X-G-ST: Free causal BRST compatibility and a variational upgrade obstruction A supported fixed-width transfer on the inherited conical host and declared Minkowski completion

We give a supported realization of Cheung's Landmark X-G on the host retained by I-B and X-A-ST, with all analytical equations posed on its independently declared Minkowski completion. The regulator width remains fixed and positive.

An independent auxiliary field and one normalized gauge fermion produce the X-E/X-F Feynman-weight graviton operator. We compute the auxiliary–ghost Green matrix and prove that the off-shell nilpotent free differential preserves both the free equations and the graded causal relations. This constructs a free causal BRST derivation, not an interacting quantum BRST operator.

Two classical obstacles are then separated. The switched scalar action has an explicit natural gauge breaking, including an off-shell equation term outside $\mathrm{supp}\mathrm{d}g$, and a nonzero coefficient detected by a free scalar and a compact odd test. Independently, post-variation multiplication of gravitational sources by the onset violates variational reciprocity. A compact pure-gauge background in the onset layer gives the strictly positive free-solution pairing

It excludes an exact source-preserving variational upgrade on the unrestricted background class, even after free-shell-zero changes of representatives. The witness also admits free inputs with compact Cauchy data. Algebraic auxiliary elimination and corrections beginning at positive formal quantum order cannot remove it.

The direct three-tree background identity of X-F remains valid on the unchanged plateau algebra. Its survival is compatible with the upgrade obstruction because the two statements test different backgrounds and different maps. Local fish extensions, an exceptional scalar wavefront configuration, and a conditional background-independent loop addition are recorded without declaring the nonlocal one-loop observable constructed.

The principal free construction and obstruction are credited to X-G. The contribution here is their supported proof, normalization and causal-sign audit, and explicit separation of completion-level witnesses from laboratory support claims. These results provide a reproducible compatibility test for proposed quantum continuations, without claiming a universal obstruction to perturbative gravity. A finite supplied-evaluation interface has a complete destructive clock and reference-preserving regular transport. The release is local Grade S only.

three verdicts are not interchangeable{ The free auxiliary–ghost causal algebra is constructed.

An exact variational upgrade preserving the entire source-switched mixed metric map is obstructed in the class defined below.

The nonlocal one-loop observable and an interacting quantum BRST differential remain unconstructed. Neither the positive free result nor the classical obstruction is renamed as a theorem about those missing quantum operations.}

The analytical source is N. Cheung's Landmark X-G: The normalized BRST multiplet and the one-loop background gate: proved compatibility conditions, an exact-upgrade obstruction, and the remaining nonlocal extension problem, version 1 XG. The supplied X-E-ST and X-F-ST are predecessor source records XEST,XFST. Their identifiers do not establish registration or independent verification.

The present proof does not use the parent obstruction or free-BRST theorem as a premise. Their required operators, signs, and arguments are stated explicitly. Conversely, the inherited scalar completion and three tree expressions are not reconstructed as additional response maps. Only the algebraic interface needed to explain their survival is checked.

The principal normalization, free differential, reciprocity argument, and positive witness are attributed to X-G. The supported transfer and the audits in this manuscript are not priority claims for those parent results.

The source is proposed as X-G-ST. No repository installation, external registration, submission, acceptance, or successful compilation is asserted.

The Enter distinction relevant here is destructive classical readout versus quantum return. A measured clock encoding is consumed from the returned quantum interface; the complete output contains its classical code and the unmeasured residual EnterI,EnterPhilosophy,En09.

We keep separate:

1. an analytically defined quantity and acquisition of unknown field data;

2. an outcome law and a complete instrument;

3. a reference-preserving transport channel and entanglement generated by gravity;

4. a prescribed spacetime and a derivation of spacetime.

I-B supplies the prescribed-host interpretation, finite clock, calibration distinctions, and transport pattern IB. It does not supply the gravitational action, the free multiplet, or the obstruction proved below.

Only explicit finite Born instruments are needed in the operational part. Their complete channel identities are proved directly. No shortened reconstruction of the Enter characterization theorem is used.

