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

This manuscript applies the stated stop rules to the proposed ramped-cubic construction. The supplied XII-A-v2 establishes a graded polynomial carrier, ordering transport, and an exclusion for the original cubic response. It does not provide the required changed-model freeze, its new Noether defect, or a decision about the changed coupled jet. The handoff gate therefore ends Unresolved, and no new cubic retarded response is defined.

A separate consistency audit identifies a mathematical issue in the proposed specification. Multiplication of a cubic source by a ramp is not the same operation as multiplication of its Lagrangian density before variation. We derive the weighted Euler formula and compute the discrepancy explicitly on the pure-trace Einstein–Hilbert sector. It contains a nonzero ramp-derivative term. A mixed-second-variation test further shows that the simply weighted source cannot be the Euler derivative of an action on the unrestricted off-shell class.

These results neither reopen XI-G nor obstruct a correctly specified density-ramped model. They identify the information and convention changes required before the completed-kernel, admission, response, and plateau gates can be evaluated. No state, loop, interacting differential, contact subtraction, or sharp-width limit is introduced.

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

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

: the B0 stop rule applies{ Constructed here: the handoff audit; the distinction between source weighting and density weighting; the weighted Euler identity; an explicit Einstein–Hilbert layer discrepancy; and a variational integrability test.

Overall ending: Unresolved. The supplied XII-A-v2 does not contain the changed-model handoff required by the production specification. Accordingly this manuscript does not define the new cubic retarded response and does not claim B1–B4 constructed.

Not inferred: a forbidden wavefront tuple for a correctly specified new insertion, a plateau obstruction for that insertion, or nonexistence of a density-ramped model.

XI-G's original class obstruction remains closed. Missing handoff data are not a physical no-go theorem.}

The intended affirmative target is admission of a scalar-completed, ramped cubic Einstein insertion and construction of one classical retarded response with a proved plateau law. The three endings are retained:

1. Constructed: the changed model is frozen, its completed kernel is microcausal, the named response is defined on its stated domain, and its plateau identity is proved.

2. Proved obstructed: a forbidden tuple is exhibited for the named completed kernel, or a nonzero interior defect is proved for the named plateau identity.

3. Unresolved: the prerequisite model, completed kernel, or response map has not been specified sufficiently to prove either alternative.

The production specification contains an additional dependency rule: XII-B consumes the changed-model freeze from XII-A; it does not supply that freeze itself. If the required XII-A statements are absent, XII-B stops and does not write the new response formula. That rule governs the present release.

XI-G proves, for its detected observation classes, that the original free-shell cubic class has no representative in ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ XIG. The inherited split null covectors are part of that predecessor result. Nothing below searches for another representative of the original ${\mathcal{C}}_3$.

A changed source would be a different model. Its admission cannot be deduced merely by negating the original obstruction, and its plateau law cannot be obtained by relabeling the original response. Likewise, absence of a new construction does not extend XI-G's obstruction to every possible changed model.

The supplied predecessor is XII-A-v2.tex, cited here as XII-A XIIA. Its actual release is a polynomial-carrier construction for the frozen XI-track multiplet. It includes:

• the separation of equation geometry from action normalization;

• a regular quantum presentation and a shell quotient;

• a smoothing-based descent argument;

• ordering transport and the named composite sector;

• delimited continuity statements;

• robustness of the detected observation class;

• a regular ambient–slab identification.

These are relevant algebraic prerequisites. They are not a definition of a new cubic action.

The production specification requires the following handoff. We distinguish requirements from statements actually supplied.

In particular, XII-A's exclusion section identifies its ${\mathcal{C}}_3$ as the frozen XI-F/XI-G functional. It does not replace that functional by a density-ramped one. Its declarations and proof ledger do not contain a changed cubic Noether theorem.

The normalization requirement H2 passes. The conjunction H1–H5 does not.

