Landmark XII-D Retarded response and carrier admission for the cubic-only density-weighted Einstein source

At fixed regulator width, we apply the XI-track inverse $-\mathcal{P}{E_0}^{\mathrm{R}}$ to the actual cubic source frozen in XII-C, including its ramp-derivative terms. Finite propagation gives a cutoff-independent smooth response without assuming source conservation. We then construct the first-order completion from both the divergence-form coordinate leg and the switched scalar-response leg.

The transposed root is smooth but need not be compact before localization. A compact ambient collar root is constructed; multiplication by the actual source coefficients produces compact effective roots supported at nonnegative times. The completed degree-$(2,1)$ kernel has a covector-sum-zero bound. This proves membership in XII-A's polynomial microcausal domain and gives carrier admission. Smooth changes of ordering add only regular linear fields, so the formally completed sector generated by the new insertion is ordering-stable.

The plateau calculation requires a sign correction. With the prescribed forcing response $u_{\mathrm{D}}={G_{\mathrm{gr}}}^{\mathrm{R}}{\mathcal{J}}_{\mathrm{D}}$, the finite two-jet term enters with a minus sign. We prove the corresponding exact variation, display the gauge-fixing companion and Einstein-shell term separately, and give a conditional companion cancellation. This is not an unqualified invariance theorem for the cubic insertion alone.

The new admission neither changes XI-G's original exclusion nor repairs XII-C's empty fixed-input jet. No quantum retarded product, interacting differential, state, loop, or sharp-width limit is constructed.

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

Figure fig:scope-XII-D records that separation. The drawing adds no hypothesis.

and proof boundary{ Constructed: the prescribed classical response, its completed compact root, the compact microcausal cubic kernel, carrier admission, ordering transport on the enlarged named sector, and the exact signed plateau variation.

Corrections: a compact root generally requires an ambient time collar; its effective source-weighted coefficients, not the raw transposed root, have nonnegative-time support. The prescribed positive-source forcing response has a negative two-jet variation. A positive two-jet law would belong to the opposite forcing sign.

Not claimed: unqualified plateau invariance of ${\mathcal{C}}_{3,\mathrm{D}}$ alone, construction of the complete second-order relational observable, or repair of the invariant coupled jet.

The original detected unramped classes remain excluded. No carrier enlargement, quantum retarded product, state, loop, interacting differential, or $\epsilon \to 0$ limit is opened.}

The endings are Constructed, Proved obstructed, and Unresolved. They apply to the object actually named. Failure of a sufficient distribution-product hypothesis would not prove nonexistence. An empty coupled solution fiber would not obstruct a prescribed gauge-fixed response.

The selected model weights the specified cubic density only:

The scalar action, free gauge-fixing coefficient, and higher unweighted gravitational coefficients are unchanged. The actual cubic Euler source is

In the pure-trace sector,

The layer coefficients are smooth and flat at their endpoints; they are not hypersurface distributions XIIC.

For the named inputs ${\varphi }_m$, the invariant coupled jet ${{\mathcal{J}}_{g,{\sigma }_{\epsilon }}}^{\mathrm{D}}({\varphi }_m)$ is empty. The proof uses a late switch defect, not a failure of a Green operator. The zero-field vacuum prevents universal jet emptiness. The completed insertion and its generated root were not constructed, and the inverse was recorded but not applied XIIC.

XII-A supplies the polynomial domain, shell quotient, product, ordering maps, and regular embedding:

Its detected original unramped classes remain excluded. XI-B supplies

XI-G's source-ramping remark is not an admission theorem for (eq:source-input) XIIA,XIB,XIG.

XII-B's weighting audit is not reopened. No X-track normalization or response identification is imported. The predecessor statements are kept separate from the new response, root, and cone calculations.

The analytic input consists of smooth wave Cauchy theory and finite propagation BGP, together with the distributional product and proper-pushforward calculus Hormander. The functional and ordering framework is comparison background only FR; XII-A's established maps are used without reconstructing its carrier.

Use

Fields are ambient smooth configurations. No spatial decay is required. All finite-slab constructions depend only on their restrictions to a neighborhood of the slab.

Retain the fixed ramp

and

We assume $U\ne \varnothing$, as in the plateau handoff. Neither external function is varied.

Trace reversal and divergence are

Parentheses include $\frac{1}{2}$.

