Landmark XI-A A switched Einstein–scalar action for the ramp relational scalar: sign conventions, Noether identities, and an empty fixed-input Einstein jet

We specify a classical Einstein–scalar model compatible with the signature, scalar switching prescription, and plateau gauge algebra of Landmark X-A. The Einstein–Hilbert density is not switched. Its sign is fixed by a positive transverse-traceless kinetic term and an attractive Newtonian comparison. We display the quadratic Fierz–Pauli action, one de Donder gauge-fixing term, and the metric Euler derivative through coupling order ${\kappa }^2$.

Several distinctions are indispensable. Minimal scalar coupling introduces the switch itself, but no switch derivatives, into the metric Euler derivative. Switch derivatives appear in the scalar equation and in the divergence of the metric source. Nonlinear Noether identities require the classical higher gravitational coefficients; the quadratic action alone is insufficient. Gauge fixing supplies a separate breaking term and is not diffeomorphism invariant on the plateau.

We prove the exact fixed-background Noether identity and the tensor-density covariance of the invariant metric Euler operator on the admitted plateau group. X-A's varied-action formula is used as a premise, not presented as a new relational covariance theorem. Finally, for a specified free scalar input, the ungauge-fixed coupled Einstein jet is empty for every nonzero compact switch of the declared class. This is not an emptiness theorem for all inputs, nor for the gauge-fixed wave equations. The manuscript stops at the classical sheet: no propagator, quantum multiplet, tree, or BRST class is constructed.

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

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

: a corrected classical sheet, with the on-shell obstruction separated{ The signed action, quadratic operator, metric Euler coefficients, Noether identity with its switching and gauge-fixing terms, and plateau covariance of the invariant metric Euler derivative are constructed.

For the fixed free input of Theorem thm:empty, the coupled Einstein jet is empty. The unrestricted jet family is not declared empty.

Neither invariance of the gauge-fixed action nor nonlinear covariance of a Fierz–Pauli-only truncation is asserted. Those versions of the proposed sheet are false. The metric remains off shell throughout the construction of the sheet.}

The objective is to give a classical action and its identities before constructing any quantum metric. The relational prescription of X-A is retained, but no quantum conclusion is inferred from its classical covariance.

A production plan cannot require both a background de Donder term and an unbroken diffeomorphism identity for the resulting action. It also cannot identify a quadratic gravitational action with the nonlinear Einstein action at the next two coupling orders. Accordingly, the model below has one declared gauge-fixed action, together with its explicitly identified invariant part:

The identities for these two functionals are different and are printed separately.

Five corrections determine the release.

1. The Einstein–Hilbert action is specified before expansion. Its quadratic coefficient is Fierz–Pauli. Its cubic and quartic coefficients are retained in the classical sheet when required by coupling order. They are not evaluated as graphs.

2. The scalar action contains no metric derivatives. Its metric variation therefore contains $g$ but no $\partial g$, including at second order. Adding such terms would change the stated minimally coupled model.

3. The nonlinear Noether identity uses the formal adjoint of the metric pullback action. A bare partial divergence is only its leading flat contribution.

4. Covariance of a metric Euler derivative means covariance as a rank-two tensor density. Gauge fixing contributes a noncovariant background-dependent term.

5. An empty jet must have a specified input. The zero scalar and flat metric provide an immediate counterexample to universal emptiness.

The broader perturbative treatment of gravity distinguishes classical gauge symmetry, gauge fixing, and subsequent BV quantization BFR. Here those distinctions are enforced without opening a BV complex. Effective-field-theory discussions likewise do not justify deleting the classical Einstein self-interactions Donoghue.

Use the X-A conventions:

The mass is fixed for each instance of the model. All coefficient fields are real and smooth on a neighbourhood of the slab.

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

Choose an open relatively compact $U$ such that

The group ${\mathcal{G}}_U$ is generated by the corresponding compactly supported flows. Every admitted transformation is the identity near $\Sigma$ and preserves $g$ and ${\sigma }_{\epsilon }$.

