Landmark XI-C One fixed-regulator graviton tree for the ramp relational scalar: the retarded symbol, its normalization, and the boundary of the free-algebra handoff

We construct the one-graviton retarded tree determined by the signed gauge-fixed inverse of Landmark XI-B and the switched stress-tensor vertex of Landmark XI-A. At fixed regulator width, the bilinear coordinate insertion and the cubic scalar output have compact distributional kernels satisfying the polynomial microcausal exclusion. The contraction is defined directly on smooth off-shell configurations, without a Hadamard covariance.

For a smooth free scalar and a switch above the ramp, the coordinate response is minus one half of a double retarded wave inverse applied to the divergence of the switched stress. It vanishes exactly when that source is conserved. The named XI-A input gives a nonzero response and does not repair its empty Einstein jet.

An essential domain qualification prevents an unconditional quantum release. XI-B constructed finite products of compactly smeared linear fields, not an extension admitting the singular polynomial kernels used here. We prove the distinction and do not substitute a wavefront estimate for an admission map. A restricted classical second-order candidate has an explicit, generically nonzero background defect; its sign and domain are specified. No quantum second-order coefficient, interacting BRST class, mixed loop, Hadamard selection, or sharp limit is constructed. The classical symbol tree is complete; its output in the unchanged quantum algebra remains unresolved.

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

Figure fig:scope-XI-C records that separation. The drawing adds no hypothesis.

: a constructed retarded symbol, with the quantum admission gate left explicit{ The joint fixed-width kernel bound, the one-graviton classical retarded contraction, its cubic output symbol, causal support, and normalization are proved.

The literal quantum target in the unchanged XI-B algebra is Unresolved: that algebra does not include a declared admission of these distributional polynomial symbols. A separate quantization or extension theorem is required.

The mixed loop is also unresolved. The empty fixed-input Einstein jet is not repaired. Neither unresolved item is a wave-equation obstruction to the retarded symbol.}

The proposed calculation involves two logically separate steps: constructing a retarded polynomial functional and admitting that functional as an element of a quantum algebra. The first step does not require a state. The second cannot be inferred solely from the existence of a causal inverse.

XI-B's Definition of the free causal polynomial algebra uses compact interior smearings of linear fields and their finite products. Its construction expressly does not extend to all local or microcausal polynomial functionals. The local vertex and the nonlocal bilinear insertion in the present problem therefore require more than the algebraic operations already handed off.

We retain that limitation rather than silently replace ${\mathcal{A}}_0$ by a Wick algebra. The additional space used below, ${\mathcal{P}}_{\mathrm{mc}}$, is a space of classical polynomial symbols. No quantum product on that space is selected. The usual distinction between regular polynomial generators, local polynomials, and microcausal functionals is relevant here FR,HRV.

For the conventions and support class below:

1. The bilinear insertion ${\mathcal{B}}_f=-\int f{\mathsf{X}}_1[\gamma ]\cdot \partial \varphi$ has a compact mixed kernel satisfying the polynomial microcausal exclusion.

2. The one-graviton retarded contraction is a well-defined continuous cubic classical scalar polynomial with a compact microcausal kernel.

3. Its intermediate metric uses precisely ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$. For free scalar input and a switch above the ramp, its coordinate response is the double retarded integral displayed in Theorem thm:normalization.

4. The classical restricted second-order candidate defined in Section sec:boundary has an explicitly nonzero background defect for suitable free inputs.

These statements do not establish an admission map into ${\mathcal{A}}_0$, a quantum second-order Ward identity, or an interacting differential.

The alternative endings remain distinct. There is no wavefront obstruction to the classical tree constructed here. There is also no proved theorem excluding every possible quantum extension. The unmet step is admission into the frozen target, not an unnamed propagator normalization.

Use

All classical configurations are smooth on a neighbourhood of the closed slab. Parentheses include the factor $1/2$.