For completeness, the coefficients in (eq:source-input) have the finite prescription

Only the displayed coefficients occur in the sums. The original $J_3$ is the quadratic coefficient $2[{\kappa }^2](\sqrt{-q}{G}^{\mu \nu })$, with $q=\eta +\kappa \gamma$ and XII-C's curvature convention.

Thus ${\mathcal{J}}_{\mathrm{D}}$ is a quadratic differential polynomial of total field-derivative order at most two, with smooth coefficients built from ${\sigma }_{\epsilon },{\sigma }_{\epsilon }^{\prime },{\sigma }_{\epsilon }^{\prime \prime }$. Its polarization is

It vanishes for $t\le 0$ and is flat there. For $t\ge \epsilon$, it equals $J_3$.

The prescribed response is

This is not the response to ${\sigma }_{\epsilon }{J}_3$ alone.

Put $L=-\square \mathcal{P}$. Equation (eq:response) solves $L{u}_{\mathrm{D}}={\mathcal{J}}_{\mathrm{D}}$. The gravitational contribution to a perturbative coefficient of the action equation instead solves $Lr=-{\mathcal{J}}_{\mathrm{D}}$. That coefficient is $r=-u_{\mathrm{D}}$. This distinction changes a forcing sign, not the coefficient $-1$ in the inverse of $L$. It is essential in Section sec:plateau.

For every smooth $\gamma$, (eq:response) defines a unique smooth zero-initial-data solution on the slab. For every compact observation region $K\subset {M_T}^{\circ }$, it can be computed using a fixed spatial cutoff equal to one near

The result on a neighborhood of $K$ is independent of that cutoff. Moreover,

The support assertion is not a layer-support assertion.

Proof. The set $C_K$ is compact. Choose a spatial cutoff $\zeta$ equal to one near its spatial projection and replace $\gamma$ by $\zeta \gamma$ for this local calculation. The modified source is spatially compact on bounded time intervals and vanishes smoothly for $t\le 0$. Smooth wave Cauchy theory gives its zero-data response.

All derivatives of the two metric configurations agree near $C_K$, so their local differential sources agree there. Finite propagation implies equality of the resulting responses near $K$. Two admissible cutoffs therefore give the same result. Applying this argument on compact regions defines a consistent smooth response throughout the slab.

The source is past compact in the causal sense: its intersection with the causal past of any compact set is compact. Consequently the ambient retarded prescription agrees with the locally constructed zero-data solution. Finite propagation gives (eq:response-support); homogeneous zero-data uniqueness gives uniqueness.

The source need not be spatially compact before localization, and it need not vanish above the ramp. The cutoff is an auxiliary local construction, not a change of the model.

We record the relevant XII-C identity before using it. For a symmetric contravariant density $v$, define

The actual source obeys

where

The apparent third ramp derivative cancels in XII-C's coefficient identity. No conservation assumption is made.

The response satisfies

Proof. Use ${\mathcal{P}}^2=\mathrm{id}$ in (eq:response), commute constant derivatives with the locally defined wave inverse, and apply (eq:source-divergence).

A nonzero value in (eq:response-divergence) is not a Green-operator obstruction. It does not turn the response into an invariant coupled Einstein solution.

The completion is written explicitly so that no unspecified predecessor formula is needed. Set

The divergence-form coordinate coefficient and the scalar equation coefficient are

They follow from the inherited coordinate prescription and the minimally coupled switched scalar action. In particular, derivatives of $g$ are retained inside the displayed divergence.

For $f\in {C_c}^{\infty }({M_T}^{\circ })$ define

The scalar field is arbitrary and smooth at this stage.

The classical completed cubic insertion is

It is quadratic in the metric and linear in the scalar. Equation (eq:CD), not an independently chosen source pairing, is its definition.

Put $v_f={E_m}^{\mathrm{A}}f$. Compact integration by parts in (eq:Aone) gives

where

Both legs are present.

If $K=\mathrm{supp}f$, define the corrected scalar support set

The map $\varphi \mapsto {\alpha }_{f,g}[\varphi ]$ depends only on the scalar near $O_{f,g}$. Its range is smooth, compact, and supported in $C_K$.

Transpose the metric inverse with its prescribed sign:

The minus sign follows from ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{E}_0^{\mathrm{R}}$ and (eq:alpha-pair); it is not an independent convention.

The smooth advanced function (eq:beta) may extend to negative times. In particular, it need not vanish at $t=0$. One cannot generally multiply it by a smooth cutoff supported in $t\ge 0$ which is identically one at every positive time.