For coefficient calculations, genuine flows of $\kappa \xi +{\kappa }^2\zeta$, $\xi ,\zeta \in {\mathfrak{g}}_U$, provide all Taylor coefficients needed below. They lift X-A's second-order formal transformations without enlarging its support class.

The field variable is one classical symmetric tensor $\gamma$:

There are no independent quantum fields $\hat{h},\hat{k}$. Write $\tau ={\eta }^{\mu \nu }{\gamma }_{\mu \nu }$ and set

Thus

These symbols are not coefficients of $\sqrt{-q}R[q]$.

The action is defined as a full formal generating expression before its Euler derivative is projected through ${\kappa }^2$. In particular, $\gamma$ may later be evaluated on a formal solution family. This does not introduce another fundamental metric field.

X-A supplies the coordinate prescription

with the fixed data ${{\mathsf{X}}_{\mathsf{q}}}^{\mu }{|}_{\Sigma }=x^{\mu }{|}_{\Sigma }$ and ${\partial }_t{{\mathsf{X}}_{\mathsf{q}}}^{\mu }{|}_{\Sigma }={{\delta }^{\mu }}_0$. The observable is the fixed label-space smearing of $\varphi \circ {{\mathsf{X}}_{\mathsf{q}}}^{-1}$.

Its plateau covariance and X-A's Route II are premises XA. They are not reproved here. The ramp does not enter the gravitational action.

X-B, X-C, and X-D are not inputs. VI and VII-A are comparison manuscripts with different signature and metric prescriptions. VII-C, X-Enter, and the cone papers supply no premise. No quantum cancellation from another relational prescription is imported Frob.

The factor $\kappa {\mathsf{X}}_1[\gamma ]$ has one metric leg. This observation alone does not eliminate a cubic Einstein vertex: that vertex has coefficient $\kappa$, so the two factors have total coupling degree two before any additional graph requirements are specified. No rooted-tree topology is selected or calculated here.

More importantly, an action identity is not a graph selection rule. The nonlinear classical identities require the relevant Einstein coefficients. The Fierz–Pauli-only alternative is explicitly audited in Section sec:quadraticfailure.

Fix the curvature convention

with the Levi-Civita Christoffel symbols and $G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}{q}_{\mu \nu }R$. Use

There is no cosmological constant and no gravitational switch.

The unchanged X-A scalar density is

Actions are interpreted through compact interior variations and compact differences. Numerical finiteness on every smooth configuration is not assumed. No metric boundary variation is taken.

Define

The single frozen gauge-fixing term is

Its coefficient has been fixed; no gauge-parameter family is used.

Modulo a total divergence, the quadratic bulk coefficient of (eq:EH) is

Its Euler derivative is $2{G_{\mathrm{lin}}}^{\mu \nu }(\gamma )$, where

Proof. Expanding (eq:ricci) to first order gives (eq:Glin). Variation with respect to the covariant metric gives

Consequently the quadratic coefficient has derivative $2G_{\mathrm{lin}}(\gamma )$. Integration by parts in (eq:FP) gives that same derivative. Both quadratic functionals vanish at $\gamma =0$; integrating their equal directional derivatives along $s\gamma$, $0\le s\le 1$, proves equality as bulk variational functionals. The linear Einstein–Hilbert coefficient is a divergence and has no bulk Euler derivative.

For ${\gamma }_{0\mu }=0$, ${\partial }_i{\gamma }_{ij}=0$, and ${\delta }^{ij}{\gamma }_{ij}=0$, the quadratic density becomes

Its time-derivative term has the same sign as $\frac{1}{2}{\dot{\varphi }}^2$.

Proof. The trace and divergence terms in (eq:FP) vanish. Raising both spatial indices contributes two minus signs, so ${\gamma }^{ij}={\gamma }_{ij}$. The stated expression follows.

