded by $O_f$, not necessarily by $K_f$. Replacing the too-small bound by (eq:Dsupport) preserves the relevant compactness arguments. It does not change the affine-slot or direct plateau identities.

In this paper X-F's direct polynomial background identity is used as a prior premise, with this support correction. The unavailable earlier manuscripts are not presented as independently verified proofs.

Introduce independent fields

with parities $(0,0,1,1,0)$ and ghost numbers $(0,0,1,-1,0)$.

Define an odd derivation commuting with spacetime derivatives:

The differential (eq:s0) is nilpotent on the independent fields and their graded polynomial differential algebra.

Proof. Its square vanishes on each generator. The square of an odd derivation is an even derivation, so it vanishes on the algebra generated by them.

Choose

The graded Leibniz rule and $CK=\square$ give

The auxiliary equation is $b=C[h]/2$. Eliminating it gives

and therefore reproduces the frozen gauge-fixing action $\frac{1}{4}\int C_{\mu }{C}^{\mu }$. In the notation $\Psi =\int \bar{c}(C+\alpha b)$, the required value is $\alpha =-1$.

Proof. Variation in $b$ gives $C-2b=0$. Completing the square proves the result.

The signs are fixed by the chosen action and by $s_0\bar{c}=b$, not by an assumed positive auxiliary measure. After eliminating $b$, ${s_0}^2\bar{c}=\frac{1}{2}\square c$ is only an on-shell identity. For off-shell nilpotence, the independent auxiliary field is retained.

The bosonic Hessian is

For sources $(J,j)$ its equations are

The retarded and advanced inverses are

Proof. The auxiliary equation gives $b=(Ch-j)/2$, hence

Apply its Green operator and substitute back. The identities

follow from (eq:Cstar), ${\mathcal{P}}^2=\mathrm{id}$, and $CK=\square$. They give the second matrix. Source elimination proves both inverse identities on compact tests. All blocks have the stated causal support.

The algebraic $b$ equation does not require imposing zero Cauchy data on every free field. Retarded inverses select responses, not a single free configuration.

Use left Grassmann variations. In the order $(c,\bar{c})$, the ghost kinetic matrix and its inverses are

The flat vector–covector fibre pairing is implicit. Matrix multiplication verifies both inverse identities.

Let

The nonzero causal blocks are

The ghost block has the required graded symmetry because ${{\delta }_0}^{\mathsf{T}}=-{\delta }_0$.

Define ${\mathcal{A}}_0$ by compactly smeared linear fields, the equations

and

The graded causal form descends to the equation quotient. Its central graded Lie algebra has an enveloping algebra, providing the corresponding associative polynomial carrier.

The differential (eq:s0) preserves the free equations and causal relations. It induces a nilpotent graded derivation of ${\mathcal{A}}_0$.

Proof. The variation of the first metric equation is $AKc=0$. The variation of $Ch-2b$ is $\square c$. The antighost equation varies into $\square b$.

The latter is a consequence of the bosonic equations: the linearized Bianchi identity and

give $\square b=0$. The remaining equations are stable.

For the causal relations, the nontrivial cancellation is the initially zero $h$–$\bar{c}$ relation:

The reversed relation follows by graded symmetry. All other variations use zero cross-blocks or the zero $cc$ and $bb$ blocks. Nilpotence descends from Lemma lem:s0nil.

For a concrete normal-ordering presentation, choose flat scalar product kernels $H_{\mu }$ of a common Hadamard orientation, normalized by

No positivity is required. A compatible translation-invariant multiplet is

All unspecified blocks vanish.

The blocks are bisolutions, their graded antisymmetric parts are $i\Delta$, and

The reversed identity follows by slot interchange. Thus the free differential is also a derivation of the compatible polynomial Hadamard product.

For compact microcausal polynomial kernels this product is the finite graded contraction expansion; the wavefront criterion is unchanged by the finite field matrix and Koszul signs FR.

The notation in (eq:Hadamard) distinguishes the scalar equation kernel from the scalar action-Hessian kernel. A prefactor alone does not reverse a wavefront orientation. Consequently no sign change from $-4$ to $+4$ is justified merely by comparing two differently normalized scalar symbols.

If inherited Hadamard data are not translation invariant, the auxiliary blocks must be defined by their actual slot derivatives and the compatible presentation transported by the smooth ordering isomorphism. An unchanged component formula is not silently asserted in a different ordering coordinate.

