Landmark XII-C One declared cubic weighting: actual Euler source, changed Noether identity, and the coupled-jet question

We freeze one density-before-variation prescription at fixed regulator width. A necessary correction is made explicit: weighting the entire Einstein–Hilbert density would change the lower-order sheet and would not retain the XI-B free operator. To preserve that handoff, only the specified cubic density coefficient is multiplied by the existing ramp. All other gravitational coefficients, the switched scalar action, and the gauge-fixing term remain those of XI-A and XI-B.

A finite, unintegrated prescription fixes the cubic density representative. Its actual Euler source is computed as ${\sigma }_{\epsilon }{J}_3+{\mathcal{D}}_{{\sigma }_{\epsilon }}$. The complete tensor formula displays all first and second derivatives of the ramp and reproduces XII-B's pure-trace layer term. The changed fixed-background Noether identity contains both the scalar-switch defect and the additional breaking caused by weighting only one gravitational coefficient. Its apparent third ramp derivatives cancel algebraically.

For the named free inputs ${\varphi }_m$, the coupled invariant metric–scalar jet is empty for every switch of the declared plateau class. This is proved anew: the compatibility condition has a layer correction, but a nonzero late switching defect survives where the ramp is already one. Arbitrary leading vacuum metric data are allowed. The zero-field vacuum remains a counterexample to universal jet emptiness.

No Green operator is applied. No completed insertion is admitted, no wavefront theorem or response plateau law is proved, and no interacting differential or sharp-width limit is introduced.

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

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

and explicit specification correction{ Selected model: density before variation, at cubic order only. This is the unique weighting used in the manuscript.

Correction at C0: the production formula weighting the entire Einstein–Hilbert density cannot also satisfy the requirement that the lower-order XI-A/XI-B sheet remain unchanged. We therefore replace that formula by the explicitly declared cubic-only action in Section sec:freeze. The literal full-density model is not certified here.

Constructed: the action freeze, actual cubic source, changed Noether identity, and successor data.

Proved obstructed: the fixed-input coupled jet for the family ${\varphi }_m$. This is an obstruction of that solution fiber, not of an observable or of XII-A's carrier.

Not opened: B1–B4, kernel admission, any new metric retarded response, a response plateau cancellation, a state, or ${\hat{s}}_{\mathrm{int}}$.}

The three endings remain:

1. Constructed: the weighting and density representative are fixed, their actual source and Noether identity are proved, and the jet question has a named verdict.

2. Proved obstructed: an explicit incompatibility is proved for the chosen specification or for the named input family.

3. Unresolved: a necessary definition or calculation is absent.

The action freeze and the solution-fiber verdict are reported separately. A constructed action need not have a solution for every prescribed scalar input.

The following statements are used with their predecessor scope. They are not conclusions about the new weighting.

The supplied carrier is

with equal equation and action kernels ${\mathcal{C}}_{\mathrm{sol}}$ and a regular quotient recovering the causal free algebra. Ordering transport is an isomorphism on the named composite sector. For detected observations, the original unramped ${\mathcal{C}}_3{|}_{{\mathcal{C}}_{\mathrm{sol}}}$ is excluded. Undetected observations, the Einstein subshell, global hypocontinuity, and enlargement remain undecided. No quantum retarded product is supplied XIIA.

The requested changed-model handoff was incomplete, so B0–B4 were left Unresolved. The proved variational distinction is

with the pure-trace evaluation

The simply weighted source is not an Euler derivative on the unrestricted off-shell class. These results distinguish source weighting from density weighting; they do not obstruct an action with its actual layer terms retained XIIB.

The inherited inverse normalization is ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$. The free rule is $s_0\gamma =\mathcal{R}c$, with $\mathcal{R}=K/2$ and $c=2\xi$ recovering $K\xi$. The compact plateau residual intersection is zero. The de Donder layer formula is conditional on $D\gamma =0$, and propagated support is not layer support XIB.

