Landmark XI-E On-shell admission of the fixed-width tree, a localized first-order identity, and the held-fixed defect

We decide three calculations separately, retaining the graviton inverse ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$. First, the fixed-width insertion and classical graviton tree are admitted to a polynomial on-shell Hadamard algebra extending the finite-generator causal algebra of XI-B. The full auxiliary and ghost blocks are derived, their graded antisymmetric parts are checked, and smooth changes of product data give algebra isomorphisms. No state or expectation value is selected.

Second, the complete coordinate word has a first-order coefficient identity for compact ghosts above the ramp. Its ghost-slot defect is localized in the ramp. The scalar-response frame adds switching terms; their advertised support bound requires an observation plateau. The affine ghost witness leaves precisely $\epsilon /2$ after the complete-word cancellation, whereas the bare insertion retains its larger response.

Third, the held-fixed second-order defect cancels against the order-$\kappa$ co-transformation of the metric in the first coefficient. The positive witness pairing verifies the sign. The omitted cubic Einstein response is then named and its classical variation is computed, including a nonzero evaluation.

These are admission and coefficient theorems, not one interacting BRST-closure theorem. Raw causal-Weyl multiplication has a separate obstruction; the Hadamard extension preserves the causal commutation relations, not that undefined contraction prescription. The mixed loop, quantum master equation, sharp-width limit, and empty Einstein jet remain outside the release.

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

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

: three separately constructed calculations{

1. Route 1: Constructed. A polynomial on-shell Hadamard extension admits ${\mathcal{B}}_f$ and $\mathcal{T}$ and embeds XI-B's algebra. The smaller raw causal-Weyl branch is obstructed under its specified continuity interface.

2. Route 2: Constructed. The plateau coefficient identity, affine recomputation, and corrected localized support statement are proved. This is not an interacting differential.

3. Route 3: Constructed at the classical coefficient level. The split defect cancels. The cubic Einstein tree has an explicit variation, not an asserted vanishing variation.

The combined release is constructed for these named calculations. A complete interacting quantum Ward identity is not constructed. No loop value, state, sharp limit, or Einstein-jet repair is claimed.}

XI-D stopped before quantum composite admission and distinguished bare closure from closure of a complete observable XID. This paper takes the admission step by specifying Hadamard product data. It does not turn the three routes into a single BRST theorem.

Three distinctions prevent incorrect conclusions.

relations versus causal multiplication. An extension may retain every causal commutation relation of XI-B while using a different symbol product. The obstruction below is to extending the raw causal-Weyl contraction prescription to singular polynomial kernels with the stated continuity. It is not an obstruction to every algebra having those commutators.

elements versus old symbol formulas. An exact algebra embedding preserves old elements and products. It need not preserve their Weyl-symbol formulas after a change to normal-ordering coordinates. Agreement on smooth finite-rank symbols is stated with that ordering convention made explicit.

identities versus differentials. The expression $s_0{\mathcal{B}}_f+\frac{1}{2}\int fc\cdot \partial \varphi$ is a first-order coefficient calculation. The second term is the scalar co-transformation in that calculation. Naming it does not construct a differential on the on-shell algebra.

The extension is restricted to polynomials with fixed compact distributional kernels. We do not assert that a pointwise wavefront condition alone gives a closed algebra of every nonpolynomial smooth functional. This restriction also avoids the broader functional-analytic issue described in HRV.

The three endings are applied to each named route: Constructed, Proved obstructed, or Unresolved. Excluded calculations are not silently incorporated into a constructed route.

Use

Parentheses include the factor $1/2$. All classical fields are smooth on a neighbourhood of the closed slab.

Fix $0<\epsilon <T$ and the ramp

Fix

and

The observation test is real $f\in {C_c}^{\infty }({M_T}^{\circ })$. An additional observation-plateau hypothesis will be stated only where it is needed.

For a symmetric tensor $r$ put

Then ${\mathcal{P}}^2=\mathrm{id}$ and $D\mathcal{R}=\frac{1}{2}\square$.

The independent multiplet is $(\gamma ,\varphi ,c,\bar{c},b)$, with parities $(0,0,1,1,0)$ and ghost numbers $(0,0,1,-1,0)$. XI-B supplies

Here ${{\mathsf{E}}_0}^{\mathrm{R}}$ is the zero-initial-data wave inverse on the slab. Its advanced counterpart has zero final data. The scalar equation inverse is ${{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$; its action-Hessian inverse is $-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}$.