Write

The complete formal background may include ${\kappa }^2{k}_{\mathrm{cl}}$; it is held fixed under the quantum-gauge differential.

Define

Odd coefficients are placed on the left in the component Lie-derivative formulas.

The rule (eq:nonlinear) is nilpotent on the off-shell graded field algebra.

Proof. Let $w^{\mu }=c^{\nu }{\partial }_{\nu }{c}^{\mu }$. The graded Lie-derivative identity is ${{\mathcal{L}}_c}^2={\mathcal{L}}_w$. It follows directly for scalars and covectors by expansion; the symmetric second-derivative term vanishes by anticommutativity of the ghosts. The identity extends to tensor products by the graded Leibniz rule.

Since $sq=\kappa {\mathcal{L}}_{c}q$,

The scalar calculation is identical. Direct expansion of $s(c^{\nu }{\partial }_{\nu }{c}^{\mu })$ gives zero after substitution of $sc$ and antisymmetrization. The nonminimal pair is nilpotent by definition.

This is a kinematical differential, not yet an interacting quantum BRST operator.

For the background de Donder vector of X-F, set

The background is $s$-inert, so

Eliminating $b$ gives exactly the bosonic gauge fixing of X-F.

Define the background ghost kinetic operator, with its Euler type, by

With X-E's curvature conventions,

Moreover,

If

then

The left Lie derivative acts on vector densities and the right one on vectors.

Proof. Trace reversal of ${\mathcal{L}}_c\bar{q}$ gives

Take its divergence and commute the two derivatives on the vector. The commutator contraction is ${\bar{R}}^{\mu }{}_{\nu }{c}^{\nu }$, giving (eq:ghostclosed).

The metric connection, Lie derivative, contractions, and volume density are natural. This proves the finite identity. Differentiate it along a compact flow through $\eta$ to obtain (eq:ghostcomm); locality removes the compactness restriction from the resulting differential identity.

For $\xi \in {\mathfrak{g}}_U$, the retarded inverse ${G_{\mathrm{gh}}}^{\mathrm{R}}=-{{\mathsf{E}}_0}^{\mathrm{R}}$ gives the analogous first-order response identity

Indeed, ${\mathcal{L}}_{\xi }c$ is compactly supported and ${G_{\mathrm{gh}}}^{\mathrm{R}}{P_{\mathrm{gh}}}^{(0)}{\mathcal{L}}_{\xi }c={\mathcal{L}}_{\xi }c$. Equation (eq:ghostjet) is a retarded equation identity. It does not establish a nonlinear interacting quantum differential.

Consider the natural reference action

where

This action is used to evaluate its natural gauge variation. Its Euler equation is not identified with X-F's post-variation source-switching prescription.

Put

For compact ghost coefficient functions, modulo integrated total derivatives,

Equivalently,

where

Proof. Naturality gives $s{\mathfrak{L}}_q=\kappa {\partial }_{\mu }(c^{\mu }{\mathfrak{L}}_q)$. The flat metric is held fixed, so its scalar density instead has variation

Weight these identities by $g$ and $1-g$ and integrate the divergence terms. This proves (eq:sSfirst). One more integration by parts and the displayed stress divergence give (eq:B).

The Einstein–Hilbert variation is a compact divergence. The variation of $s{\Psi }_{\bar{q}}$ is zero by Lemma lem:nonlinear.

The full off-shell breaking is not supported only in $\mathrm{supp}\mathrm{d}g$: its equation term with $1-g$ is also present.

Every nonzero compact scalar switch with a plateau admits a free scalar and a compact odd ghost test for which the coefficient of $\kappa$ in (eq:sS) is nonzero.

Proof. Use the spatially homogeneous free solution

It has ${T_0}^{00}=1/2$ and ${T_0}^{0i}=0$. At the leading metric $q=\eta$, the density difference is zero.

Along a vertical line through a plateau point, $g$ rises from zero to one. Therefore ${\partial }_{t}g>0$ on some open interior region. Choose a nonnegative nonzero $\zeta \in {C_c}^{\infty }$ there and an odd Grassmann generator $\vartheta$. Set $c=\vartheta \zeta {\partial }_t$. Then

An ordinary canonical BV extension generating (eq:nonlinear) and having $S_{\mathrm{ref}}$ as its antifield-independent action cannot satisfy the classical master equation with the scalar switch and transformations unchanged. A correction starting strictly at $\hslash$ cannot cancel this classical breaking.