XI-A's empty fixed-input Einstein jet concerns its original unswitched gravitational density. XI-G's detected unramped shell class remains excluded; its source-ramping remark is not a completed-kernel admission theorem XIA,XIG.

Nothing below searches for a representative of the old ${\mathcal{C}}_3$. X-E/X-F's normalization, X-G's reciprocity pairing, VII-C's word, the IX sharp commutator, and any cone or Enter readout are not inputs.

The general variational treatment of curvature with a scalar prefactor and of nondynamical background breaking provides useful comparison literature SF,BS. The model-specific formulas below are derived in the displayed conventions.

The nonvariational forcing option is not selected.

Use

All fields are smooth on a neighborhood of the closed slab. The ramp is exactly the inherited function

The scalar switch satisfies

and

Neither $g$ nor ${\sigma }_{\epsilon }$ is varied. The class ${\mathfrak{g}}_U$ is not subsequently reduced.

Put

Parentheses in indexed formulas include $\frac{1}{2}$. The curvature convention is

Multiplication of the entire signed Einstein–Hilbert density by ${\sigma }_{\epsilon }$ does not retain the XI-A/XI-B lower-order model. Its metric Euler derivative at the flat background contains

This expression is nonzero for the declared ramp.

Proof. The compact-variation calculation proved in Lemma lem:weighted-curvature below gives the displayed expression at $q=\eta$. Its spatial components are $-2{\delta }^{ij}{\sigma }_{\epsilon }^{\prime \prime }/\kappa$. The ramp has derivative zero at both endpoints but changes from zero to one, so ${\sigma }_{\epsilon }^{\prime \prime }$ cannot vanish identically. Thus flat space is not stationary even with zero scalar input. In particular, the original zero-background free sheet is not preserved.

This is a consistency check on a rejected formula, not a second selected model. Keeping the entire weighted density would require a different lower-order freeze and a different background audit.

Let ${\ell }_3$ be the precise cubic density fixed in Section sec:density. Define

The unchanged scalar action is

Write

Actions mean formal bulk variational functionals, tested with compact interior variations. Numerical finiteness on all smooth configurations is not required. The unweighted linear Einstein–Hilbert divergence is treated as in XI-A. The selected cubic representative is not changed by a later integration by parts.

Equation (eq:selected-action) weights only the cubic coefficient. The quadratic coefficient and all quartic and higher coefficients remain unweighted. This fixes the higher-order completion rather than leaving it unspecified.

The quadratic gravitational and gauge-fixing coefficients of $S_{\mathrm{D},\mathrm{gf}}$ are the XI-A/XI-B coefficients. Their metric operator is

The auxiliary and ghost normalization of XI-B is unchanged.

Proof. The added functional in (eq:selected-action) is cubic in $\gamma$. Its first and second derivatives vanish at $\gamma =0$. It therefore changes neither the free metric Hessian nor the independent quadratic auxiliary and ghost terms.

This cubic-only correction is a definition made in XII-C. It is not attributed retrospectively to XII-A or XII-B.

Write $\tau ={\mathrm{tr}}_{\eta }\gamma$ and set

For $n=1,2,3$, define

The declared representative is

It is the coefficient of $\kappa$ in the unintegrated signed Einstein–Hilbert density. Equivalently, it is $-2$ times the coefficient of ${\kappa }^3$ in $\sqrt{-q}R[q]$.

Every term is local, cubic, and of jet order at most two. The formulas specify the density itself, not only its Euler equivalence class.

For clarity, the original source can also be evaluated finitely. Put

Then

Compact variation of (eq:ell3) gives ${\mathsf{EL}}_{\gamma }({\ell }_3)=J_3$ with the XI-A sign.

For a fixed smooth scalar $\sigma$, compact covariant metric variation gives

This identity is used for coefficient extraction, not to select the full-density model.