Retain exactly the X-A-ST instance used by X-D-ST, X-E-ST, and the supplied X-F-ST:

1. the regular time-oriented conical product $({\mathcal{M}}^{\circ },G)$, its stipulated worldsheet $\mathcal{W}$, and its supplied future relation;

2. the supported injective developing chart

   $$\iota :\mathcal{L}⟶{\mathcal{M}}^{\circ },$$

   with its retained compact reserve disjoint from $\mathcal{W}$;

3. the independently declared analytical completion

   $$\hat{M}={\mathbb{R}}^{1+3},\eta =\mathrm{diag}(1,-1,-1,-1),M_{T_{\ast }}=[0,T_{\ast }]\times {\mathbb{R}}^3,\Sigma =\{t=0\};$$

4. the signature and time ledger

   $${\iota }^{\ast }G=-\eta ,t=c_{\mathrm{host}}(T-T_{\Sigma });$$

   the host has signature $(-,+,+,+)$, whereas the analytical equations use $(+,-,-,-)$;

5. a nonempty observation region

   $$\mathcal{O}⋐(0,T_{\ast })\times {\mathbb{R}}^3,D_{\ast }:=J_{\eta }^{-}(\overline{\mathcal{O}})\cap M_{T_{\ast }}⋐\mathcal{L}.$$

   Compact laboratory tests are transported and extended by zero on the completion.

The regular conical product is not used as a globally hyperbolic Cauchy spacetime. Every global free field and Green operator below belongs to $\hat{M}$.

The defect remains a worldsheet, not a worldline. There is no second cone, global conical inverse, worldsheet gluing construction, or host-update map IB,STRecords.

The package comprises the inherited supported host; the named free identities and obstruction with their normalization and support contracts; a finite dictionary of supplied evaluations; a complete destructive-clock instrument including untouched references; and scheduled locality with a nonempty regular reference-preserving route under declared finite permissions.

It does not include Grade E, spacetime emergence, or Einstein dynamics derived from records.

Fix

Let $U$ be a nonempty relatively compact open plateau:

Write

The symbol $K$, without a subscript, is reserved for the Killing operator.

Choose $\delta >0$ with

The onset ${\sigma }_{\epsilon }(t)=\chi (t/\epsilon )$ is smooth, zero for $t\le 0$, one for $t\ge \epsilon$, and flat at both endpoints. The width is fixed. No limit $\epsilon \to 0$ is taken.

The classical coefficients $h,k$ remain arbitrary smooth real symmetric tensors. In particular, they need not vanish in $0<t<\epsilon$. The metric jets below are not updates of the host metric $G$.

Set

The positive equation inverses are

Their causal support, Cauchy uniqueness, finite propagation, and continuity are used on the completion BGP.

For a covariant symmetric tensor $r$, put

and

All component identifications in flat formulas use $\eta$ and the coordinate volume density.

The formal transpose is bilinear, not a Hilbert-space adjoint. Compact integration by parts gives

Let

These are the retained X-E/X-F normalization identities XE,XF. They can also be checked by substituting the standard linearized Einstein tensor into (eq:operators). In particular,

The metric Green inverses remain

The scalar action Hessian is $-P_m$, whose inverses are $-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}},-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$. It is not the positive equation operator $P_m$.

Metric inputs are covariant tensors. Metric Euler outputs are contravariant tensor densities. Denote their Lie derivatives by ${{\mathcal{L}}^{\mathrm{cov}}}_{\xi },{{\mathcal{L}}^{\mathrm{dens}}}_{\xi }$. Thus

Their compact pairing satisfies

Keep the curvature convention and local action of X-E:

Euler derivatives mean compact variations. Neither global action integrability nor decay is assumed. The Palatini variation and transverse-traceless sign are retained from the stated action, not selected anew XE,XEST.

For $\bar{q}=\eta +\kappa h$, retain the background de Donder vector

and the fluctuation gauge-fixing density

Only the fluctuation is gauge fixed.