Fix $0<\epsilon <T$, with the same smooth ramp ${\sigma }_{\epsilon }(t)=\chi (t/\epsilon )$ as XI-A and XI-B:

Fix $g\in {C_c}^{\infty }({M_T}^{\circ };[0,1])$, with nonempty plateau $P_g=\mathrm{int}\{g=1\}$, and

The observation test is $f\in {C_c}^{\infty }({M_T}^{\circ };\mathbb{R})$, and $K=\mathrm{supp}f$. No limit of $\epsilon$ is taken.

For a symmetric tensor $r$, put

XI-B supplies

Here ${{\mathsf{E}}_0}^{\mathrm{R}}$ is the zero-initial-data inverse of $\square$. Its advanced counterpart has zero final data. Their causal support and smooth Cauchy properties are those used in XI-B XIB,BGP.

The coefficient of $\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$ is $-1$, not $-1/2$. The scalar equation inverse is ${{\mathsf{E}}_{\mathsf{m}}}^{\pm }$; its action-Hessian inverse is $-{{\mathsf{E}}_{\mathsf{m}}}^{\pm }$. Thus

The auxiliary field is retained, with

The classical XI-A parameter $K\xi$ corresponds to $c=2\xi$. Eliminating $b$ gives only on-shell nilpotence of the reduced antighost rule. Ghosts and antifield source slots are spectators in the contraction below.

XI-B also gives

Consequently the plateau algebra is not a nontrivial residual action on free de Donder solutions. Its classical background action and the free graded differential are different objects.

The only contracted vertex is

where

It is a classical local polynomial before any quantum admission.

Write

The coordinate map is the linear operator

The layer formula alone applies only when $Dr=0$. It cannot be imposed on a sourced response with nonzero divergence.

The cubic Einstein vertex is not contracted here. Its coupling degree permits contributions at the same order as the present tree. Its omission is a scope restriction, not a consequence of counting one metric leg.

Let ${\mathcal{P}}_{\mathrm{mc}}$ be the space of finite polynomial functionals on smooth configurations whose compact distributional derivative kernels satisfy

Here ${\tilde{V}}_{\pm }$ are the closed causal covector cones with zero covectors included; the total zero covector is omitted from a wavefront set. Field indices are finite dimensional and do not change the criterion.

This is a classical symbol declaration. No Hadamard product, locally convex quantum completion, or inclusion into ${\mathcal{A}}_0$ is contained in the definition.

Let ${G_0}^{\mathrm{R}}(x-z)$ be the retarded wave kernel and define

Integration by parts, with the flat ramp removing the initial boundary term, gives

All integrals are restricted to the slab where necessary.

The bare insertion is

Ordering the variables as scalar slot $x$ and metric slot $z$, its kernel is

The mixed kernel is compactly supported and satisfies

In particular, ${\mathcal{B}}_f\in {\mathcal{P}}_{\mathrm{mc}}$. Its metric support is contained in

and its scalar support is contained in $K$.

Proof. The retarded wavefront has opposite endpoint covectors: diagonal conormals and the off-diagonal null propagation relation. Smooth multiplication by ${\sigma }_{\epsilon }$ and $f$, followed by differentiation, introduces no new wavefront directions BDH.

If $p$ and $-p$ both belong to the same closed causal cone, then $p=0$, which is excluded. Thus the mixed kernel obeys (eq:microcausal). Lower derivative kernels are obtained by smearing one slot with a smooth configuration. Their wavefront bounds have zero covector in that integrated slot, and hence zero in the remaining slot; they are smooth.

Retarded support gives $z\in J^{-}(x)$ and the ramp removes negative-time dependence. The causal past of a compact observation set within a bounded-time closed slab is compact. Therefore the mixed kernel is compactly supported.

The factor ${\sigma }_{\epsilon }$ is smooth even at $t=0$. No sharp step, boundary delta, or reflected sharp-Cauchy branch has been inserted. This proof uses the actual joint kernel, not smooth scalar tests evaluated at a fixed metric.

Let ${\mathcal{P}}_{\mathrm{fin}}$ denote the off-shell polynomial algebra generated by compact smooth linear smearings. Every degree-$n$ kernel in that algebra is a finite sum of smooth tensor products:

The classical limit of XI-B's construction uses precisely this finite-generator presentation before its equation quotient.