Proof. For $h_{\mu \nu }=\delta q_{\mu \nu }$, write $h=q^{\mu \nu }{h}_{\mu \nu }$ and $h^{\mu \nu }=q^{\mu \alpha }{q}^{\nu \beta }{h}_{\alpha \beta }$. The Christoffel variation gives

Together with $\delta \sqrt{-q}=\frac{1}{2}\sqrt{-q}h$, two compact integrations by parts give

Multiply by $-2/{\kappa }^2$ and use $h_{\mu \nu }=\kappa \delta {\gamma }_{\mu \nu }$.

The coefficient of $\kappa$ on the left is the Euler derivative of $\sigma {\ell }_3$. Thus coefficient extraction supplies its complete layer correction without changing the chosen density.

Define

The three coefficients are explicitly

Also define the quadratic tensor

The selected cubic vertex

has Euler source

For the time-only ramp,

These are all ramp-derivative terms in the cubic source.

Proof. The extra term in (eq:weighted-full-variation) is

Its coefficient of $\kappa$ is (eq:actualsource), using (eq:Ccoefficients) and (eq:M2). The Einstein term at that coefficient is ${\sigma }_{\epsilon }{J}_3$. No derivative of the fixed weight beyond order two occurs.

The correction is quadratic in $\gamma$ and its derivatives. It vanishes outside the closed layer $0\le t\le \epsilon$ and is smooth and flat at its endpoints. Its nonzero values occur inside the open layer; its closed support need not avoid the endpoints. There is no hypersurface distribution.

XII-B's identity (eq:input-trace) is a consistency requirement, not the general source formula. For $\gamma =\psi \eta$, algebraic substitution gives $C_1=C_2=0$ and ${\Gamma }_2=-\psi {\Gamma }_1$. Consequently (eq:actualsource) gives

Contracting with ${\eta }_{\mu \nu }$ reproduces (eq:input-trace). Thus the predecessor's nonzero layer term is retained.

Density representatives cannot be changed silently. If ${\ell }_3$ is replaced by ${\ell }_3+{\partial }_{\mu }{B}^{\mu }$, the weighted source changes by $-{\mathsf{EL}}_{\gamma }(({\partial }_{\mu }{\sigma }_{\epsilon })B^{\mu })$, as proved in XII-B. Equation (eq:ell3) eliminates that ambiguity.

Define

The scalar equation is unchanged. Explicitly,

Put $w={\sigma }_{\epsilon }-1$ and

The exact formal metric derivative is

where

In particular, for coupling-independent fields,

For a symmetric contravariant density $v$, set

Thus ${\mathcal{N}}_{\nu }(v)=N_{0,\nu }(v)+\kappa Q_{\nu }(v)$.

The unchanged scalar-background defect is

The derivative-of-switch term remains visible. There are no derivatives of $g$ in the metric variation of the minimally coupled scalar density itself.

Define

For the declared ramp this is

The following identities hold off shell:

In particular, differentiating the actual source introduces no surviving third derivative of ${\sigma }_{\epsilon }$.

Proof. The contracted Bianchi identity, written for $\sqrt{-q}{G}^{\mu \nu }$, gives ${\mathcal{N}}_{\nu }(2\sqrt{-q}G/\kappa )=0$. Its coefficient of $\kappa$ is (eq:original-cubic-noether).

Now differentiate (eq:Z). The terms from $w{J}_3$ are

Differentiation of (eq:actualsource) gives the remaining terms of (eq:Rampdef) and an apparent term

For every symmetric three-covector tensor $S_{\mu \alpha \beta }$,

the two products of inverse metrics become equal after permuting the three contracted indices. Every coefficient $C_r$ has the same property. The apparent third derivative therefore vanishes. Substitute (eq:original-cubic-noether).

For the selected action, off shell,

All ramp derivatives are displayed in (eq:actualsource) and (eq:Rampdef). The $Q_{\nu }({Z_{\sigma }}_{\epsilon })$ term is the explicitly defined first-order operator (eq:Noperators) applied to the displayed quadratic source.