Let $\mathcal{E}[q]=J_q{G}^{\mu \nu }[q]$, with its Euler density type, and define

This fixes $Q_w$ without an unspecified vertex. It is a local formally symmetric differential operator, linear in the two-jet of $w$, with $\mathrm{supp}(Q_{w}r)\subset \mathrm{supp}w$.

Naturality of both terms in (eq:Q) gives

Indeed, differentiate $Q[F_s^{\ast }\eta ](F_s^{\ast }r)=F_s^{\ast }(Q[\eta ]r)$. This verifies the typed commutator used later; it does not rebuild the three response kernels.

The frozen mixed source map is

For general smooth inputs the source is prepared by its smooth onset and interpreted in the past-compact Green domain. For the compact backgrounds used in the obstruction, it is a compact smooth source.

We record the insertion target to make the counting unambiguous XC,STRecords,XFST. For $q=\eta +\kappa l+{\kappa }^{2}j$, define $a_1[l]$, $A_2[l,j]$, $b_1[l]$, and $B_2[l,j]$ by

In particular $a_1[l]=-{\mathsf{P}}_{\mathrm{tr}}l$ and $b_1[l]=\frac{1}{2}{\mathrm{tr}}_{\eta }l$.

The coordinate coefficients $w[l],v[l,j]$ have zero position and velocity on $\Sigma$ and solve

Put

The affine decomposition is

With ${\hat{A}}_1[l]=\mathbf{\varphi }(d_1[l])$, the frozen three tree coefficients are

Here $G_2$ is the polarized quadratic coefficient of $J_q{G}^{\mu \nu }[q]$. These symbols mean the compact distributional kernels of the fixed-width tree sheet, not pointwise multiplication of singular fields.

The inherited expression is

The matter response equals $-2{\mathsf{P}}_{\mathrm{tr}}{{\mathsf{E}}_0}^{\mathrm{R}}(g{T}_0[\mathbf{\varphi }])$. The scalar second-order pair in (eq:tests) and the three coefficients in (eq:trees) are retained once.

The inherited support distinctions remain: the scalar slot lies in $C_f$; the matter cubic has support in ${C_f}^3$; graviton slots of the other trees can reach $D_f$. Full kernels are distributions. Suitable smooth scalar-slot contractions are elements of ${\mathcal{D}}_{C_f}$. No kernel-admission theorem is inferred from a scalar-test topology.

is new below{ The new free construction adds independent auxiliary and ghost fields and proves compatibility of their causal relations.

The new obstruction tests whether the fixed source map (eq:mixed) can be the classical part of a variational retarded map on all free inputs.

Neither task requires constructing the predecessor trees again. Their admission is not used to prove the reciprocity obstruction.}

Use independent fields

Their parities are $0,0,1,1,0$, and their ghost numbers are $0,0,1,-1,0$. The odd fields have coefficients in a Grassmann algebra. All fields are ambient smooth configurations before equations are imposed. Compactness belongs to smearing tests and variations, not to a separate antighost configuration space.

Define an odd derivation $s_0$, commuting with spacetime differentiation, by

The graded Leibniz rule is $s_0(FG)=(s_{0}F)G+(-1{)}^{|F|}F(s_{0}G)$.

The derivation $s_0$ is nilpotent on the independent fields and their graded polynomial differential algebra.

Proof. Its square is zero on each generator in (eq:s0). The square of an odd derivation is an even derivation. It therefore vanishes on every polynomial in generators and their derivatives. No equation of motion is used.

This is the free normalization of X-G, not the differently scaled XI-B multiplet XG (:s0nil, :gfnorm).

Take the gauge fermion

The word “fermion” here denotes an odd gauge-fixing functional; no physical spinor is introduced. The free action is

As elsewhere, these are local actions with compact variations.