Free configurations satisfy

Ghost Cauchy data are unrestricted.

Define

The inherited symbols are

where

XI-C's physical one-graviton response of ${\mathcal{B}}_f$ is $-\mathcal{T}$, at coupling degree ${\kappa }^2$ XIC.

The factor in (eq:free) is $-1$. Neither $-2\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$ nor $-4\mathcal{P}{{\mathsf{E}}_0}^{\mathrm{R}}$ is used. VII-C's Hadamard matrix, sharp word, and reflected-residue calculation are not inputs.

Let $Q_{\epsilon }=L_{\epsilon }{G_{\mathrm{gr}}}^{\mathrm{R}}$ denote the coordinate response kernel, with its tensor indices written ${Q_{\epsilon }}^{\mu }{}_{ab}(x,y)$. Then

For a compact scalar test $w$ put $\chi ={\Delta }_{\varphi \varphi }w=-{\delta }_{m}w$, where ${\delta }_m={{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{A}}$. The formal first commutator with a linear generator is

It has the explicit classical coefficient

The first summand is a local quadratic stress polynomial with smooth compact coefficient

Smoothness follows by transposing the composed retarded operators onto the smooth observation test.

Such local quadratic representatives are not smooth finite-rank kernels. More generally, a nonnegative local square $\int q{\varphi }^2$, with $q$ positive on an open set, has infinite rank on free scalar solutions: its quadratic form is positive on the infinite-dimensional space of solution restrictions to that set. A quadratic polynomial in finitely many linear smearings has finite rank.

Thus a finite-generator presentation does not automatically contain the composites demanded by (eq:comm). This is a domain warning, not the asserted impossibility of every bimodule extension. Every algebra can be placed in larger algebraic modules; leaving ${\mathcal{A}}_0$ is not a proof to the contrary. Nor does one summand of (eq:comm), without analysis of the whole coefficient, establish every possible on-shell nonrepresentation claim.

There is, however, a direct obstruction to the proposed unmodified causal contraction prescription.

Consider an extension containing compact local quadratic polynomials and requiring their products to be obtained by continuous distributional limits of the raw causal-Weyl contractions. Such an extension does not exist without changing that prescription.

Proof. For two local squares the second contraction requires, up to a nonzero constant,

Choose two small disjoint neighbourhoods of distinct null-related points. In suitable coordinates transverse to the light cone, the leading singularity of ${\delta }_m$ is a nonzero smooth multiple of $\delta (u)$. The massive tail does not remove it.

Let $\rho \ge 0$ be a smooth mollifier of integral one. The regularizations ${\rho }_a(u)=a^{-1}\rho (u/a)$ converge to $\delta (u)$, but

for $\psi (0)>0$. The second contractions therefore have no distributional limit. In wavefront language, both signs of the conormal occur and the diagonal pullback of the tensor square is not defined by the multiplication criterion.

A prescription requiring this limit cannot assign the proposed product. Renormalized or reordered products are different prescriptions and are not excluded by the argument.

The quantifier includes all local quadratic symbols, as does the proposed microcausal extension. It does not exclude an abstract extension preserving the linear causal commutators. The Hadamard construction below is precisely such an extension.

The example also shows why disjoint endpoint supports are not a universal multiplication theorem: distinct null-related points already exhibit the problem.

We use the surrounding Minkowski space $M={\mathbb{R}}^4$ to define distributional kernels. The observation and switch remain on the slab. The ramp is smooth on $M$ and zero for $t\le 0$. All insertion kernels below are compact on $M^n$.

Smooth free slab solutions extend to global free solutions by their Cauchy data. Thus restricting these compact kernel functionals to global solutions gives the same interpretation on ${\mathcal{C}}_{\mathrm{sol}}$ as the original slab prescription. No boundary test is silently replaced by an interior test.

For $\mu \ge 0$ set

and define the scalar bidistribution

It is interpreted distributionally, including the massless case. The spatial Fourier formula for ${\delta }_{\mu }$ gives

To match the proposed block notation put

These are bisolutions with one common oriented Hadamard wavefront relation. Their normalization follows directly from (eq:W); it is not copied from another metric action.

We use these distributions as product data. No functional $\omega$ on the resulting algebra is selected, and no expectation value is computed.