Proof. For an arbitrary compact interior $\xi$, use the scaled metric–scalar variation

Compact integration by parts gives

For the unweighted Einstein–Hilbert part, naturality gives zero variation. Variation of the unchanged switched scalar action gives $\kappa \int {\xi }^{\nu }{\mathcal{B}}_{g,\nu }{\mathrm{d}}^{4}x$. This follows by expanding the flat scalar variation as

and the curved scalar variation as a density divergence, then integrating the factors $1-g$ and $g$.

The added cubic functional in (eq:selected-action) varies as

Comparison of the two variation expressions, followed by Lemma lem:divergence, proves (eq:changed-noether).

This is not XI-A's identity with a relabeled source. The additional term $-w{Q}_{\nu }(A\gamma )$ is important: weighting only the cubic coefficient, while leaving the quadratic coefficient unchanged, produces breaking even where the weight is a constant different from one. Thus the new breaking is not exclusively a ramp-derivative term. Omitting this bulk coefficient mismatch would falsely treat a cubic-only modification as weighting the complete natural gravitational density.

For the gauge-fixed action, add the separate contribution ${\mathcal{N}}_{\nu }(e_{\mathrm{gf}})$ to the right, where

Gauge fixing is not made invariant by the ramp.

For $\xi \in {\mathfrak{g}}_U$, the new action correction and scalar-switch breaking vanish in the contracted variational identity. Thus

Proof. On a neighborhood of $\mathrm{supp}\xi$, one has $g={\sigma }_{\epsilon }=1$ and all their derivatives vanish. Hence ${\mathcal{B}}_g$, $w$, ${{\mathcal{D}}_{\sigma }}_{\epsilon }$, ${Z_{\sigma }}_{\epsilon }$, and ${{\mathfrak{R}}_{\sigma }}_{\epsilon }$ vanish there.

This local action statement supplies no propagated cancellation. No source has been evolved, and no plateau ledger is computed.

For a fixed free scalar ${\varphi }_0$, let ${{\mathcal{J}}_{g,{\sigma }_{\epsilon }}}^{\mathrm{D}}({\varphi }_0)$ consist of smooth formal families

for which the coefficients through ${\kappa }^2$ of $e_{\mathrm{D}}=0$ and $E_{\varphi }=0$ vanish. Positive-order scalar coefficients have the inherited zero initial data. Metric Cauchy data are unrestricted.

This is a jet of $S_{\mathrm{D}}$, not of the gauge-fixed action. In particular, ${\gamma }_0$ is not required to vanish.

Every element of ${{\mathcal{J}}_{g,{\sigma }_{\epsilon }}}^{\mathrm{D}}({\varphi }_0)$ satisfies

For $t>\epsilon$, this reduces to

Proof. The leading metric equation is $A{\gamma }_0=0$. The next metric equation is

Apply $N_0$. The linearized Bianchi identity gives $N_0(A{\gamma }_1)=0$. Lemma lem:divergence also gives

Since $P_m{\varphi }_0=0$, ${\partial }_{\mu }{T_0}^{\mu }{}_{\nu }[{\varphi }_0]=0$. The resulting equation is (eq:new-compatibility). Above the ramp, (eq:Rampdef-time) vanishes identically.

The new compatibility condition permits a layer contribution from leading metric data. It is therefore not obtained by transferring XI-A's unswitched condition to the whole slab.

For every $m\ge 0$, set

It is a free scalar and

There are $c>0$ and a nonempty interior open set $O\subset \{t>\epsilon \}$ on which ${\partial }_{t}g\le -c$.