The auxiliary equation gives

Eliminating $b$ in the action produces exactly $\frac{1}{4}\int (\mathsf{C}r{)}_{\mu }(\mathsf{C}r{)}^{\mu }$. Thus the reduced metric operator is ${\mathsf{P}}_{\mathrm{gr}}$, and the ghost term in this convention is $-\int {\bar{c}}_{\mu }\square c^{\mu }$, without an extra normalization factor.

The independent $b$ is required for off-shell nilpotence. After its elimination, ${s_0}^2\bar{c}=\frac{1}{2}\square c$ is only an on-shell identity.

Proof. Complete the square:

Equation (eq:operators) gives the reduced operator. The minus sign of the ghost term follows from the odd parity of $\bar{c}$ in the variation of (eq:Psi0). Finally, $s_0(\mathsf{C}r)=\mathsf{C}Kc=\square c$.

The free action is $s_0$-invariant modulo integrated divergences: $\mathsf{A}K=0$ handles its metric term, the scalar is $s_0$-inert, and ${s_0}^2{\Psi }_0=0$.

The bosonic Hessian in the order $(r,b)$ is

For compact sources $(J,j)$, its equations are

For either retarded or advanced support,

\mathcal G_{rb}^{\mathrm R/\mathrm A}
  &=
  \begin{pmatrix}
   \Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}
       &\tfrac12\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}\mathsf C^*\\
   \tfrac12\mathsf C\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}
       &\tfrac14\mathsf C\Delta_{\mathrm{gr}}^{\mathrm R/\mathrm A}
                                      \mathsf C^*-\tfrac12I
  \end{pmatrix}
 \nonumber\\
  &=
  \begin{pmatrix}
     -4\mathsf P_{\mathrm{tr}}\mathsf E_0^{\mathrm R/\mathrm A}
          &K\mathsf E_0^{\mathrm R/\mathrm A}\\
     -2\mathsf C\mathsf P_{\mathrm{tr}}\mathsf E_0^{\mathrm R/\mathrm A}&0
  \end{pmatrix}.
 \label{eq:Ghb}

Both inverse identities hold on compact tests. The operators have the indicated causal support.

Proof. The second equation gives $b=(\mathsf{C}r-j)/2$. Substitution into the first gives

Apply the appropriate metric Green inverse and substitute back for $b$. This yields the first matrix.

The identities

follow from (eq:Cstar) and $\square {{\mathsf{E}}_0}^{\mathrm{R}/\mathrm{A}}=1$. They give the second matrix. Substitution verifies the right inverse; applying the same elimination to a compact field pair verifies the left inverse. Uniqueness reduces to that of the free metric wave equation. Derivatives and the cancelled algebraic block preserve causal support.

This Green matrix is the auxiliary normalization of X-G XG (:Ghb). It is not a Green matrix on the regular cone.

Use left Grassmann variations, placing variations on the left. The ghost Euler derivatives of $-\int \bar{c}\square c$ are

Consequently, in the order $(c,\bar{c})$,

Ordinary matrix multiplication verifies the two inverse identities with the indicated fibre pairings.

For the complete order $(r,b,\varphi ,c,\bar{c})$, use

Put

Its nonzero blocks are

\begin{array}{c|c@{\qquad}c|c}
 AB&\Delta_{AB}&AB&\Delta_{AB}\\
 \hline
 rr&-4\mathsf P_{\mathrm{tr}}\delta_0&\phi\phi&-\delta_m\\
 rb&K\delta_0&br&-2\mathsf C\mathsf P_{\mathrm{tr}}\delta_0\\
 c\bar c&-\delta_0&\bar c c&\delta_0
\end{array}
 \label{eq:causalblocks}

and ${\Delta }_{bb}=0$. The scalar sign comes from its action Hessian. The ghost form is graded antisymmetric: its odd–odd blocks are symmetric under simultaneous exchange of species and spacetime arguments, because ${{\delta }_0}^{\mathrm{T}}=-{\delta }_0$.

Let ${\mathfrak{A}}_0$ be the unital associative graded algebra over $\mathbb{C}[[\hslash ]]$ generated by compactly smeared linear fields, modulo the free equations

and the causal relations

No inverse of $\hslash$ is adjoined.