Proof. The antifield-independent coefficient of the classical master equation requires $s{S}_{\mathrm{ref}}=0$. Theorem thm:matter violates this necessary condition. Coefficient extraction at ${\hslash }^0$ proves the final assertion.

This is not a theorem that every observable quantum Ward coefficient is nonzero. Nor does it exclude a modified spurion rule, extra compensators, or a different local-coupling construction. Those would change the sheet.

For a ghost test supported inside $P_g$, the two terms in (eq:sSfirst) vanish. The restricted symmetry therefore survives.

Under a simultaneous background pullback supported in $U$, the metric, fluctuation, scalar, ghost, antighost, and auxiliary fields transform in their tensor types. The fermion and ghost density are natural, $g\circ F^{-1}=g$, and the flat scalar term is unchanged where its coefficient $1-g$ is nonzero. Thus the integrated reference action is exactly invariant under this simultaneous pullback.

This background covariance is different from quantum-gauge invariance with $\bar{q}$ held fixed.

If an observation point belongs to $P_g$, its vertical past curve crosses a region where $g$ changes from zero. Hence $\mathrm{supp}\mathrm{d}g$ meets its causal past. Without an observation-plateau or comparable causal hypothesis, compactness alone does not prove the asserted intersection.

Even a nonempty causal intersection does not prove a nonzero ghost contraction with a particular observable. Propagator support, index contractions, and smearings still have to be evaluated.

For a fixed background $w$ define

Since $Q_w$ is formally self-adjoint,

Thus source multiplication generally destroys the symmetry of the action Hessian.

The following on-shell test excludes an equation-zero representative repair.

Let an even variational interaction have reduced mixed metric Hessian $L_w$, with compact coefficients. If its classical response agrees with $-{{\Delta }_{\mathrm{R}}}^{\mathrm{gr}}{\mathsf{N}}_{w}u$ on all free metric solutions, then

for all such solution pairs.

Proof. Symmetry of second directional derivatives gives

Equality of the retarded responses, followed by application of $P_{\mathrm{gr}}$, gives $L_{w}u={\mathsf{N}}_{w}u$ on every free input. Substitute into the symmetric pairing. Compactness of the coefficients removes all boundary terms.

A smooth ramp changing from zero to one has ${\sigma }_{\epsilon }^{\prime \prime }\not\equiv 0$ on $(0,\epsilon )$. Indeed, otherwise it would be affine there, and endpoint smoothness would force zero slope and hence a constant.

Choose a nonnegative $\zeta \in {C_c}^{\infty }((0,\epsilon ))$ supported where ${\sigma }_{\epsilon }^{\prime \prime }\ne 0$, with $\zeta \not\equiv 0$, and put

Choose

Set

This is a permitted classical background. The vector field $\xi$ here is not asserted to belong to the plateau algebra: it parametrizes the chosen background, not an admitted plateau transformation.

Let $e_{00}$ be the tensor whose only nonzero covariant component is $(e_{00}{)}_{00}=1$, and take

Both satisfy $P_{\mathrm{gr}}u=P_{\mathrm{gr}}v=0$.

For the background (eq:rampgauge),

Proof. By (eq:commutator), on a free input, $Q_{w}u=-P_{\mathrm{gr}}{\mathcal{L}}_{\xi }u$.

Directly,

and

For a tensor with no spatial–spatial components,

It follows that

After spatial integration, the Laplacian terms vanish. Since

we obtain

Twice integrating by parts gives (eq:Dnonzero).

The homogeneous solutions can be replaced by free solutions with compact Cauchy data. Choose their initial data equal to those of $u,v$ on a ball containing the initial causal shadow of $\mathrm{supp}w$, and multiply by a larger compact cutoff. Finite propagation preserves equality on a neighbourhood of $\mathrm{supp}w$. Since $Q_w$ is local and supported there, the value (eq:Dnonzero) is unchanged.

Together with $b=C[u]/2$ or $C[v]/2$ and zero scalar and ghost fields, these are configurations of the full free multiplet.

There is no upgrade satisfying Definition def:upgrade on X-F's unrestricted background class.

The contradiction remains if off-shell representatives are changed by terms vanishing on all free configurations. It cannot be removed by the algebraic auxiliary field, ghost loops, or counterterms beginning strictly at quantum order.