Relative to the supplied XII-A-v2 and the stated production stop rule, the changed-model handoff is incomplete. Gate B0 ends Unresolved. Consequently no new cubic retarded response, completed-kernel admission, or plateau theorem is released in this manuscript.

Proof. An affirmative B0 ending requires all the changed-model obligations, not only the inherited free inverse. The supplied text contains neither the changed density/source freeze nor its new Noether identity and coupled-jet decision. The production rule explicitly makes these omissions a stop. Thus the dependent construction is not authorized. This is a dependency conclusion, not a wavefront or plateau nonexistence argument.

The proposed handoff also requires a mathematical clarification before it can be supplied consistently: its phrases “multiply the source” and “multiply the density before variation” do not specify the same operation. Sections sec:weighted–sec:trace prove this point.

The inherited geometry is

Fix $0<\epsilon <T$ and

The ramp is the one supplied in XI-B and XI-E:

It is smooth and flat at both endpoints. Moreover ${\sigma }_{\epsilon }^{\prime }(t)>0$ for $0<t<\epsilon$. Indeed, on $(0,1)$,

The switch and plateau satisfy

This class is not reduced after the audit.

Trace reversal and the differential normalization are

Parentheses include $\frac{1}{2}$. The equation inverse and action inverse remain distinct as in XI-B XIB. The factor $-4$ is not imported.

The original cubic source is

with the curvature convention and Einstein–Hilbert action

These data specify the original local coefficient, not its new weighted variational completion.

The scalar completion ${\mathcal{A}}_1$ and the coordinate ramp are those of XI-E XIE. The new completed insertion would need both its scalar-response leg and its coordinate leg. Neither is discarded merely because a local metric source has a smooth weight. No completed new kernel is defined here, because B0 has stopped.

This section is a variational consistency calculation, not a construction of the prohibited new retarded response.

Let ${\psi }^a$ denote finitely many even field components, and let $\ell (x,j^r\psi )$ be a local density of finite jet order. For a multi-index $I$, write

Total derivatives act on local jet expressions; when applied to a function of $x$ alone, they are ordinary derivatives.

Compact variations define the Euler expression

Thus

for compact variations. This is the ordinary local variational calculus; see also KO. No global convergence of an unlocalized action on arbitrary configurations is needed.

For a fixed smooth scalar function $\sigma$,

where

For a time-dependent ramp, all terms in ${\mathcal{D}}_{\sigma ,a}$ contain a positive-order time derivative of $\sigma$.

Proof. Since $\sigma$ is independent of the field jets,

Apply the multi-index Leibniz rule to every term of (eq:Euler). The terms with $J=0$ give $\sigma {\mathsf{EL}}_a(\ell )$. The others give (eq:layerEuler). Spatial derivatives of a time-only weight vanish.

For a first-order density this becomes

For a second-order density, first and second derivatives of the ramp may occur.

With the fixed ${\sigma }_{\epsilon }$, these layer terms vanish outside the closed layer $0\le t\le \epsilon$ and are smooth and flat at its endpoints. Their nonzero values lie in $0<t<\epsilon$. They are not sharp-surface distributions.

Let ${\ell }_3$ be a cubic density normalized so that $\mathsf{EL}({\ell }_3)=J_3$. There are then two distinct prescriptions:

1. Source weighting: prescribe the local expression ${\sigma }_{\epsilon }{J}_3$ as the new forcing term.

2. Density weighting: prescribe ${\sigma }_{\epsilon }{\ell }_3$ before variation, whose Euler expression is ${\sigma }_{\epsilon }{J}_3+{{\mathcal{D}}_{\sigma }}_{\epsilon }[{\ell }_3]$.

The production sheet asks for a density-ramped vertex but also writes a response driven only by ${\sigma }_{\epsilon }{J}_3$. That combination needs justification; it is not a consequence of multiplication by a smooth ramp.

The next section shows that the omitted term is genuinely nonzero for the displayed Einstein–Hilbert convention. It is not merely a possible correction for unrelated models.