The Green identities imply that the causal form vanishes on equation tests, so it descends to their linear quotient. Its central graded Lie algebra has an enveloping algebra with the prescribed central relation. This gives the associative polynomial causal carrier. It does not require a state or a time-ordered product.

The derivation $s_0$ preserves the free equation ideal and all relations (eq:CCR). It induces a nilpotent graded derivation of ${\mathfrak{A}}_0$.

Proof. First consider the equations. Their variations are

The antighost equation varies into $\square b$. Taking the divergence of the first equation and using (eq:dividentities) shows that $\square b=0$ belongs to the equation ideal. The scalar and ghost equations are otherwise inert.

For the causal relations, the only cancellation between nonzero blocks occurs in the initially zero $r$–$\bar{c}$ relation. In distributional component notation,

The exchanged relation follows by graded antisymmetry. Variations of the $r$–$r$, $r$–$b$, and $r$–$c$ relations involve zero ghost cross blocks or the zero $cc$ block. Variations of the $\bar{c}$–$\bar{c}$ and $b$–$\bar{c}$ relations use zero $b$-ghost blocks and the zero $bb$ block. All scalar cross relations are inert. These checks exhaust the generator pairs.

Thus $s_0$ is well defined on the quotient. Lemma lem:s0nil proves its nilpotence there.

The nontrivial causal cancellation is the normalized free mechanism of X-G XG (:freeBRST). The construction is a free graded causal statement, not a time-ordered hierarchy.

The supplied X-F-ST allows a stipulated Wick presentation. The action-normalized causal algebra above is not identified with that presentation by ignoring an antisymmetric-part or scalar-sign difference. A smooth change of normal ordering cannot change a causal commutator.

The predecessor tree identity is retained at ${\hslash }^0$, where it is an identity of normal symbols modulo equations. Any later use at quantum order must specify a compatible common product presentation and its transport. No such identification is hidden in Theorem thm:freeBRST.

VII-I's prepared free algebra and XI-B's differently normalized auxiliary multiplet remain comparisons, not premises or identical carriers FreeComparisons.

This section distinguishes formal gauge kinematics from the released free causal derivation.

Write $q=\bar{q}+\kappa r$, with $\bar{q}$ held fixed. Define the formal kinematical rule

Equivalently, $s_{\mathrm{kin}}r={\mathcal{L}}_c\bar{q}+\kappa {\mathcal{L}}_{c}r$. Odd coefficients are placed on the left. Its leading flat rule is (eq:s0).

The rule (eq:skin) is nilpotent on the off-shell graded tensor differential algebra. This is not a claim that it acts on an interacting quantum algebra.

Proof. For $w^{\mu }=c^{\nu }{\partial }_{\nu }{c}^{\mu }$, expansion with the odd Leibniz rule gives ${{\mathcal{L}}_c}^2={\mathcal{L}}_w$. The terms with symmetric second spacetime derivatives cancel against the antisymmetric ghost coefficients. Thus

The scalar calculation is the same. In ${s_{\mathrm{kin}}}^{2}c$, the two terms involving first derivatives cancel after relabelling, and the second-derivative term is zero by ghost antisymmetry. The nonminimal pair is nilpotent by definition.

The natural background fermion is

Since $s_{\mathrm{kin}}\bar{q}=0$,

Auxiliary elimination gives the retained bosonic background gauge fixing.

Define the vector-density Euler operator

For the retained curvature convention,

The operator is natural. Its first background coefficient satisfies

where the two Lie derivatives act on vector densities and vectors, respectively.

Proof. Trace reversal of ${\mathcal{L}}_c\bar{q}$, with its indices raised by $\bar{q}$, is

Take the divergence. The commutator of the two covariant derivatives contributes the displayed Ricci term. Every operation is natural under simultaneous pullback, including the volume density. Differentiate the finite naturality identity at $\eta$ to obtain (eq:ghostcomm).