Proof. Choose $(t_{\ast },{\mathbf{x}}_{\ast })\in U$. Then $t_{\ast }>\epsilon$ and $g(t_{\ast },{\mathbf{x}}_{\ast })=1$. Compact interior support gives a time $t_1\in (t_{\ast },T)$ with $g(t_1,{\mathbf{x}}_{\ast })=0$. The mean value theorem yields a point between them where ${\partial }_{t}g<0$. Smoothness gives the claimed open neighborhood and a strictly positive $c$.

For every mass $m\ge 0$ and every switch of the declared plateau class,

The obstruction already occurs in the order-one compatibility condition, independently of the allowed leading metric data.

Proof. For $\nu =0$, equation (eq:late-compatibility) becomes $\frac{1}{2}{\partial }_{t}g=0$ above the ramp. This contradicts Lemma lem:late-switch.

More explicitly, choose $f\ge 0$ supported in $O$ with $\int f{\mathrm{d}}^{4}x=1$. Then

because the ramp defect is zero on $O$. No leading vacuum metric configuration can alter this pairing. Higher-order coefficients cannot cancel a nonzero lower-order compatibility coefficient.

This proof is new to the selected action. The layer correction is present in (eq:new-compatibility); it simply cannot cancel the late switch defect. The witness is in the switching region, not an asserted nonzero plateau variation of an observable. The plateau algebra has not been shrunk.

The unrestricted jet family of the selected action is not empty. The family with $\gamma =0$ and $\varphi =0$ is a solution.

Proof. The flat Einstein tensor and scalar stress vanish. Both $J_3$ and ${{\mathcal{D}}_{\sigma }}_{\epsilon }$ are quadratic in the metric field, so their values at $\gamma =0$ vanish. The scalar Euler expression also vanishes.

This vacuum statement depends on the cubic-only freeze. It would be false for the flat background of the rejected full-density formula in Proposition prop:fullweight.

Other scalar inputs are not classified. The necessary condition (eq:new-compatibility) is not promoted to a general sufficiency theorem. Gauge-fixed solvability is a different equation because $e_{\mathrm{gf}}$ changes the divergence constraint. No gauge-fixed solution is constructed here, and none is declared an Einstein-jet repair.

The action supplies a local quadratic source and its polarized form

This definition applies equally to the displayed layer terms. It contains no propagator.

A successor may prescribe a compact root datum: a continuous linear map

whose range is supported in a fixed compact set

Here the range support and the map are given data. They are not claimed to arise from a retarded construction already performed in this paper.

For such a datum, the classical trilinear source pairing is

It is well defined on smooth configurations: the test has compact support and the source is a local smooth polynomial.

Equation (eq:source-pairing) is an actual classical functional with an independently supplied root datum. It is not asserted to be the completed insertion of XI-E. No claim about its distributional kernel or admission is made.

The notation ${\mathcal{C}}_{3,\mathrm{D}}(f,g)$ is reserved for the eventual scalar-completed insertion. Its intended signature is the inherited completion ${\mathcal{A}}_1[f,g](r;\varphi )$ with a new metric-response slot $r$. That slot has not been constructed.

A successor must determine the completed root datum associated with both the coordinate and scalar-response legs, establish its support, and then identify the resulting functional. We do not turn this unconstructed slot into a response formula.

This is a necessary type distinction. A source pairing is scalar valued; an intermediate compact test is tensor valued; a metric response is a smooth metric configuration. Naming one does not construct either of the others. The action freeze alone cannot supply the compact root test generated by an unperformed evolution.

The successor receives the normalization

No occurrence of this identity in the manuscript is an application to $J_{3,\mathrm{D}}$, ${\sigma }_{\epsilon }{J}_3$, or any other source.

The successor also receives XII-A's existing carrier and the unchanged exclusion of the original detected unramped classes. It does not receive membership of a new insertion in that carrier.

The selected source is not XI-F's $J_3$: it is ${\sigma }_{\epsilon }{J}_3+{{\mathcal{D}}_{\sigma }}_{\epsilon }$. It is not the simply source-weighted expression ${\sigma }_{\epsilon }{J}_3$. It is not XII-A's excluded original ${\mathcal{C}}_3$ and is not identified with X-F's mixed map or VII-D's triangle.