If ${\ell }_3^{\prime }={\ell }_3+{\partial }_{\mu }{B}^{\mu }$, then

but

Proof. Euler variation annihilates a total divergence under compact variations. Use

and apply that fact.

Thus an unweighted source alone does not identify a density-ramped vertex. A density representative, or an explicit boundary convention giving the same information, must be named. A silent integration by parts after the ramp is introduced can change the model.

The following calculation uses the pure-trace sector already displayed in XI-G, but concerns density variation rather than XI-G's plane-wave obstruction.

Set

All expressions are formal near $\kappa =0$.

For the displayed Einstein–Hilbert density, its coefficient of ${\kappa }^1$ on this sector is

Its Euler derivative is

Proof. Computing the Christoffel symbols of $e^{2w}\eta$ gives

With $e^{2w}=1+\kappa \psi$, this is

Therefore the signed density is exactly

Its coefficient of $\kappa$ is (eq:tracedensity).

Direct variation gives

Variation along $\delta {\gamma }_{\mu \nu }={\eta }_{\mu \nu }\delta \psi$ contracts the full cubic Euler source with ${\eta }_{\mu \nu }$, proving the last equality. It agrees with the pure-trace coefficient in XI-G XIG.

Density weighting gives

The second term is not identically zero on the unrestricted off-shell configuration class.

Proof. Apply (eq:firstorderlayer) to (eq:tracedensity). To detect the discrepancy, choose a spacetime box with $0<t<\epsilon$ and a smooth $\psi$ equal to $t$ near that box. Let $h\ge 0$ be a nonzero compact variation supported inside it. The discrepancy pairing is

The strict sign follows from $t>0$ and ${\sigma }_{\epsilon }^{\prime }>0$ there. All variations are compact and supported away from the layer endpoints, so no boundary convention enters this pairing.

This is a defect in the identification of the proposed vertex source, not a computed B4 plateau defect. The test $h$ is a compact metric variation used to audit an Euler derivative; it is not a replacement for the stipulated plateau parameter class ${\mathfrak{g}}_U$.

The preceding discrepancy is stronger than the failure of one chosen density representative.

For nonconstant ${\sigma }_{\epsilon }$, the full source ${\sigma }_{\epsilon }{J}_3$ cannot be the Euler derivative of a twice differentiable action on an off-shell neighborhood containing the unrestricted pure-trace configurations and compact metric variations.

Proof. If such an action existed, restrict it to $\gamma =\psi \eta$. Its first derivative would be

Symmetry of mixed second variations requires

for every pair of compact variations $h,k$.

Choose $\psi =1$ on a neighborhood of their supports. There

Compact integration by parts gives

Take a nonzero real $h$ supported strictly inside the layer and put $k=th$. Both variations are compact, and

Since the ramp depends only on time, (eq:Hessianantisymmetry) equals

This contradicts symmetry of the Hessian.

Consequently the simultaneous demands “an action vertex varied before response” and “Euler source exactly ${\sigma }_{\epsilon }{J}_3$ on unrestricted off-shell tests” are not interchangeable specifications. Adding an unnoticed total divergence cannot evade Proposition prop:integrability.

A source-weighted forcing prescription may still be considered as a nonvariational model. A density-ramped action may also be considered, with its actual layer terms included. Neither possibility is ruled out. Choosing between them, and freezing the relevant identities, is outside this stopped construction.

The relevant comparison criterion from XI-G is

Its role is comparison after a new compact kernel has been defined. It does not define that kernel.

For a corrected density-ramped model, the completed kernel must include:

1. both metric slots of the actual cubic Euler source, including its ramp-derivative terms;

2. the metric retarded leg with the XI-track normalization;

3. the coordinate leg with its own divergence-form ramp;

4. the scalar-response part of ${\mathcal{A}}_1$, including the switch and massive advanced scalar leg after transposition;

5. every support and properness condition used in the composition.

Checking only a smooth weight on the local metric diagonal would leave the scalar completion untested.