This is an equation-level construction of X-G XG (:ghost). For compactly supported $\xi$, the retarded inverse ${G_{\mathrm{gh}}}^{\mathrm{R}}=-{{\mathsf{E}}_0}^{\mathrm{R}}$ therefore gives

Indeed ${\mathcal{L}}_{\xi }c$ is compactly supported and the retarded left inverse applies. This identity does not construct a nonlinear interacting differential or a quantum master operation.

For this calculation only, define the natural reference action

where

It is used to evaluate a natural gauge variation. Its Euler equation is not identified with the post-variation source prescription (eq:mixed).

The flat stress satisfies

For compact ghost coefficient functions, modulo integrated divergences,

where

The full off-shell breaking is not supported only on $\mathrm{supp}\mathrm{d}g$.

Proof. The curved scalar density is natural: $s_{\mathrm{kin}}{\mathcal{L}}_q=\kappa {\partial }_{\mu }(c^{\mu }{\mathcal{L}}_q)$. For the flat density, $\eta$ is fixed, so

Multiplying by $g,1-g$ and integrating gives the first integral in (eq:breaking). Integrating its stress term once more and using the stress-divergence identity gives (eq:B).

The Einstein–Hilbert variation is a compact divergence. Lemma lem:kin gives ${s_{\mathrm{kin}}}^2{\Psi }_{\bar{q}}=0$. Neither contributes a breaking here.

Where $g=0$ and $\mathrm{d}g=0$, the off-shell term $-(P_m\varphi ){\partial }_{\nu }\varphi$ can still be nonzero.

This is Obstacle A of X-G XG (:breaking). In particular, the noncompact set $\mathrm{supp}(1-g)$ has not been replaced by a compact source.

For every nonzero compact switch $g$ with a plateau, there are a smooth free scalar and a compact odd ghost test on the completion for which the coefficient of $\kappa$ in (eq:breaking) is nonzero.

Proof. Take

Then $P_m{\varphi }_m=0$, ${T_0}^{00}[{\varphi }_m]=1/2$, and ${T_0}^{0i}[{\varphi }_m]=0$.

A vertical line through a plateau point has $g=0$ at sufficiently early times and $g=1$ at that point. The fundamental theorem of calculus gives an open region with ${\partial }_{t}g>0$. Choose a nonnegative nonzero $\beta \in {C_c}^{\infty }$ there and a nonzero odd Grassmann generator $\theta$. Put $c=\theta \beta {\partial }_t$.

At the leading metric $q=\eta$, the density difference in (eq:B) is zero, and the free scalar equation term vanishes. Therefore

The even coefficient of $\theta$ is strictly positive.

The witness is the parent mechanism XG (:matter). The compact ghost coefficient is an off-shell test of the action identity, not a nonzero compactly supported free ghost solution.

Theorem thm:matter is a completion-level theorem for every such switch. To call its detecting test a retained laboratory test, one must also verify

and choose $\beta$ there. This follows, for example, when the vertical segment from $\Sigma$ to a plateau point lies in the retained reserve.

The host assumption alone does not place every transition of an arbitrary compact $g$ inside $\mathcal{L}$. No such support claim is silently added. The completion-level breaking and the supported version are thereby kept distinct.

An ordinary canonical BV extension generating (eq:skin), with $S_{\mathrm{ref}}$ as its antifield-independent action and with the switch rule unchanged, cannot satisfy the classical master equation for all these inputs. A correction beginning at positive formal powers of $\hslash$ cannot cancel this classical breaking.

Proof. The antifield-independent coefficient of that master equation requires $s_{\mathrm{kin}}{S}_{\mathrm{ref}}=0$. Theorem thm:matter violates this necessary condition. Formal powers of $\hslash$ have independent coefficients.

This is a necessary-condition argument, not the construction or evaluation of a quantum BV operator. It does not establish that a particular observable's one-loop Ward coefficient is nonzero.

For a compact ghost coefficient supported where $g=1$, both terms in the first integral of (eq:breaking) vanish. Restricted plateau symmetry and unrestricted breaking are compatible.