Choose $\delta >0$ and $\vartheta \in {C_c}^{\infty }((-\delta ,T+\delta ))$ with $\vartheta =1$ on a neighborhood of $[0,T]$. Set

For fixed $f,g,\vartheta$, the map

is continuous and linear, with range in a fixed compact support contained in

It reproduces the completed insertion:

The effective roots ${\sigma }_{\epsilon }{a}_{f,g}$, ${\sigma }_{\epsilon }^{\prime }{a}_{f,g}$, and ${\sigma }_{\epsilon }^{\prime \prime }{a}_{f,g}$ are compact and supported in $C_K$. The insertion is independent of the collar cutoff.

Proof. The first root (eq:alpha) is supported in $C_K$. Advanced support gives $\mathrm{supp}\beta \subset J^{-}(C_K)\subset J^{-}(K)$. Intersecting this past with the bounded time collar makes its support compact.

Multiplication by fixed smooth coefficients, differentiation, and wave propagation on fixed causal compact sets are continuous smooth-function operations. This proves continuity of the root map into its fixed compact support stage.

Transpose the metric response in (eq:CD). The resulting integral is $\int \beta {\mathcal{J}}_{\mathrm{D}}$. Only nonnegative times below the latest observation time contribute: the source vanishes for $t\le 0$, and the advanced root vanishes above the observation. Hence insertion of $\vartheta$ changes nothing and gives (eq:root-identity). All three effective roots vanish smoothly at nonpositive times and have support in $C_K$.

Thus the independent source pairing of XII-C now becomes a derived equality. It was not renamed as the insertion. The raw-root support requirement has been corrected explicitly, without adding a contact term or altering the source.

The set $O_{f,g}$ controls the scalar slot. It does not control the entire functional support: the metric slots can occur in its causal past. This distinction prevents a false support claim for a nonlocal response functional.

All covectors are compared using the fixed Minkowski coordinate trivialization. For $n\ge 2$, write

Since the future and past causal cones are pointed, $Z_n$ does not meet ${\tilde{V}}_{+}^n\cup {\tilde{V}}_{-}^n$.

Write

Its two components are explicitly

In the first term of $A_{\mathrm{sc}}$, the coefficient depends on $z$, so differentiation acts only on the displayed delta.

Translation invariance of the Minkowski Green kernel gives

This includes arbitrary diagonal Green covectors; no positive-frequency restriction is used.

Let $B_{f,g}(y,x)$ be the kernel of the compact root (eq:compact-root):

The composition (eq:B-kernel) exists, has compact support, and satisfies

Its scalar projection is contained in $O_{f,g}$.

Proof. The $z$ support of $A$ is compact. A forbidden product cancellation would have zero covectors in the external variables $y,x$. A Green wavefront covector has the form $(p,-p)$, so the zero $y$ covector forces that Green covector to vanish. Then (eq:A-WF) and the zero $x$ covector force the root covector to vanish as well. The product is therefore defined.

The $z$ projection is proper on the relevant support. For a pushforward wavefront point, cancellation of the internal $z$ covectors leaves output covectors $(p,-p)$. This proves the bound. The time collar makes the support compact; the scalar projection follows directly from (eq:A-kernels).

Let $j_{\mathrm{D}}(y;y_1,y_2)$ be the symmetric bilinear kernel defined by

It is a finite sum of differentiated diagonal deltas with smooth coefficients built from ${\sigma }_{\epsilon },{\sigma }_{\epsilon }^{\prime },{\sigma }_{\epsilon }^{\prime \prime }$. Consequently

Flatness at zero means that these are ambient smooth coefficients. No step function is inserted.

The polarized completed kernel is

The product and pushforward in (eq:t-kernel) exist. The kernel is compact and symmetric in its metric slots, with

After species symmetrization it defines ${\mathcal{C}}_{3,\mathrm{D}}(f,g)\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$.

Proof. A forbidden cancellation in the product would have zero external covectors in $x,y_1,y_2$. The root bound forces its $y$ covector to be zero. The source bound then forces its $y$ covector to be zero. There is no nonzero total cancellation, so the product exists.

The root is compact and the source is supported on the local diagonal, making the $y$ projection proper. For an output wavefront point, write the root covectors as $(p,-p)$ and the source covectors as $(q,p_1,p_2)$. Internal cancellation gives $q=-p$; the local source relation gives $p_1+p_2=p$. The scalar output covector is $-p$. Thus the three output covectors sum to zero.