No X-track normalization is introduced. The scalar switch and coordinate ramp are unchanged; neither has been adjusted to remove a defect. The original XI-F ledger is not recomputed for this model.

The selected action is a coherent density-before-variation cubic modification preserving the XI-A/XI-B free handoff. Its actual source and changed Noether identity are those of Theorems thm:actualsource and thm:noether. Its fixed-input family ${\varphi }_m$ has no coupled invariant jet, by Theorem thm:empty.

The freeze is Constructed. If a single ending is assigned by the worst-gate convention, the named fixed-input release is Proved obstructed at C3. Neither statement is a kernel-admission or observable no-go.

Proof. The action and density are fixed by (eq:selected-action) and (eq:ell3). The cubic modification preserves the free Hessian. The cited source and identity theorems prove its variational and Noether consistency, while the jet theorem supplies the specified incompatibility. No admission or response premise is used in these arguments.

The next paper receives:

1. the explicit cubic-only action, including the unweighted higher gravitational coefficients;

2. the density representative (eq:ell3);

3. the actual source $J_{3,\mathrm{D}}={\sigma }_{\epsilon }{J}_3+{{\mathcal{D}}_{\sigma }}_{\epsilon }$;

4. the changed identity (eq:changed-noether);

5. the new jet verdict and its late-switch witness;

6. the unchanged XI-track inverse normalization;

7. XII-A's carrier and the unreopened old exclusion;

8. the classical source-pairing signature and the unconstructed scalar-completion slot.

It receives no completed-kernel theorem, wavefront estimate, admission map for this source, response plateau cancellation, or quantum retarded product. B1–B4 were not opened.

The nonzero jet defect is not a failure of the polynomial carrier. Retaining the layer source is not admission. The changed action is not a repaired representative of the old cubic class.

correction. This manuscript selects density weighting at cubic order only. It explicitly amends the full-density formula in the production brief because that formula is incompatible with the required unchanged lower-order handoff. No claim is made that the two actions are equivalent.

scope. All new products and variations are local classical expressions on smooth configurations. No new quantum product is defined.

scope. XII-B's weighted Euler distinction and pure-trace layer identity are copied as inputs and used as checks. The new general tensor source, changed identity, and coupled-jet verdict are proved here. XI-A's original jet theorem is not used as the new jet verdict.

scope. The cubic representative is fixed before weighting. All variational proofs use compact interior variations. No boundary action, boundary charge, or additional variational principle is constructed.

work. No metric retarded map, kernel admission, wavefront test, response ledger, interacting differential, antibracket, ghost loop, state, enlargement, or $\epsilon \to 0$ limit is included.

. No successful pdflatex execution, proof-assistant verification, independently executed symbolic computation, 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, ordering transport, and original-class exclusion input.

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. Weighted Euler identity, pure-trace layer term, and off-shell integrability distinction.

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. Original signed action, scalar switch, curvature convention, and original jet quantifier.

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. Free normalization, plateau class, and support distinctions.

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. Inherited scalar-completion signature only; no new completion is attributed to that manuscript.

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. Closed original obstruction and non-reopening boundary.

SF

T. P. Sotiriou and V. Faraoni, “$f(R)$ theories of gravity,” Reviews of Modern Physics 82 (2010), 451–497, arXiv:0805.1726. Comparison for metric curvature variation and scalar-prefactor derivative terms; the present coefficient calculation is explicit.

BS

R. Bluhm and A. Sehic, “Noether identities in gravity theories with nondynamical backgrounds and explicit spacetime symmetry breaking,” Physical Review D 94 (2016), 104034, arXiv:1610.02892. Comparison for fixed-background Noether defects; the selected cubic-only identity is derived here.