For suitable nonzero $f$ supported above the ramp, the actual off-shell mixed kernel $b_f$ is not a kernel of the form (eq:finite). The local vertex (eq:vertex) likewise does not have a smooth finite-rank off-shell kernel. Consequently the joint bound does not place either input in XI-B's declared off-shell generator algebra.

Proof. Choose $f$ nonzero near a point above $\epsilon$, where ${\sigma }_{\epsilon }=1$. Microlocally at a noncharacteristic diagonal covector, the wave inverse has principal symbol proportional to $({\eta }^{\alpha \beta }{p}_{\alpha }{p}_{\beta }{)}^{-1}$. The two endpoint derivatives in $b_f$ give principal symbol proportional to

For $p=\mathrm{d}t$ and $ab=00$, the numerator is nonzero: ${\mathfrak{a}}^{00,00}=-1/2$. The kernel therefore has a nonzero diagonal singularity and is not smooth. Every kernel in (eq:finite) is smooth.

The vertex kernel contains differentiated diagonal delta distributions linking its three field slots. Its nonzero principal derivative coefficients cannot be represented by a smooth kernel. This gives the same distinction for the vertex.

The proposition is about actual off-shell representatives. It does not prove that every possible on-shell cubic symbol has no alternative finite-generator representation. Such an assertion would require a separate argument.

It does prove that the production plan cannot use an off-shell kernel bound as a substitute for a supplied admission map. XI-B has neither a topology for taking these kernel limits nor a declared Wick-polynomial extension. No algebraic limit operation may be inferred from its CCR.

Put

The source is smooth and compact for every smooth $\varphi$. The intermediate tensor is an actual zero-data sourced solution:

It is not a homogeneous free graviton class. Its auxiliary response is $b_g=D{r}_g$, not an imposed zero.

Define

This is the positive dressing quantity specified in the production sheet.

The retarded response of a functional $F$ to a vertex $V$ has the classical sign $-F^{\prime }{G}^{\mathrm{R}}{V}^{\prime }$. Since ${\delta }_{\gamma }{V}_{1,g}=-S$ and the insertion (eq:B) has a minus sign, its one-graviton component is

Thus $\mathcal{T}$ and the contribution to the actual bare relational coefficient have opposite signs. The physical coupling degree of (eq:physicalsign) is ${\kappa }^2$.

The functional $\mathcal{T}(f,g)$ is a continuous homogeneous cubic polynomial on smooth scalar configurations. Its polarized kernel is compact and obeys (eq:microcausal). Hence it belongs to the classical cubic symbol space ${{\mathcal{P}}_{\mathrm{mc}}}^{(3)}$. The construction requires no Hadamard covariance.

Proof. For smooth $\varphi$, the compact smooth stress source in (eq:metricresponse) has a unique smooth retarded response. Applying $L_{\epsilon }$ gives a smooth coordinate response. Multiplication by the smooth scalar derivative and integration against $f$ therefore defines a number. Smooth Cauchy continuity on fixed causal compacts proves continuity. Homogeneity is quadratic in the stress and linear in the remaining scalar factor.

For the distributional kernel, compose the coordinate kernel and the graviton inverse:

Before integrating $z$, each propagator has endpoint covectors summing to zero. The two factors share only $z$. A violation of the product criterion would force the unshared $x$ and $y$ covectors to vanish, and then force the shared covector to vanish as well. The product is therefore defined. Proper integration over the bounded causal intermediate region gives a distribution with opposite $x,y$ covectors.

Attach the local stress kernel at $y$, using independent scalar slots $u,v$, and the final scalar slot at $x$. The stress diagonal kernel has total covector sum zero. Each attachment shares only one slot with the preceding kernel, so the same product argument applies. Derivatives do not change the zero-sum property.

After the proper pushforwards, every external covector tuple in the cubic kernel satisfies $p_1+p_2+p_3=0$ in the fixed Cartesian trivialization. A nonzero tuple with all entries future causal or all entries past causal cannot satisfy this identity. This proves (eq:microcausal). Permuting and symmetrizing scalar slots preserves the conclusion.