The support assertion follows from the root support, the diagonal source support, and its nonnegative-time coefficients. Permutation of the three species slots preserves the sum-zero bound. A nonzero tuple wholly in either causal cone cannot sum to zero. This is precisely XII-A's microcausal exclusion.

The theorem includes the ${\sigma }^{\prime \prime }{C}_2$ and ${\sigma }^{\prime }{M}_2$ kernels, not only the $\sigma J_3$ kernel. The proof is an explicit product–pushforward calculation; it is not an inference from smoothness of the response. There is no failed collinear product hypothesis to reinterpret as a no-go.

The bound is special to this finite Minkowski composition: constant fiber maps, translation-invariant Green kernels, local differential vertices, and smooth external weights supply covector balance. No corresponding theorem for an arbitrary nonlocal or curved-background kernel is claimed.

This admission concerns the changed functional (eq:CD). It does not supply a new representative of XI-G's original excluded shell class.

The new insertion has a well-defined class

It has even parity and ghost number zero.

Proof. Theorem thm:admission supplies an off-shell microcausal polynomial lift. XII-A's already proved free-shell descent then applies. The insertion uses two even metric slots and one even scalar slot.

No conservation constraint on ${\mathcal{J}}_{\mathrm{D}}$ is needed for this quotient operation. Admission to the gauge-fixed free-shell carrier is not solvability of the invariant coupled equation.

Let $T_{\mathrm{D}}$ denote the fully species-symmetric degree-three kernel, normalized so that

with no additional factorial.

For an admissible smooth change $d=H^{\prime }-H$,

Every component of ${\ell }_d$ is compact and smooth. Only admissible metric–metric and metric–scalar contractions can contribute.

Proof. The second functional derivative of a cubic contributes $3\cdot 2$; the ordering generator contains $\hslash /2$. No higher even derivative remains.

A nonzero wavefront point of the projected linear kernel would require a tuple $(0,0,p)$ in the cubic wavefront set. The sum-zero bound forces $p=0$, a contradiction. Compactness follows from proper projection of the compact cubic support. There is only one scalar slot, and no odd species, so the stated pair types exhaust the possibilities.

Define ${{\mathcal{S}}_{\mathcal{H}}}_{\mathrm{D},H}$ to be the coefficientwise $(\hslash ,\kappa )$-adic completion of the algebra generated by XII-A's named sector and all $C_{\mathrm{D},H}(f,g)$. This is formal completion, not analytic closure.

The inherited ordering isomorphism maps ${{\mathcal{S}}_{\mathcal{H}}}_{\mathrm{D},H}$ onto the corresponding ${\star }_{H^{\prime }}$-sector generated by the unchanged new formulas:

Proof. XII-A transports the old sector onto its counterpart. By Proposition prop:ordering, each new generator is transported to the unchanged new generator plus a regular linear field. Regular linear fields already belong to the old sector. The inverse ordering map gives the reverse inclusion. Formal-order preservation gives equality of the coefficientwise completions.

Continuity of the inherited ordering and quotient maps is used only at XII-A's released level. No completeness or global product hypocontinuity claim is added. No interacting differential is inferred from this sector construction.

For reference, XII-C's changed action identity is

where

The complete ramp defect is (eq:ramp-defect). The gauge-fixed identity has the additional term ${\mathcal{N}}_{\nu }(e_{\mathrm{gf}})$. These terms are displayed before setting $g={\sigma }_{\epsilon }=1$ on the parameter support.

For $\xi \in {\mathfrak{g}}_U$, locality gives

Indeed every term in the difference contains a derivative of $\xi$ or $\xi$ itself, hence is supported in $U$. There ${\sigma }_{\epsilon }=1$ and all ramp derivatives vanish. The layer has not been discarded from the response: its variation under these parameters is zero.

For a contravariant rank-two density define

Set

This tensor is compactly supported in $U$.

For every smooth $\gamma$ and every $\xi \in {\mathfrak{g}}_U$,

Proof. Equation (eq:layer-variation) reduces the assertion to the unweighted local coefficient identity. For the flow generated by $\kappa \xi$,

Naturality of the Einstein tensor density gives the same pullback for its Euler derivative. Comparing the coefficient of $\kappa$ in

gives

This proves (eq:finite-source).