Let $A=2G_{\mathrm{lin}}$. The metric–auxiliary Hessian is

with

Flat tensor and vector pairings are understood throughout.

Define the nonzero blocks of a bidistribution matrix $H$ by

Vector identities accompany the ghost blocks. All other cross-blocks vanish.

For the translation-invariant data (eq:h),

The matrix $H$ solves the full free equations in both slots. Its graded antisymmetric part is

where $\Delta$ is exactly XI-B's causal matrix.

Proof. Differentiating $-\mathcal{P}{h}_0$ in its second tensor slot and using ${\partial }_y{h}_0=-{\partial }_x{h}_0$ gives ${\mathcal{R}}_x{h}_0$. In contrast, differentiation in the first slot gives the displayed $D_x$ formula. Applying both divergences produces a constant multiple of ${\square }_x{h}_0$, hence zero.

The auxiliary equations hold because $b=D\gamma$ in each slot and $-\square \mathcal{P}{H}_{\gamma \gamma }=0$. The scalar and ghost equations follow from the corresponding bisolution equations.

Equation (eq:h) gives

Differentiation gives the mixed auxiliary blocks $i\mathcal{R}{\delta }_0$ and $iD(-\mathcal{P}{\delta }_0)$. For the odd pair,

which is $i{\Delta }_{c\bar{c}}$ since ${\Delta }_{c\bar{c}}=-i{\delta }_0$. The reversed pair similarly gives $-{\delta }_0$.

In particular,

has the supplied sign. The indefinite tensor factor causes no problem for this algebraic construction; positivity is not a hypothesis.

For non-translation-invariant product data, the safe definitions are the actual slot derivatives in (eq:Haux). Replacing $D_y{H}_{\gamma \gamma }$ by ${\mathcal{R}}_x{h}_0$ without translation invariance is not valid.

Let ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ consist of finite graded polynomials

where the kernels are compact, graded symmetric distributions on $M^n$, with finite field indices, and satisfy

The cones include zero covectors; a total zero covector is never a wavefront point. Define ${{\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}}^{\mathrm{sol}}$ by restriction to the linear solution space (eq:shell).

This fixed-kernel polynomial class is sufficient here. Wick polynomials also belong to the more restrictive equicausal class discussed in HRV (Theorem 5.9).

Let ${\Gamma }_H$ be the graded contraction operator between two functional factors, normalized so that

Define

Each contraction series terminates for fixed polynomials. The graded derivative convention includes the Koszul signs for odd slots.

Equation (eq:star) defines an associative unital graded algebra. All its polynomial contractions exist. The product descends to the free shell and has the causal relations (eq:antisym).

Proof. Every nonzero wavefront pair of $H$ has the same orientation: one endpoint is in one causal cone and the other in the opposite cone. Derivatives and constant field matrices do not alter this fact.

In a contraction, cancellation of the propagator covectors would require the contracted slots of the first kernel all to point in one cone, and those of the second all to point in the opposite cone. If every slot is contracted, this is excluded by (eq:mc). With uncontracted slots the same argument excludes the dangerous zero covector in a pushforward. It also excludes an output wavefront tuple wholly in either causal cone. Compact kernel supports make the integrations proper. Thus contraction gives another compact kernel satisfying (eq:mc).

Associativity follows from the exponential formula: on three factors the contraction operators ${\Gamma }_{12},{\Gamma }_{13},{\Gamma }_{23}$ have constant coefficients and commute with the graded signs already included. Both bracketings give their common exponential followed by multiplication.

For descent, the ranges of $H$ satisfy the free equations in both slots. On the reduced variables $(\gamma ,\varphi ,c,\bar{c})$, with $b=D\gamma$, all contraction directions are therefore tangent to the linear solution space. A polynomial vanishing on that space has all tangential derivatives zero there. Its contractions consequently vanish there as well. The kernel version follows by the same distributional contractions just established. Hence the vanishing ideal is a two-sided star ideal.

Finally, the graded commutator of linear generators is $\hslash$ times (eq:antisym).

The scalar version of this construction and its normal-ordering interpretation are described in FR (Section 5); the block normalization and graded auxiliary calculation above are specific to this sheet.

Map each XI-B linear generator to its linear functional in ${\mathcal{E}}_H$. The equations and commutators agree, so the map extends to a homomorphism

It is injective. Indeed, PBW gives the polynomial presentation of the old central graded relations. Independent old test classes are separated by smooth solutions of the reduced wave equations. On any finite-dimensional span this gives the ordinary symmetric/exterior polynomial algebra as the classical limit. No nonzero old polynomial can acquire a zero leading symbol; the same argument, coefficientwise in $\hslash$, proves injectivity.