The support cutoffs $f$ and $g$, and retarded propagation inside the finite slab, make all these pushforwards proper. The distributional operation criteria are used with their support requirements, not merely as formal kernel multiplication BDH.

In particular, a source-diagonal stress tensor does not produce a square of a propagator on identified endpoints in this tree. Two successive retarded kernels are composed through an independent intermediate variable.

The output is not a linear test class in $\mathcal{D}/P_m\mathcal{D}$. It is a cubic classical polynomial symbol. A quantum admission into ${\mathcal{A}}_0$ is a separate step not supplied by Theorem thm:tree.

The response is evaluated before any equation quotient. For unrestricted $\varphi$,

Dropping the first term is legitimate only after selecting a smooth free input for a classical reduction.

Likewise, replacing $r_g$ by an unspecified homogeneous metric class discards its sourced equation and generally changes $D{r}_g$. These two warnings are the scalar representative warning of X-B-v3 and the sourced-metric warning of XI-B in their respective slots.

Assume for this section that

This condition is automatic for sufficiently small fixed width relative to a fixed compact interior switch, but is not imposed retroactively on switches overlapping the ramp.

For every smooth free scalar ${\varphi }_0$,

The complete vector in (eq:double) vanishes identically on the slab if and only if the switched free source is conserved:

Proof. Trace reversal is involutive, so $\mathcal{P}{r}_g=-\frac{1}{2}{{\mathsf{E}}_0}^{\mathrm{R}}(g{T}_0)$. Differentiation commutes with the retarded inverse on these compact interior sources. Taking the divergence and using ${\partial }_{\nu }{T_0}^{\nu \mu }[{\varphi }_0]=0$ proves (eq:Dresponse).

Retardation makes $r_g$ and $D{r}_g$ zero below $\delta$. Thus the layer term in (eq:coordinate) is zero, and ${\sigma }_{\epsilon }D{r}_g=D{r}_g$ wherever this response can be nonzero. Consequently ${\mathsf{X}}_1[r_g]={{\mathsf{E}}_0}^{\mathrm{R}}D{r}_g$, proving (eq:double).

Applying ${\square }^2$ to (eq:double) recovers minus one half of the displayed source. Zero output therefore implies zero source. Conversely, zero source gives zero response by the zero-data inverse.

This is an identity of whole response fields. A fixed smearing can vanish even when the response does not; the “if and only if” statement is not a universal nonvanishing assertion for every $f$.

When (eq:above) fails, the correct formula is still (eq:coordinate), including both the factor ${\sigma }_{\epsilon }$ and the layer term. The unrestricted tree remains defined, but the simplified double-integral normalization need not hold.

For

XI-A gives ${T_0}^{00}=1/2$ and ${T_0}^{0i}=0$. Hence

A nonzero compact interior switch cannot be time independent. Injectivity in Theorem thm:normalization proves the nonzero claim.

Both $r_g$ and its divergence exist. The failure is the conservation condition needed for the ungauge-fixed Einstein equation. Therefore ${\mathcal{J}}_g({\varphi }_m)=\varnothing$ remains the XI-A result XIA. The tree does not provide a missing Einstein jet.

Take $m=0$, ${\varphi }_0=t$, and a nonnegative temporal bump $g_0$ above the ramp. Use a compact spatial cutoff equal to one on a sufficiently large causal cylinder containing the observation dependence region. In that cylinder,

Indeed, ${{\mathsf{E}}_0}^{\mathrm{R}}F(t)={{\int }_0}^t(t-s)F(s)\mathrm{d}s$ for a spatially constant source, and integration by parts in ${{\mathsf{E}}_0}^{\mathrm{R}}{{\mathsf{E}}_0}^{\mathrm{R}}({\partial }_t{g}_0)$ gives (eq:temporal).

A nonnegative observation bump after the source detects a strictly negative value of $\mathcal{T}$. The bare response $-\mathcal{T}$ has the opposite sign. Replacing the inverse coefficient $-1$ by $-1/2$ would halve this number and fail the copied normalization.

This is a classical evaluation on a smooth free solution, not a quantum expectation value or an admitted cubic element of ${\mathcal{A}}_0$.