Proof. Linearize the claimed reduced action response at zero fluctuation, zero scalar, and zero ghosts, in the background (eq:rampgauge). Its mixed metric Hessian must satisfy Lemma lem:reciprocity. Proposition prop:Dsigned contradicts that condition.

Terms vanishing on all free configurations do not change the witness. Algebraic auxiliary elimination yields another action, whose second variation is still symmetric. Ghost loops and positive-$\hslash$ counterterms do not change the classical coefficient used in the contradiction.

The quadratic functional

has Hessian

not ${\sigma }_{\epsilon }{Q}_w$. Actual switching of a selected density before variation similarly supplies ramp-derivative terms, with details depending on the density representative.

These are classical corrections. They cannot be hidden inside a finite renormalization that starts at $\hslash$.

For $w=K\xi$ with $\xi \in {\mathfrak{g}}_U$, the relevant support lies where ${\sigma }_{\epsilon }=1$. The witness of this section does not occur in X-F's particular plateau variation. Therefore the direct three-tree identity of X-F is not refuted.

What fails is the stronger identification of its entire source-switched field map with the classical part of one variational Bogoliubov map for all permitted backgrounds.

For an actual variational theory, write

and let the bare relational insertion be

At first order,

The scalar completion $-\varphi ({\mathsf{L}}_{1,g}[u]{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f)$ is generated by the scalar response. It must not also be inserted as an independent bare term and then regenerated by the same scalar retarded map.

With

the coefficient at order ${\kappa }^2$ is

This identity is combinatorial. It does not identify a density-switched tree coefficient with X-F's source-switched one.

For the inventory below:

1. free cross-blocks join only matching field species, apart from the ghost–antighost block;

2. the auxiliary field has been eliminated algebraically;

3. interaction vertices and insertion monomials use a fixed normal-ordering convention with no self-contractions at a single vertex;

4. connected contraction graphs are expanded coefficientwise in the flat free carrier; backgrounds occur through the explicit vertex coefficients;

5. no additional explicit quantum counterterm is included in the inventory.

At order $\kappa$, the relevant leg types are

Only $h\varphi \varphi$ carries both a metric and a scalar leg without a background coefficient.

Under Assumption ass:graphs, the connected ${\kappa }^2\hslash$ coefficient of (eq:fullcoefficient) has exactly two contraction types:

1. ${\mathsf{L}}_{\alpha }$: two lines between the $h\varphi$ part of $O_1$ and one $h\varphi \varphi$ vertex;

2. ${\mathsf{L}}_{\beta }$: the $F_f$ scalar line attaches to one $h\varphi \varphi$ vertex, and two lines, one metric and one scalar, join that vertex to a second $h\varphi \varphi$ vertex.

Each type has one external scalar leg. Its unextended flat contraction coefficient contains no $h_{\mathrm{cl}},k_{\mathrm{cl}}$, ghost field, gravitational cubic vertex, or background gauge-fixing coefficient.

There are no connected contraction terms at ${\kappa }^2{\hslash }^r$ with $r\ge 2$ under these assumptions.

Proof. For $n$ interaction vertices and $E$ contraction lines, the quantum power is $E-n$. Connectedness requires $E\ge n$.

For $n=0$, the insertion is normal ordered, so no quantum contraction occurs.

For $n=1$, the order-one insertion has either one scalar leg or one scalar and one metric leg. Quantum order one requires $E=2$. Both lines must end on the two-leg insertion. The only order-one vertex with the required species is $h\varphi \varphi$. It leaves one scalar leg. A single $V_2$ attached to the one-leg $F_f$ has at most one line and cannot contribute at quantum order one.

For $n=2$, the insertion is $F_f$ and has degree one. Quantum order one requires $E=3$. One line attaches $F_f$ to a scalar-bearing vertex; the other two lines must run between the two interaction vertices. The vertex attached to $F_f$ must therefore have two remaining legs. The background $\varphi \varphi$ vertex does not. The $h\varphi \varphi$ vertex does, leaving $h\varphi$; the only second vertex matching both species is another $h\varphi \varphi$ vertex. Again one scalar leg remains.

The same degree bounds give $E\le 2$ for $n=1$ and $E\le 3$ for $n=2$. Thus $E-n\le 1$. Orders $n\ge 3$ require at least ${\kappa }^3$.