Here is the precise finite-rank interface. Put

On regular finite-rank kernels, $S$ is graded symmetric and

intertwines the causal-Weyl and $H$ products. Thus an old Weyl-symbol polynomial $F$ has image ${\alpha }_{S}F$. Equivalently, the old $H$-normally ordered polynomial ${{\alpha }_S}^{-1}F$ has image $F$.

Admission below uses these $H$-normal symbol coordinates. It agrees exactly with old inclusion in that presentation and has the required unchanged classical symbol. We do not assert the stronger, generally divergent claim that ${\alpha }_S$ extends unchanged to every singular kernel. That stronger statement would confuse normal ordering with an extension of raw causal multiplication.

Let ${G_0}^{\mathrm{R}}(x-z)$ be the full retarded wave kernel and put

Then

and the mixed insertion kernel is

Its support is compact: $x\in \mathrm{supp}f$ and $z\in J^{-}(x)\cap \{t\ge 0\}$. The ramp is flat at $t=0$, so extension to the ambient space adds no boundary conormal.

The kernels of ${\mathcal{B}}_f$ and the polarized cubic $\mathcal{T}$ belong to the polynomial domain of Definition def:domain. Their free-shell classes therefore define elements

Their leading symbols are the supplied classical functionals restricted to ${\mathcal{C}}_{\mathrm{sol}}$.

Proof. The wave kernel has opposite endpoint covectors, including its diagonal conormals. Smooth multiplication and differentiation give

No nonzero opposite pair lies entirely in either causal cone.

For the tree, compose

Before integrating, the two kernels share only $z$. A violation of the multiplication criterion would force the unshared $x$ and $y$ covectors to vanish, and then force the shared covector to vanish. The product is defined. Its pushforward is proper in the compact causal region selected by $f$, $g$, and the finite slab.

Attach the stress kernel at $y$ with its two scalar slots, and the final scalar slot at $x$. At each attachment the local diagonal conormal has total covector sum zero. The same one-shared-slot argument applies. Every resulting external triple satisfies

in the fixed Cartesian trivialization. A nonzero triple entirely in either closed causal cone cannot satisfy that identity. Polarization and slot symmetrization preserve the bound.

Lower derivatives are obtained by smearing slots with smooth configurations and obey the corresponding exclusion. Ghost slots introduced by the linear free rule replace a metric slot by a differentiated ghost slot; they do not enlarge the wavefront set. Thus the graded polynomial domain also covers the free variation of ${\mathcal{B}}_f$.

For compact kernel stages with a fixed allowed wavefront cone, we use the normal distribution topology. The operations just used satisfy the applicable continuity or hypocontinuity statements of BDH. In $H$-normal coordinates admission is the kernel inclusion followed by the on-shell quotient. This supplies fixed-width kernel-limit continuity on those stages. It is not VII-C's sharp-admission package.

Let $H^{\prime }$ be another full bisolution matrix with the same graded antisymmetric part and the same Hadamard singularity class. Its auxiliary blocks obey the actual slot equations. Then $d=H^{\prime }-H$ is smooth and graded symmetric.

The map

is an algebra isomorphism ${\mathcal{E}}_H\to {\mathcal{E}}_{H^{\prime }}$, fixing the linear generators and having inverse ${\alpha }_{-d}$.

Proof. Smooth self-contractions against compact kernels exist and preserve the polynomial domain. The second-order operator has the graded product rule whose cross term is twice ${\Gamma }_d$. Exponentiation gives

The bisolution property makes the map descend to the free shell. The two exponentials with opposite $d$ compose to the identity.

This proves independence of the algebra up to isomorphism, not equality of all normal-ordering formulas. In particular, a cubic normally ordered symbol can acquire an order-$\hslash$ linear term under ${\alpha }_d$. A transported admission is ${\alpha }_d{Q}_H(F)$; the separately written $Q_{H^{\prime }}(F)$ need not be that same abstract element without a normalization convention. At ${\hslash }^0$ their symbols agree. The scalar extended-Wick comparison is also expressed in HW (Lemma 2.1).

No state was needed for these statements. Conversely, a loop coefficient can depend on the smooth product data even when no expectation value is taken.