Let $H=q-\eta$ denote the physical metric perturbation. For an auxiliary static conserved source $T_{00}=\rho \ge 0$, $T_{0i}=T_{ij}=0$, $\rho$ smooth compactly supported and nonzero, the linear unswitched equation is solved by

where

Thus ${\kappa }^2=32\pi G_N$ gives the attractive Newtonian sign.

Proof. The scalar metric variation computed below is negative, so the unswitched metric equation is $G_{\mu \nu }={\kappa }^2{T}_{\mu \nu }/4$. For the displayed perturbation, ${\bar{H}}_{00}=4{\Phi }_N$, ${\bar{H}}_{ij}=0$, and $D(H)=0$. Equation (eq:Glin) gives $G_{\mathrm{lin},00}(H)=2\Delta {\Phi }_N$ and zero spatial components. The Newton inverse of $\Delta$ gives (eq:Newton).

This static calculation is a sign diagnostic, not a solution of the compact-time switched model and not an adiabatic-limit construction. The canonically normalized field is $\gamma =H/\kappa$; confusing ${\gamma }_{00}$ with $H_{00}$ would change the normalization check.

Put $p_{\mu }={\partial }_{\mu }\varphi$ and define

Write $T_0=T_{\eta }$ and ${\ell }_0={\mathfrak{L}}_{\eta }$.

For a covariant metric variation,

In particular,

Proof. Use $\delta \sqrt{-q}=\frac{1}{2}\sqrt{-q}{q}^{\mu \nu }\delta q_{\mu \nu }$ and $\delta q^{\alpha \beta }=-q^{\alpha \mu }{q}^{\beta \nu }\delta q_{\mu \nu }$. For the second assertion,

The second matter coefficient is fixed, not guessed:

One switch multiplies this coefficient; replacing it by $g^2$ would change X-A's prescription.

Define the Einstein tensor-density coefficients by

Here ${\mathcal{H}}_1=G_{\mathrm{lin}}(\gamma )$. Appendix app:coefficients gives finite explicit formulas for all three coefficients, without an external curvature convention.

Let

Direct expansion gives

With $e^{\mu \nu }=\delta S_{\mathrm{inv}}/\delta {\gamma }_{\mu \nu }$, the exact formal derivative and its order-two expansion are

The gauge-fixing contribution is separately

Thus $\delta S_{\mathrm{tot}}/\delta \gamma =e+e_{\mathrm{gf}}$. There are no $\partial g$ terms in (eq:eexact) or (eq:eexp).

Proof. Combine the Einstein variation in Proposition prop:FP, Lemma lem:vertex, and $\delta q=\kappa \delta \gamma$. The flat scalar density is independent of $\gamma$. Equations (eq:Hdef) and (eq:J1) give the expansion. Varying $D_{\nu }{D}^{\nu }$ and integrating by parts gives (eq:egf); taking its divergence gives the last identity. No integration by parts in a metric variation of the scalar density is needed because that density has no metric derivatives.

For $\varphi ={\varphi }_0+\kappa {\varphi }_1+\cdots$, the order-one scalar source is exactly $-\frac{1}{2}g{T}_0[{\varphi }_0]$. At order two, replace $J_1[\gamma ,\varphi ]$ in (eq:eexp) by

where

This distinguishes the metric variation of the stress density from the correction due to the scalar solution.

The scalar Euler derivative is

with

Also,

These are the switch-derivative terms required by the stated model. They must not be moved into its metric Euler derivative.

For a compactly supported diffeomorphism $F$ in the interior, X-A's Route II gives

where $F_{\mathrm{dens}}^{\ast }r-r={\partial }_{\mu }{Z_F}^{\mu }[r]$. For the flow of $\kappa \xi +{\kappa }^2\zeta$,

The proof of (eq:RouteII) is not repeated.