In the tree kernel, the intermediate coordinate-source variable satisfies

The vertex variable satisfies $y\in \mathrm{supp}g\cap J^{-}(K)$. The external scalar support of $\mathcal{T}$ is contained in

All support bounds are compact in the closed slab.

Proof. The first retarded propagation requires $y⪯z$. The coordinate propagation requires $z⪯x$, with $x\in K$. The factor ${\sigma }_{\epsilon }$ gives the remaining restriction in (eq:zsupport). Transitivity gives the vertex bound. Two scalar slots come from the vertex and one from the observation, proving (eq:outputsupport). Finite time and compact endpoint supports give compactness.

The closed support of the smooth ramp can meet $t=0$. It is not a claim that ${\sigma }_{\epsilon }$ is nonzero there. Nor is (eq:zsupport) a ramp-layer support statement: the tree can propagate and remain nonzero long after $\epsilon$.

A pure-gauge diagnostic must be stated correctly:

For compact plateau $\xi$, the second expression is not generally zero. In fact

Thus invisibility to the invariant linearized Einstein tensor does not imply invisibility to the gauge-fixed operator.

Moreover, the tree (eq:T) has no external metric argument to set equal to $K\xi$. Its source is the scalar stress. A claimed vanishing of this tree merely because an unrelated classical background is pure gauge is therefore not a diagnostic of the constructed contraction.

To state a background transformation, introduce a prescribed classical tensor $h$, a second coefficient $k$, and an independent smooth tensor configuration $\gamma$. Put $H=h+\gamma$.

For this subsection the copied scalar-only completion is a classical polynomial functional:

where

For free $\varphi$ and $\xi \in {\mathfrak{g}}_U$,

More explicitly, the off-shell difference is the scalar equation test $⟨P_m(X_{\xi }{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}f),\varphi ⟩$. This is the first-order background identity proved in X-B-v3 XB; it is also verified by ${\mathsf{L}}_{1,1}[K\xi ]=D_{\xi }{P}_m-P_m{X}_{\xi }$, with $D_{\xi }v={\partial }_{\mu }({\xi }^{\mu }v)$.

Replacing the quantum metric by a fixed classical $r$ before taking the scalar variation gives

These are exactly the scalar-only maps. Setting $r=0$ gives their zero specialization. It does not recover a nonzero background response without a retained background.

Conversely, $\mathcal{T}$ is independent of an external $\gamma$. Setting $\gamma =0$ after taking the graviton response does not remove $\mathcal{T}$. Restriction of a configuration and taking its response do not commute in that fashion.

Use X-C-v3's completed scalar-only coefficient

Here $c_2$ is the copied bare second-coordinate test, and ${\mathsf{L}}_{2,g}[H,k]$ uses $a_1[k]+a_2[H]$ and $b_1[k]+b_2[H]$. All entries are evaluated on actual smooth configurations. This is not a quantum admission assertion.

Its dependence on the second metric coefficient is

Indeed, the $k$ contribution to ${\mathsf{X}}_2$ is ${\mathsf{X}}_1[k]$, and the $k$ contribution to ${\mathsf{L}}_{2,g}$ is ${\mathsf{L}}_{1,g}[k]$.

Define the restricted candidate

It names the scalar-only completion together with the single graviton component constructed here. It is not declared to be the full interacting-gravity coefficient.

For smooth free $\varphi$, hold $\gamma$ fixed and transform the classical coefficients by

Then the ordinary forward variation is

Equivalently, the backward defect has the production sheet's positive sign:

Proof. The full transformation of $H=h+\gamma$ would require

The supplied X-C-v3 background identity applies to these smooth coefficients and the free scalar XC. It identifies ${{\mathcal{A}}_2}^{\mathrm{sc}}[H^{\prime },{k^{\prime }}_{\mathrm{full}};\varphi ]$ with ${{\mathcal{A}}_2}^{\mathrm{sc}}[H,k;\varphi ]$. Subtracting the omitted second coefficient and using (eq:klinear) gives (eq:forwarddefect). The tree is independent of $h,k,\gamma$ and has zero variation under this particular background operation.