The schematic mixed retarded loop contains a scalar Hadamard line together with retarded metric and coordinate lines. If $\mathrm{supp}f\cap \mathrm{supp}g=\varnothing$, the endpoint coincidence $x=y$ is absent. This removes that particular diagonal, not every singular configuration.

At distinct null-related endpoints, a retarded kernel has both conormal orientations while a Hadamard kernel has one. Their naive product can therefore contain opposite covectors whose sum is zero. Failing this criterion is not by itself a universal nonexistence theorem, but it invalidates the claimed automatic multiplication.

A properly defined retarded quantum product would need its ordering combinations and extension data specified. The existence of ${\star }_H$ does not supply those data for this nonlocal retarded loop. Its value remains unresolved and is not used in admission, nor made finite merely by disjoint supports.

For a $\kappa$-independent scalar define

This is the coordinate expansion of $\varphi \circ {\mathsf{X}}^{-1}$. The scalar responses ${\rho }_1,{\rho }_2$ and the graviton response $-\mathcal{T}$ are solution-map outputs, not terms added to (eq:word). No admission of (eq:O2) is needed for the first-order identity below.

For this coefficient calculation let

This is notation for the proposed first scalar co-transformation. We do not thereby define $d_1$ on every generator or construct $\hat{s}$.

For every odd $c\in {C_c}^{\infty }(\{t>\epsilon \};TM)$,

off shell in $\gamma$ and $\varphi$.

Proof. The tensor $a_1[\mathcal{R}c]$ is supported where ${\sigma }_{\epsilon }=1$. Consequently

Both fields have zero initial data because $c$ vanishes near $\Sigma$. Cauchy uniqueness gives the first identity. The second follows from $s_0{\mathcal{B}}_f=-\int f(L_{\epsilon }\mathcal{R}c)\cdot \partial \varphi$.

No free ghost equation is used. The plateau condition involving $g$ is unnecessary for this coordinate identity.

Let $\vartheta$ be an odd Grassmann generator and take

Then

and

Ramp symmetry $\chi (1-s)=1-\chi (s)$ gives

Therefore

A nonnegative test supported where $\cos (mt)$ has a fixed nonzero sign detects both formulas.

The complete-word coefficient does not vanish on this witness. It has exactly the stated width-dependent factor, while the bare insertion retains $S_{\epsilon }(t)$. There is no limit statement. The witness is not in the compact above-ramp ghost class of Theorem thm:plateau.

Put $a_{\mu }=f{\partial }_{\mu }\varphi$ and let $L_{\epsilon }^{\ast }$ be the formal transpose into the metric slot. The complete first coefficient is

At fixed smooth $\varphi$, the ghost-slot coefficient in (eq:Wlocal) is a smooth compact kernel on the ambient space, with support contained in

Proof. Transposition of the divergence-form coordinate prescription gives

where $Cr=Dr$. The advanced response is smooth and has compact support in the relevant closed-slab causal past. Multiplication by the ramp makes the kernel flat at $t=0$. Applying the differential operator ${\mathcal{R}}^{\ast }$ gives another smooth compact kernel.

Theorem thm:plateau says that its pairing with every compact ghost supported above $\epsilon$ is zero. Below $t=0$ both terms in (eq:Wlocal) vanish. The asserted support follows.

A decomposition into homogeneous initial-data terms and layer sources is possible, but it must not be mistaken for an additional delta term in this final kernel. For the action-level ramp, the transpose above is smooth; artificial initial-surface terms in an integrated decomposition cancel appropriately.

Define

For $\xi =c/2$ supported in $U$,

where $D_{\xi }u={\partial }_{\mu }({\xi }^{\mu }u)$ and $X_{\xi }={\xi }^{\mu }{\partial }_{\mu }$. Zero data for $X_{\xi }\varphi$ and the preflight identity give

Thus it vanishes for a free scalar.

If one names an external scalar antifield slot and the single Koszul–Tate rule ${\delta }_{\mathrm{KT}}{\varphi }^{‡}=-P_m\varphi$, the last expression is

This identifies the equation term without constructing a full antibracket or an interacting differential.

The relation between (eq:plateau) and (eq:KT) is explicit: the variation of the scalar response supplies $X_{\xi }\varphi$ and an equation term, whereas the coordinate word displays its scalar co-transformation separately. No full quantum solution-map intertwiner is inferred from this coefficient comparison.