Define the zero-data gauge-fixing companion parameter

It is a propagated compact-source field, not a compact residual plateau parameter.

The prescribed response obeys

Proof. Apply ${G_{\mathrm{gr}}}^{\mathrm{R}}$ to (eq:finite-source). For compact $s$, the identities in the input sheet imply

The term with ${{\mathcal{L}}_{\xi }}^{\mathrm{dens}}(A\gamma )$ is $-\mathcal{P}{E}_0^{\mathrm{R}}{{\mathcal{L}}_{\xi }}^{\mathrm{dens}}(A\gamma )$. Substitution gives (eq:response-law).

The positive-two-jet formula sometimes associated with the action's coefficient belongs to $-u_{\mathrm{D}}$, not to (eq:response). Multiplying the whole law by minus one recovers that distinction; no factor $-4$ or change of Green normalization is involved.

The sign is not merely a choice of names. Choose a compact $\xi \in {\mathfrak{g}}_U$ which near an interior point is

and put $\gamma =0$. In that neighborhood, writing $F=\lambda (t-t_0)(x^2-{x_0}^2)$ gives $s_{\xi }=-\partial F\otimes \partial F$. At the central point, $(A{s}_{\xi }{)}^{11}=(A{s}_{\xi }{)}^{33}=-2{\lambda }^2$. Equation (eq:finite-source) gives $J_3[K\xi ]=-A{s}_{\xi }$. The proposed positive-two-jet response $s_{\xi }-K{E}_0^{\mathrm{R}}D{s}_{\xi }$ has $L$-image $A{s}_{\xi }$, whereas the prescribed response has $L$-image $-A{s}_{\xi }$. They cannot agree. This diagnoses an incompatible response law, not a failure of kernel admission.

Keep the scalar field fixed in this finite metric shift. By linearity of ${\mathcal{A}}_1$ in its metric slot,

The three contributions are

No row has been assigned an automatic numerical value.

For every $\xi \in {\mathfrak{g}}_U$,

This is the proved plateau variation. The algebraic companion on the left is explicit; it has not been identified with an unconstructed full second-order observable. The Einstein-shell term stays named whenever $A\gamma$ does not vanish near the parameter support.

A support hypothesis on observations, not a shrinkage of ${\mathfrak{g}}_U$, makes a useful conditional cancellation precise.

Assume $P_m\varphi =0$ and that $g=1$ on a neighborhood of

Then

If also $A\gamma =0$ on a neighborhood of $\mathrm{supp}\xi$, then

Proof. The source $D{s}_{\xi }$ is supported above $\epsilon$, so ${\rho }_{\xi }$ and its derivatives vanish before that time. Consequently the coordinate ramp is one wherever $K{\rho }_{\xi }$ can contribute. The coordinate equation gives

with zero past data.

For any smooth vector $v$, direct differentiation of (eq:completion-legs) with $g=1$ gives

On the free scalar shell this is $-P_m(v^{\mu }{\partial }_{\mu }\varphi )$. The support condition (eq:causal-plateau-condition) makes the switched and unswitched scalar sources agree throughout the causal past relevant to $K$. Finite propagation and zero past data therefore give

on $K$. The two legs in (eq:Aone) cancel.

If $A\gamma$ vanishes near $\mathrm{supp}\xi$, the Einstein-shell source in (eq:plateau-identity) vanishes. The final assertion follows.

Without the stated causal condition the gauge-fixing row is not deleted. Without the Einstein condition the last row is not deleted. In particular, no universal zero variation of ${\mathcal{C}}_{3,\mathrm{D}}$ alone is asserted.

These are classical identities for named polynomial lifts. The compact shift $K\xi$ need not preserve the gauge-fixed free shell, so no action of ${\mathfrak{g}}_U$ by automorphisms of ${\mathcal{E}}_H$ is inferred. There is no interacting differential or quantum Ward identity hidden in the ledger.

For the source and completion specified in this manuscript, $u_{\mathrm{D}}$ and its completed root exist, ${\mathcal{C}}_{3,\mathrm{D}}(f,g)\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$, and $Q_H{\mathcal{C}}_{3,\mathrm{D}}(f,g)\in {\mathcal{E}}_H$. The enlarged named sector is stable under admissible ordering transport. Its classical finite plateau variation is (eq:plateau-identity), with the additional cancellation hypotheses stated in Proposition prop:causal-plateau.