For a fixed compact background $w$, the unswitched operator $Q_w$ is formally symmetric. Hence

No equality between the two sides is automatic.

The quadratic functional

has Hessian

It is not in general the source operator $N_w$. Switching a chosen local density before variation likewise produces derivatives of the onset, with the result depending on that density representative.

Proof. Vary the two copies of $r$ and transpose the operator in the first variation. The two second-variation contributions are $N_w$ and ${N_w}^{\mathrm{T}}$. For a density depending on derivatives of $r$, the Euler integration by parts differentiates its switch as well as its original coefficients.

This is Obstacle B, independent of (eq:B) XG,XFST. The difference is classical, supported in the onset layer for the commutator in (eq:symmetric), and can propagate retardedly. It is not a loop counterterm.

Off-shell failure of symmetry alone would not exclude a symmetric operator agreeing on the free shell. The next section supplies precisely that stronger test.

An upgrade in the class considered here must satisfy all of the following:

1. the free metric operator is ${\mathsf{P}}_{\mathrm{gr}}$, with the retarded inverse ${{\Delta }_{\mathrm{gr}}}^{\mathrm{R}}$ and its frozen causal relations;

2. the interaction is even and has a symmetric bosonic second variation after any algebraic auxiliary elimination;

3. for a compact background $w$, its reduced mixed metric Hessian $L_w$ has compact coefficients, or more generally a compact two-point kernel, so its bilinear pairing on smooth free inputs is defined and symmetric;

4. its classical retarded mixed field response obeys

   $$-{{\Delta }_{\mathrm{gr}}}^{\mathrm{R}}{L}_{w}u=-{{\Delta }_{\mathrm{gr}}}^{\mathrm{R}}{N}_{w}u$$

   for every smooth free metric solution ${\mathsf{P}}_{\mathrm{gr}}u=0$ and every allowed smooth background, including compact backgrounds in the onset layer;

5. quantum corrections are formal series in nonnegative powers of $\hslash$, with the classical response in (U4) unchanged.

Equality in (U4) is equality of fields, equivalently equality after every compact smooth linear metric smearing. It is stronger than equality of one chosen relational observable.

The compact pairing condition makes explicit the domain required by the reciprocity proof. It is satisfied by local mixed interactions with compact background coefficients. No exclusion is claimed for a proposed object whose bilinear pairing is undefined or whose boundary data change the prescribed response.

For an upgrade in Definition def:upgrade,

for every pair of smooth free metric solutions and every compact background $w$.

Proof. Apply ${\mathsf{P}}_{\mathrm{gr}}$ to (eq:exactresponse). The right-inverse identity yields $L_{w}u=N_{w}u$ for each free $u$. Symmetry of the bosonic second variation gives

Both pairings exist by (U3). Substitute the equalities on the free inputs to obtain (eq:reciprocity).

This is a necessary free-shell test, not just a test on arbitrary off-shell variations XG (:reciprocity).

A nonconstant smooth onset with the stated endpoint extensions has ${\sigma }_{\epsilon }^{\prime \prime }\not\equiv 0$ on $(0,\epsilon )$. Otherwise it would be affine there, and smoothness at the endpoints would force zero slope.

Choose

Choose

Set

where $e_{00}$ has only covariant component $00$ equal to one. Both $u,v$ solve ${\mathsf{P}}_{\mathrm{gr}}u={\mathsf{P}}_{\mathrm{gr}}v=0$.

The vector field ${\xi }_{\mathrm{on}}$ parametrizes an allowed background. It is not asserted to lie in the plateau algebra ${\mathfrak{g}}_p$.

For (eq:witnessdata),

Proof. The typed commutator (eq:commutator), evaluated on a free input, gives $Q_{w}u=-{\mathsf{P}}_{\mathrm{gr}}{\mathcal{L}}_{{\xi }_{\mathrm{on}}}^{\mathrm{cov}}u$, and the same formula for $v$.

For these tensors the nonzero Lie components are

There are no spatial–spatial components. For any tensor $a$ of this form,