For a compact interior $\xi$, the free scalar identity gives

where

Proof. Use ${\mathsf{V}}_{1,g}[K\xi ]\varphi =-g{P}_m{X}_{\xi }\varphi +(\partial g)a_1[K\xi ]\partial \varphi$ on the free scalar shell. Since $g{X}_{\xi }\varphi$ has zero initial data,

Subtract the coordinate term and group $\xi -L_{\epsilon }K\xi$.

After transposition and integration by parts in the ghost, (eq:generaldefect) has ghost-slot support contained in

intersected with its compact causal dependence region. The support of derivatives of $g$ is contained in the closed support of $\mathrm{d}g$.

If $\mathrm{supp}f\subset P_g$, the last term in (eq:supportgeneral) is absent. The scalar-response first-order defect is localized in

Without an observation plateau the shorter statement is false. For example, with $m=0$, $\varphi =t$, and an above-ramp observation outside $\mathrm{supp}g$, a compact ghost supported there gives the nonzero local dressing variation, even though it misses both the ramp and $\mathrm{supp}\mathrm{d}g$.

In a spatially constant massless diagnostic, $c=2\vartheta t{\partial }_t$ gives

Thus the scalar-response coefficient, after the ramp, is

A sufficiently large spatial cutoff realizes this formula in an observation dependence cylinder. It is a diagnostic of the different frame, not (eq:completeaffine).

Finally,

This is XI-B's Cauchy-uniqueness lemma XIB. The plateau identity is nontrivial for off-shell ghost test parameters; its restriction to compact plateau free ghosts has only the zero parameter. Unrestricted free ghosts, including the affine witness, are not thereby made closed.

Let ${{\mathcal{A}}_2}^{\mathrm{sc}}[H,k;\varphi ]$ be the complete classical scalar-only second coefficient from X-A: it includes the second scalar response, mixed response–coordinate term, inverse-coordinate term, and quadratic Taylor term XA. Its dependence on the second metric coefficient is

Indeed, the $k$ change in ${\mathsf{X}}_2$ is ${\mathsf{X}}_1[r]$, and the $k$ change in the second scalar response is $-{{\mathsf{E}}_{\mathsf{m}}}^{\mathrm{R}}{\mathsf{V}}_{1,g}[r]\varphi$.

Set $H=h+\gamma$ and

For a free scalar, XI-C's forward held-fixed difference is

where

In

restore the order-$\kappa$ shift $\gamma \mapsto \gamma +\kappa {\mathcal{L}}_{\xi }\gamma$ while applying the stated background transformation. The order-${\kappa }^2$ held-fixed defect cancels exactly.

Proof. Linearity gives

The first coefficient's $K\xi$ difference vanishes for a free scalar by (eq:KT). Changes of the second coefficient caused by this additional order-$\kappa$ shift start at ${\kappa }^3$. Its remaining difference is (eq:held), which has exactly the opposite sign. The tree has no external metric argument to vary here.

For a single metric field $q=\eta +\kappa \gamma$, $k=0$ describes the original parametrization, not a prohibition on $\kappa$ dependence after transformation. The genuine flow of $\kappa \xi +{\kappa }^2\zeta$ gives

Its infinitesimal law is $\delta \gamma =K\xi +\kappa {\mathcal{L}}_{\xi }\gamma$. No independent second metric field is needed for this bookkeeping identity.

Take

in a sufficiently large causal cylinder, with $a\ge 0$, $a\not\equiv 0$, compactly supported above the ramp. A compact spatial cutoff and plateau make $\xi$ admissible without changing the calculation.

For $r={\mathcal{L}}_{\xi }\gamma$,

Hence

and

For a nonnegative observation after the support of $a$,

The held-fixed change contributes $-I$; the restored first coefficient contributes $+I$. Their sum is zero.

The same tensor $r$ is not generally free. In the cylinder its spatial wave operator contains $-a^{\prime \prime }{\delta }_{ij}$, which is not identically zero for a nonzero compact bump. Thus the co-transformation is a legitimate off-shell coefficient operation, not an automorphism of XI-B's free solution algebra.

The preceding cancellation is not the cubic Einstein calculation. We now name that missing contribution and compute its classical variation.

Retain XI-A's curvature convention and

Write

For completeness, a finite prescription for ${\mathcal{H}}_2$ is as follows:

Put