No new $u^{\sigma }$ or $W_{u^{\sigma }}$ is defined here. Accordingly no wavefront inclusion, admission criterion evaluation, or forbidden tuple for that new object is asserted.

XII-A supplies an inherited regular-algebra embedding into its named polynomial shell carrier:

This is the old embedding, not admission of an unnamed new functional. Its Hadamard matrix is product data, not a selected state.

The existence of the old carrier does not decide which changed cubic action is being used. Nor does it imply membership of the completed insertion before B1 is discharged. Thus B2 has an available carrier prerequisite but no new admission conclusion.

There is also a type issue to resolve in a future B3 statement. For the original quadratic insertion ${\mathcal{B}}_f$, its metric derivative at a smooth scalar configuration is a linear covector in the metric slot, with scalar field degree one. A cubic vertex has a quadratic Euler source in the metric.

Their single rooted contraction is therefore a scalar-valued functional of field degree $(2,1)$, not an element of ${C_c}^{\infty }(M)$. This agrees with the distinction between classical response functionals and their functional derivative kernels FR.

A smooth compact test can instead be an intermediate rooted coefficient after fixing the smooth fields and transposing the propagator, possibly with source localization. Those are different types. A complete B3 theorem must state:

• the domain of fields and external tests;

• whether its codomain is a test space, a polynomial functional space, or a continuous dual pairing;

• which intermediate coefficient is claimed compact;

• the exact support statement for that coefficient.

This observation does not prove that the eventual map leaves a test space. The map and test-space identification are not yet fixed. Under the B0 stop rule, no new cubic retarded formula is supplied to fill that gap.

If a correctly specified cubic density has a layer correction, that correction is locally supported where derivatives of ${\sigma }_{\epsilon }$ are nonzero. Retarded propagation of a layer-supported source need not be layer-supported, as the explicit XI-B layer diagnostics show XIB.

Consequently a support statement about a local Noether breaking term is not, by itself, a theorem that its completed response vanishes on the plateau. Conversely, compact plateau parameters do not justify discarding a term before its variation and propagation have been calculated.

The stipulated class remains ${\mathfrak{g}}_U={C_c}^{\infty }(U;TM)$. No contact term is selected, no layer term is deleted, and this class is not shrunk. No new B4 variation has been computed because the new vertex has not passed B0.

The nonzero pairing in Proposition prop:tracelayer is not relabeled as a nonzero plateau pairing. It establishes a source-definition mismatch only. Ending Proved obstructed for B4 would require a separate, correctly typed interior witness for the actual completed new response.

The following boundaries remain explicit.

X-track comparison. No X-E or X-F inverse, density switch, or mixed-tree prescription is imported. No field map is identified with $k_{\mathrm{mix}}$. The reciprocity constraint mentioned in the production brief is a constraint on a proposed identification, not a permission to remove ramp derivatives from a new action.

The X-G manuscript is not among the supplied dependencies. Its numerical reciprocity witness is therefore not rederived or used as a premise of any theorem here.

original coupled jet. The original empty fixed-input Einstein jet ${\mathcal{J}}_g({\varphi }_m)=\varnothing$ is retained as the predecessor result for its original action, as cited by XI-B and XI-E. A new action requires its own coupled equation and conservation analysis.

The statement “no jet repair is claimed” in XII-A is not a decision about a model that XII-A never defines. This manuscript does not turn it into one. The changed jet is unresolved.

original ledger. The classical ledger $-I+I+0=0$ belongs to the original XI-F response and its stated coefficient comparison. It is not recomputed here, not promoted to a quantum relation, and not used as the plateau law of an undefined changed response.

quantum constructions. No second graviton line, fish graph, triangle subtraction, ${\mathsf{T}}_2$, reflected residue, ghost loop, state, or expectation value is formed. No interacting differential, master equation, or interacting BRST class is claimed.

width. The ramp width remains $\epsilon >0$ throughout. No limiting argument is used to suppress the layer terms. Smoothness at fixed width does not identify the new model with a sharp insertion.

additional readout. No X-Enter record, cone token, conical current, or independent readout is assigned to the kernel. None is a premise of this audit.