The unswitched Einstein–Hilbert density is natural. The same divergence lemma therefore shows ${\delta }_F{S}_{\mathrm{EH}}=0$ for compact interior $F$. Consequently (eq:RouteII) is also the finite variation of $S_{\mathrm{inv}}$. There is no analogous assertion for $S_{\mathrm{gf}}$.

For a symmetric contravariant density $u^{\mu \nu }$ define

For an arbitrary interior compact vector field $\xi$, the scaled infinitesimal transformation is

Thus ${\delta }_{\xi }\gamma =K\xi +\kappa {\mathcal{L}}_{\xi }\gamma$.

Introduce the explicit background-breaking density

Off shell, the invariant action satisfies

The gauge-fixed action instead satisfies

On $U$, ${\mathcal{B}}_{\nu }=0$. This removes the switching breaking, not the gauge-fixing contribution.

Proof. By symmetry of $e$ and integration by parts,

Take the infinitesimal coefficient of the imported (eq:RouteII). The flat scalar variation is

After integrating these two terms, the coefficient of $\kappa {\xi }^{\nu }$ is exactly (eq:B). The fundamental lemma for arbitrary compact interior $\xi$ proves (eq:noether). Adding the separate gauge-fixing variation proves (eq:noether-gf). On $U$, both $1-g$ and $\partial g$ vanish.

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

This is the contraction of (eq:noether) and the infinitesimal form of X-A's Route II. It is not a new covariance theorem for the ramp observable.

For a $\kappa$-independent scalar write $e=e_0+\kappa e_1+{\kappa }^2{e}_2+\cdots$ and

Then (eq:noether) gives

where ${\mathfrak{L}}^{(1)}=\frac{1}{2}({a_1}^{\alpha \beta }{p}_{\alpha }{p}_{\beta }-m^2{b}_1{\varphi }^2)$. These formulas expose the missing coupling and nonlinear metric terms in a bare partial-divergence identity. The source sign in (eq:N1) agrees with $-\frac{1}{2}g{T}_0$ in the metric derivative.

Replacing $S_{\mathrm{EH}}$ by $S_{\mathrm{FP}}$ does not preserve the nonlinear Noether identity even on the plateau.

Proof. Set $\varphi =0$. For the complete Einstein action, ${\mathcal{N}}_{\nu }(e_{\mathrm{EH}})=0$. Its order-one identity requires

Deleting the cubic action deletes $2{\mathcal{H}}_2$ but does not delete $Q_{\nu }(2{\mathcal{H}}_1)$. The latter is not identically zero. For example, where $\gamma =\psi (t)\eta$,

Choose a smooth compact tensor supported in $U$ which has this form near a point where $\dot{\psi }\ddot{\psi }\ne 0$. The proposed quadratic nonlinear identity fails there.

The fixed term (eq:gf) is not invariant under ${\mathcal{G}}_U$. At the linear level its breaking is $-\square D^{\nu }$, as in (eq:egf).

Proof. Take $\xi =f{\partial }_1$, with nonzero $f\in {C_c}^{\infty }(U)$. Directly,

A nonzero compact interior $f$ cannot satisfy $\square f=0$, by Cauchy uniqueness with zero past data. Starting from $\gamma =0$, a genuine pure-gauge metric has ${\gamma }^{\prime }=K\xi +O(\kappa )$, so the nonzero displayed term is the leading gauge-fixing action. The Einstein action on that flat pullback is unchanged. Hence the discrepancy is gauge fixing, not a switching defect.

No compact plateau gauge algebra is recovered by simply demanding $\square \xi =0$: compact support to the future of $\Sigma$ and Cauchy uniqueness then force $\xi =0$.

Let $F_{(2,0),\mathrm{dens}}^{\ast }$ denote pullback of a contravariant rank-two density of weight one:

All admitted flows preserve orientation.