The proposition does not construct either required nonlocal retarded operation. It also does not classify every allowed finite renormalization. An explicitly added order-${\kappa }^2\hslash$ counterterm is outside Assumption ass:graphs and must be tested separately.

The absence of higher-loop contraction graphs is therefore not an unconditional all-$\hslash$ theorem about every possible normalized observable.

In the declared convention, a one-vertex tadpole is absent. A background-dependent graviton, ghost, or scalar one-point graph requires a second vertex. Feeding that order-${\kappa }^2$ metric one-point response through the order-$\kappa$ relational metric insertion first contributes at ${\kappa }^3$.

For example, the possible pairs are

The two legs between the vertices form the loop, while the metric leg reaches the relational insertion.

A different normal-ordering convention can generate one-vertex tadpole normalization terms. Such terms must be included explicitly, rather than declaring the two-graph inventory universal. In particular, the word “one-point function” must not silently mean an expectation value in an unspecified state.

The main lesson is narrower than the original plan: ghosts are necessary for the BRST carrier, but the non-tadpole scalar coefficient at ${\kappa }^2\hslash$ does not contain a ghost loop.

Let ${D_F}^{\mu }$ denote flat scalar Feynman product data and

The metric–scalar fish between two matter vertices has finite sums of component kernels of the form

Tensor coefficients and external scalar derivatives are suppressed.

The kernels (eq:fish) are well defined away from $r=0$ and have scaling degree at most six in four dimensions. They admit extensions with that scaling-degree bound. Two such extensions differ by

with the relevant tensor coefficients and lower-degree restrictions for the less singular terms.

Proof. Away from coincidence, null-related Feynman singularities in the two factors have the same first-slot orientation, so their nonzero wavefront covectors cannot add to zero. At spacelike separation both factors are smooth.

A scalar Feynman kernel has scaling degree two. At most two derivatives occur on the contracted scalar line, so the product has scaling degree at most $2+2+2=6$.

The finite-scaling-degree extension theorem gives an extension without increasing that bound BF. A difference is supported at $r=0$. A point-supported distribution of scaling degree at most six in four dimensions has the form (eq:fishambiguity).

The derivative order two in (eq:fishambiguity) is the internal extension ambiguity. The external scalar factors may already carry derivatives. It would be incorrect to infer that every resulting local quadratic density has total derivative order at most two.

Let $k(y,y^{\prime })$ be a distribution with $\mathrm{WF}(k)\subset {\Sigma }_2$, where

For a compact smooth $a(y)$, the properly supported pairing $\int k(y,y^{\prime })a(y)\mathrm{d}y$ is smooth in $y^{\prime }$.

Proof. The pushforward wavefront bound can only produce an output covector ${\eta }^{\prime }$ from a kernel covector with $\eta =0$. The sum-zero condition then forces ${\eta }^{\prime }=0$, which is not a wavefront covector.

A translation-invariant extension of (eq:fish), lifted as $k(y,y^{\prime })=t(y-y^{\prime })$, has wavefront set in ${\Sigma }_2$. Applying Lemma lem:smoothing to the compact scalar switch and the smoothly smeared insertion leg shows that the isolated local two-vertex contribution has the form

with compact support supplied by the remaining vertex switch. This is a local graph construction with the stated finite normalization freedom. It is not a construction of the nonlocal mixed retarded product.

The coordinate part of the bare insertion contains a derivative of ${{\mathsf{E}}_0}^{\mathrm{R}}(x,z)$. Consequently a representative time-ordered scalar contraction has the underlying product

The actual insertion has derivatives, tensor contractions, and smooth coefficient functions. Equation (eq:triangle) is used only to audit the scalar product criterion, not to assert the full loop coefficient equals this one term.

Choose the Hadamard orientation so that the first-slot Feynman covector is future-pointing at the later endpoint and past-pointing at the earlier endpoint. Reversing the Fourier convention reverses all these signs without changing the argument.

Define

It includes the corresponding partial coincidences and the total diagonal.

For the scalar kernels in (eq:triangle), the sufficient wavefront product criterion holds off $\mathcal{N}$. At a strictly ordered null-collinear triple $z\prec y\prec x$ it fails.

This failure does not prove nonexistence of a distributional extension, nor nonexistence of the fully contracted retarded coefficient.

Proof. The three pulled-back wavefront covectors have the forms