Proof. The response is Theorem thm:response, the compact root is Theorem thm:root, and the cone verdict is Theorem thm:admission. Section sec:descent applies the inherited carrier and ordering maps. The finite variation is Theorem thm:response-law and Corollary cor:plateau.

Thus the corrected response, admission, and signed-identity target is Constructed. An alternative requirement of unqualified invariance of ${\mathcal{C}}_{3,\mathrm{D}}$ alone is not part of this release. The literal positive-two-jet response law is not silently transferred to the opposite forcing sign.

The successor receives:

1. $u_{\mathrm{D}}=-\mathcal{P}{E}_0^{\mathrm{R}}({\sigma }_{\epsilon }{J}_3+{{\mathcal{D}}_{\sigma }}_{\epsilon })$;

2. the completed two-leg functional (eq:CD);

3. the generated root (eq:compact-root), its collar qualification, and its effective support bounds;

4. the explicit compact-kernel admission theorem;

5. the inherited quotient class and regular ordering correction;

6. the signed plateau ledger and its exact conditions.

It receives no interacting BRST class, full second-order relational construction, loop value, quantum retarded product, state, or sharp limit.

• The original detected unramped shell classes remain excluded. The admitted functional is a different model-dependent insertion.

• XII-C's empty fixed-input invariant jet is not repaired. The constructed object solves a prescribed gauge-fixed forcing equation.

• The layer source was included before evolution. No extra ramp, contact prescription, subtraction, or moved initial surface was introduced.

• The inverse coefficient is $-1$, not $-4$. No identification with an X-track mixed map is made.

• Propagated support is not layer support. No $\epsilon \to 0$ limit is taken.

• No state, loop, antibracket, master identity, cone token, or Enter record is imported.

The source pairing is a derived identity, not a renamed definition. Smooth response and carrier admission have separate proofs. Neither is an Einstein-jet repair.

response scope. The retarded map is a map on smooth configurations, defined locally by finite propagation. It is not a quantum retarded product or an interacting field map.

scope. The cone calculation proves membership of the new compact polynomial kernel in XII-A's domain. It neither enlarges that domain nor removes the predecessor obstruction.

correction. The raw advanced root can extend below the initial surface. A compact ambient collar representative is used only in pairings with the flat, nonnegative-time source coefficients. No sharp cutoff or boundary distribution is introduced.

correction. The forcing convention $L{u}_{\mathrm{D}}={\mathcal{J}}_{\mathrm{D}}$ is distinguished from the action coefficient convention $Lr=-{\mathcal{J}}_{\mathrm{D}}$. The signed variation is proved for the former. A complete second-order relational observable has not been constructed by naming its algebraic companion.

scope. XII-C supplies the action and actual source. XII-A supplies the carrier and ordering maps. XI-B supplies the inverse normalization. XI-G supplies the unreopened old exclusion. The completed root and new cone bound are proved here.

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

XIIA

N. Cheung, Landmark XII-A: One graded polynomial carrier for the finite-ramp multiplet, with ordering-stable composites and a robust observation exclusion. Supplied manuscript XII-A-v2.tex. Carrier, shell descent, regular sector, and ordering inputs.

XIIB

N. Cheung, Landmark XII-B: Admission and one retarded response of the ramped cubic Einstein source, at fixed regulator width: handoff audit, variational consistency, and the unresolved release. Supplied manuscript XII-B.tex. Density/source distinction and pre-freeze stop rule.

XIIC

N. Cheung, Landmark XII-C: One declared cubic weighting: actual Euler source, changed Noether identity, and the coupled-jet question. Supplied manuscript XII-C.tex. Selected action, complete local source, changed identity, and fixed-input jet verdict.

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. Inverse normalization, free identities, and layer qualifications.

XIG

N. Cheung, Landmark XI-G: The free-shell class of the cubic Einstein kernel: the onset covector is inherited by every representative. Supplied manuscript XI-G.tex. Original class exclusion; no source-ramping admission statement 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. Smooth Cauchy theory, causal Green operators, and finite propagation.

Hormander

L. Hörmander, The Analysis of Linear Partial Differential Operators I, second edition, Springer, 1990. Wavefront bounds for differential operators, distribution products, kernel composition, and proper pushforward.

FR

K. Fredenhagen and K. Rejzner, Perturbative algebraic quantum field theory, arXiv:1208.1428. Functional and deformation-quantization background; the present admission is proved for its explicit compact polynomial kernel.