For $F\in {\mathcal{G}}_U$, put $q^{\prime }=F^{\ast }q$, ${\varphi }^{\prime }=F^{\ast }\varphi$, and ${\gamma }^{\prime }=(q^{\prime }-\eta )/\kappa$. Then

The identity is formal and holds after expansion through ${\kappa }^2$, provided all coefficients needed before projection are retained. It does not hold with $e$ replaced by $e+e_{\mathrm{gf}}$ in general.

Proof. The Einstein tensor-density in (eq:eexact) is natural. The scalar stress density $\sqrt{-q}{T_q}^{\mu \nu }$ is natural as well. Since $F^{\ast }g=g$, its switched multiple transforms by (eq:tensorpull). This proves (eq:covariance). The counterexample in Proposition prop:gfbad, or the explicit formula (eq:egf), excludes the gauge-fixed assertion.

If covariant indices are preferred, define

It transforms as a covariant rank-two density; its two indices also acquire Jacobian factors. Treating it as four-by-four scalar densities would be incorrect.

For a general compact $F$ which need not preserve $g$, the exact metric covariance defect is

Its infinitesimal form involves $X_{\xi }g$. The divergence defect involves $({\partial }_{\mu }g)T^{\mu \nu }$, as in (eq:N1); these two defects are related but not identical formulas.

In particular, $\partial g$ is a source-conservation issue, not an extra term in the minimally coupled metric Euler derivative. The restriction to ${\mathcal{G}}_U$ removes the covariance defect. It does not imply global solvability of the metric equation in the switching region.

For a fixed free scalar ${\varphi }_0$, let ${\mathcal{J}}_g({\varphi }_0)$ consist of smooth formal coefficient families

such that the coefficients through ${\kappa }^2$ of $e=0$ and $E_{\varphi }=0$ vanish. Positive-order scalar coefficients have the X-A zero Cauchy data. No condition is imposed on metric Cauchy data.

This is the jet of the invariant Einstein equation, not of the gauge-fixed Euler equation. Requiring $D=0$ in addition is a possible further restriction, not a replacement of that equation.

If ${\mathcal{J}}_g({\varphi }_0)$ is nonempty, then

For the source-response subclass ${\gamma }_0=0$, this is already the compatibility condition for

Proof. The linearized Bianchi identity ${\partial }_{\mu }{G_{\mathrm{lin}}}^{\mu \nu }=0$ follows directly from (eq:Glin). Also ${\partial }_{\mu }{T_0}^{\mu \nu }=(P_m{\varphi }_0){\partial }^{\nu }{\varphi }_0=0$. Taking the divergence of (eq:source-jet) proves the assertion in that subclass.

For arbitrary ${\gamma }_0$, evaluate the order-one coefficient of the exact Noether identity on a purported jet. Since the metric Euler coefficients vanish, its left side vanishes. Since $P_m{\varphi }_0=0$, the order-zero scalar Euler term vanishes. The order-zero value of ${\mathcal{B}}_{\nu }$ is precisely $({\partial }_{\mu }g){T_0}^{\mu }{}_{\nu }[{\varphi }_0]$. Equation (eq:noether) therefore forces (eq:compatibility). This argument permits arbitrary vacuum leading metric data and does not presume a harmonic section.

For the chosen mass define

It is a smooth free solution for every $m\ge 0$.

For every nonzero compact switch of the frozen class,

The obstruction occurs in the first sourced metric coefficient, before the second-order scalar or metric coefficients can repair it.

Proof. The input has no spatial derivative. A direct computation gives

and

Consequently the $\nu =0$ component of (eq:compatibility) is

A nonzero function compactly supported in the slab interior cannot be independent of time: for each fixed spatial point it vanishes near both time boundaries. Since the switch has a nonempty plateau, ${\partial }_{t}g$ is not identically zero. The necessary compatibility condition fails.