Thus the sign depends on the explicitly named orientation of the difference. Writing a positive forward variation without changing that convention would be incorrect. A hatted quantum coefficient with this identity has not been constructed.

There are admissible plateau data and smooth free multiplet configurations for which ${\mathcal{A}}_1[{\mathcal{L}}_{\xi }\gamma ;\varphi ]\ne 0$. Therefore this classical defect does not belong to the ideal of functionals vanishing on all free configurations.

Proof. Take $m=0$ and the free tensor and scalar

They satisfy $-\square \mathcal{P}\gamma =0$ and $\square \varphi =0$. Set $b=D\gamma$ and the ghosts to zero; these choices satisfy the auxiliary free equations.

Choose a nonnegative nonzero $a\in {C_c}^{\infty }((\epsilon ,T))$. On a sufficiently large causal cylinder let $\xi =a(t){\partial }_t$, with a compact spatial cutoff outside the observation dependence region. Choose $U$ and $g=1$ near its support, and observe after the support of $a$. All calculations in that region are unaffected by the cutoff, by finite propagation.

Put $r={\mathcal{L}}_{\xi }\gamma$. Its components there are

Consequently

There is no layer contribution, and

Direct differentiation gives

and its scalar retarded response is

Thus the completed first coefficient has density

A nonnegative nonzero $f$ in the described observation region gives a strictly positive pairing. An equation-ideal functional would vanish on these free configurations, proving the assertion.

Failure of ${\mathcal{L}}_{\xi }$ to preserve the wave equation is also visible in this example, but that failure alone would not prove the proposition. The evaluated observable is the necessary witness.

The shift $\gamma \mapsto \gamma +\kappa {\mathcal{L}}_{\xi }\gamma$ does not generally preserve the free gauge-fixed solution space. It therefore cannot simply be declared an automorphism of ${\mathcal{A}}_0$ to cancel the defect. The ghost rule $s_0\gamma =\frac{1}{2}Kc$ is not that shift.

The insertion is bilinear, while the vertex is cubic. After one graviton cross-contraction, a scalar cross-contraction is still available. Two contractions consume four of the five legs and leave a scalar-linear coefficient. With the retarded prefactor, this is a potential ${\hslash }^1{\kappa }^2$ contribution.

A schematic term involves

with the appropriate stress derivatives and mass terms. Here $H_m$ denotes a not selected scalar Hadamard bisolution. A full retarded-product construction would also have to specify its covariance matrix and ordering conventions. XI-B supplied none.

We do not evaluate this expression. Its off-diagonal wavefront compatibility and its extension near observation-vertex coincidence require separate analysis. No divergence, subtraction, or unique diagonal extension is inferred from the drawing. Missing Hadamard data are a stop for this loop, not a no-go for Theorem thm:tree.

In particular, tree-exactness cannot be imported from X-C-v3, whose insertions were scalar-linear. The present one-line coefficient is designated a tree because it is the classical retarded coefficient. It is not a claim that the full response of $V_{1,g}$ and ${\mathcal{B}}_f$ has no positive-$\hslash$ terms.

There is also a distinct one-scalar-line classical response of this pair, with metric fields remaining. It is not the single graviton component defined by (eq:physicalsign). Equation (eq:A2sc) names that scalar sector for the background comparison; one graviton component alone cannot establish a complete second-order Ward theorem.

The comparison concerns the supplied VII-C, not an identification of kernels or signatures VIIC.

The word $B_f\star W_g\star B_{f^{\prime }}$ is not formed. No reflected light-ray residue is subtracted. X-D's sharp-limit theorem is not used. Neither the present kernel bounds nor the absence of a chosen state establish VII-C's four assumptions.

The literal frozen-algebra target ends at Unresolved, not Constructed. The reason is concrete: its singular polynomial inputs and output need an admission theorem beyond XI-B's finite linear-generator algebra. A wavefront bound establishes eligibility for an appropriate extension; it is not that extension.

The classical retarded problem, by contrast, is completely constructed. It has one normalized graviton line, a nonzero de Donder dressing for the named source, and an explicit background order boundary.