For the supplied dependency set, the requested affirmative XII-B release is not established. Its ending is Unresolved at B0, with the dependent gates left unresolved.

The audit additionally proves that a density-ramped cubic Einstein action and a forcing prescription whose source is exactly ${\sigma }_{\epsilon }{J}_3$ are different specifications on the unrestricted off-shell class. This does not obstruct a correctly defined new action or a separately defined source-weighted model.

Proof. The B0 conclusion is Theorem thm:B0. The distinction is Lemma lem:weighted, Proposition prop:tracelayer, and Proposition prop:integrability. No theorem here exhibits a forbidden completed-kernel tuple or an interior defect of a specified new plateau identity. Therefore neither of the stated grounds for a new Proved obstructed release has been established.

A consistent updated handoff must:

1. choose a density-ramped action or an independently source-weighted forcing prescription;

2. if it chooses an action, name the cubic density representative and retain its actual Euler source, including the terms generated by differentiating the ramp;

3. keep the XI-track free inverse and state exactly which quadratic and lower-order coefficients remain unchanged;

4. derive the changed Noether identity before propagating or completing it;

5. state the changed coupled-jet conclusion, or amend the handoff contract to permit that question to remain unresolved;

6. define the full scalar-completed insertion;

7. specify the response codomain and the intermediate compact test to which its support theorem applies.

These are missing data, not a newly supplied freeze. They must not be inserted retroactively into XII-A while continuing to cite the old version as if it contained them.

At this stage XII-C receives an audit and the exact variational consistency calculations. It does not receive an admitted ramped insertion, a new retarded response, or a proved new plateau law. Opening those objects remains conditional on a corrected handoff and completion of B1–B4.

The old XI-G question stays closed. The density/source mismatch is displayed rather than hidden. No new response is written before its prerequisites are supplied.

scope. The proofs given here concern the dependency audit, local Euler variation, the explicit pure-trace Einstein–Hilbert coefficient, and variational integrability. They do not constitute the proposed admission and retarded response construction.

scope. XI-B, XI-E, XI-G, and XII-A-v2 are used in their supplied forms. Their inherited statements are distinguished from the calculations proved here. The changed-model freeze is not attributed to a predecessor that does not contain it.

-go scope. The integrability result concerns the source ${\sigma }_{\epsilon }{J}_3$ as an Euler derivative on the unrestricted off-shell class. It is not a no-go for density-ramped actions with their actual layer terms, nonvariational forcing prescriptions, larger algebras, or future admission constructions.

. No successful pdflatex execution, proof-assistant check, independent referee review, or journal acceptance is asserted. The source uses standard theorem environments and defines all internal reference targets.

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. Used for the actual carrier handoff and the B0 audit.

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 free inverse, ghost normalization, plateau class, ramp, and distinction between layer source support and propagated support.

XIE

N. Cheung, Landmark XI-E: On-shell admission of the fixed-width tree, a localized first-order identity, and the held-fixed defect. Supplied manuscript XI-E.tex. Used for the scalar completion, coordinate prescription, polynomial carrier, and distinction between product data and a state.

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. Used for the closed original obstruction, the class criterion, and comparison with the pure-trace cubic coefficient. Its plane-wave proof is not repeated.

KO

I. A. Kogan and P. J. Olver, “Invariant Euler–Lagrange equations and the invariant variational bicomplex,” Acta Applicandae Mathematicae 76 (2003), 137–193. Background for local Euler operators and variational integration by parts. The weighted identities used here are proved explicitly.

FR

K. Fredenhagen and K. Rejzner, Perturbative algebraic quantum field theory, arXiv:1208.1428. Background for the distinction between classical response functionals, functional derivative kernels, and test functions. No quantum retarded product is imported.