The same obstruction can use compact scalar Cauchy data. Choose a spatial cutoff equal to one on a ball containing the initial causal shadows of $\mathrm{supp}g$, and prescribe ${\varphi }_0(0)=0$, ${\dot{\varphi }}_0(0)$ equal to that cutoff. Finite propagation and uniqueness imply that this compact-data solution equals (eq:input) on a neighbourhood of $\mathrm{supp}g$. Thus its stress gives the same witness. This uses the classical Cauchy theory already admitted in X-A, not a gravitational propagator.

Because $g$ rises from zero to one somewhere, there is an interior open bump region on which ${\partial }_{t}g\ge c>0$. For nonnegative $f$ supported there with $\int f{\mathrm{d}}^{4}x=1$,

This is an evaluated conservation obstruction, not an Einstein solution.

In the massless diagnostic ${\varphi }_0=t$, take also $\gamma =\psi \eta$, $\psi$ smooth. Then

Choosing a conformal test tensor does not alter the nonzero number (eq:bump) and does not solve the coupled metric equation. This is a check of the compatibility gate only.

For every smooth vector field,

Substitution into (eq:Glin) verifies the first identity. For $\xi \in {\mathfrak{g}}_U$ this is the required pure-gauge diagnostic. It checks the linearization, not the existence of a sourced jet.

The unrestricted family of Einstein jets is not empty: ${\varphi }_0=0$ and all metric and scalar coefficients zero solve the equations. Other inputs satisfying (eq:compatibility) are not classified here. Necessity is not being promoted to a general sufficiency theorem.

No second-order correction can cancel the nonzero order-one compatibility coefficient. The stress tensor is not modified, and the Einstein–Hilbert density is not switched. A possible removal or controlled comparison of compact cutoffs belongs to a successor local-coupling or adiabatic problem; no such limit is taken.

At quadratic order the sum of (eq:FP) and (eq:gf) is

with Euler operator

In the source-response expansion, the gauge-fixed equation is

Componentwise smooth Cauchy theory permits such an equation with a nonconserved smooth compact source. Taking its divergence gives

For the witness input, $D$ therefore cannot vanish identically.

Thus a gauge-fixed solution is not an Einstein solution in de Donder gauge unless the source compatibility is restored. Gauge fixing does not remove the Bianchi obstruction; it changes the equation being solved. Equation (eq:quadraticoperator) specifies an operator only. No graviton Green function, two-point function, propagator normalization, or state is constructed.

The corrected sheet reaches Constructed. The specified on-shell Einstein reading reaches Proved obstructed. The literal demand for a nonlinear covariant, gauge-fixed Fierz–Pauli-only sheet cannot receive a construction theorem.

XI-B receives:

1. the signed quadratic operator $2G_{\mathrm{lin}}+e_{\mathrm{gf}}=-\square \bar{\gamma }$;

2. the explicit de Donder term and its breaking;

3. the stress-tensor vertex (eq:vertex);

4. the unswitched Einstein action and the order bookkeeping needed for its nonlinear identities;

5. the Noether identity (eq:noether), including its background-switch defect;

6. the empty fixed-input Einstein fiber and its distinction from gauge-fixed wave solvability.

It receives no propagator, ghost, antifield, tree, or quantum state. The algebra ${\mathfrak{g}}_U$ acts on classical configurations; it has not been shown to act on a quantum graviton. Establishing a BRST description of gauge fixing would be a later construction, not a consequence of naming the de Donder term BFR.

A coherent classical sheet has been frozen without putting the metric on shell. Its invariant and gauge-fixed derivatives are distinguished, and its source obstruction has a named input:

The ramp relational prescription remains the inherited X-A object. No quantum-gravitational observable has been constructed.

This appendix makes (eq:Hdef) an explicit coefficient prescription. All contractions use $\eta$ unless a coefficient $u_i$ is written.

Set

For $n=1,2,3$ define

These are precisely the first three coefficients of the Christoffel symbols of $q=\eta +\kappa \gamma$.

Define, for $1\le n\le 3$,