XI-D receives:

1. the cubic classical tree symbol and the sign $R_{\gamma ,\mathrm{tree}}=-\mathcal{T}$ for the actual bare insertion;

2. the compact kernel bounds and causal support;

3. the double-retarded source-divergence normalization;

4. the restricted candidate and its forward defect $-{\mathcal{A}}_1[{\mathcal{L}}_{\xi }\gamma ]$, or the equivalent positive backward defect;

5. the demonstrated need for quantum composite admission;

6. the statement that $s_0$ is a free differential, not an interacting differential.

It does not receive an admitted quantum cubic coefficient, a Hadamard bisolution, a closed insertion, a loop extension, or a sharp limit. Supplying any of those would be an additional construction rather than a reinterpretation of the causal inverse.

The proved implication is

The stronger implication to the unchanged quantum algebra has not been established. Meanwhile the empty Einstein jet and the unresolved mixed loop remain separate facts. No construction is certified by silently changing its output algebra.

scope. The polynomial kernel bounds, causal contraction, normalization, support, and classical defect are proved on smooth configurations. Quantum composite admission is explicitly not included.

. XI-A supplies the signed interaction and the Einstein-jet obstruction. XI-B supplies the multiplet, normalized causal inverse, residual lemma, and limited algebraic domain. X-B-v3 and X-C-v3 supply the scalar-only comparison and completed classical background identity. X-D is not used.

. The insertion sign, forward versus backward defect sign, finite-generator versus microcausal domain, pure-gauge diagnostic, and scalar reduction are distinguished. The cubic Einstein vertex is not eliminated by a graph count. A second contraction is not declared absent.

. No successful pdflatex execution, proof-assistant verification, executed symbolic calculation, independent referee review, or journal acceptance is asserted.

XIA

N. Cheung, Landmark XI-A: A switched Einstein–scalar action for the ramp relational scalar: sign conventions, Noether identities, and an empty fixed-input Einstein jet. Supplied manuscript XI-A.tex. Used for the signed vertex and the Einstein-jet obstruction.

XIB

N. Cheung, Landmark XI-B: The free de Donder multiplet for the ramp Einstein–scalar sheet: causal propagators, residual gauge, and the layer dressing. Supplied manuscript XI-B.tex. Used for the causal inverse, auxiliary multiplet, free differential, and stated finite-generator algebra.

XB

N. Cheung, Landmark X-B: Fixed-regulator admission and one retarded correction for the ramp relational scalar, with the metric kept classical. Supplied manuscript X-B-v3.tex. Used for the scalar-only first response and background identity, with its representative warning.

XC

N. Cheung, Landmark X-C: The missing second-order retarded maps for the ramp relational scalar, with the metric kept classical. Supplied manuscript X-C-v3.tex. Used for the explicitly named scalar-only completed coefficient and its classical background identity.

VIIC

Landmark VII-C: A mixed-Wick obstruction to a normal-topology boundary extension for sharp Cauchy harmonic insertions. Supplied manuscript VII-C.tex. Comparison only; no fixed Wick matrix or sharp insertion is imported.

BGP

C. Bär, N. Ginoux, and F. Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society, 2007. arXiv:0806.1036. Used for smooth Cauchy theory, finite propagation, and causal Green operators.

BDH

C. Brouder, N. V. Dang, and F. Hélein, “Continuity of the fundamental operations on distributions having a specified wave front set (with a counter example by Semyon Alesker),” Studia Mathematica 232 (2016), 201–226. arXiv:1409.7662. Used for wavefront multiplication and pushforward criteria.

FR

K. Fredenhagen and K. Rejzner, Perturbative algebraic quantum field theory, arXiv:1208.1428v2, 2013. Used for the distinction between regular polynomial generators and Hadamard-based polynomial extensions; no such extension is constructed here.

HRV

E. Hawkins, K. Rejzner, and B. Visser, A novel class of functionals for perturbative algebraic quantum field theory, arXiv:2312.15203v2, 2024. Background for keeping polynomial kernel statements separate from broader functional and continuity claims.