Landmark XIII-A Classical cancellation on a fixed subclass, and the obstruction to a coefficientwise restriction quotient

We distinguish restriction of classical evaluations from restriction compatible with the inherited quantum product. One nonempty open set $W$ is fixed before parameters are selected. On the complete gauge-fixed free shell, impose $D^{\ast }b=0$ on $W$. Under a uniform causal plateau hypothesis, XII-F's auxiliary remainder vanishes on this subclass, including for the full displayed second-order coefficient. This is an evaluation identity, not an action on a quantum quotient.

We then test the coefficientwise restriction kernel. The constraint smearing $L_q=\int q^{\mu \nu }(D^{\ast }b{)}_{\mu \nu }$ is an auxiliary-field functional off shell. Its commutator with a metric trace smearing must therefore use the inherited $b\gamma$ causal block. We compute that commutator and construct compact smearings with a strictly nonzero scalar coefficient for every nonempty $W$. Consequently the restriction kernel is not a two-sided ideal. A quotient killing all these constraints while retaining the unchanged brackets forces $\hslash 1=0$.

An admissible smooth rank-one change of ordering independently fails to preserve coefficientwise vanishing on the subclass. We also display the free differential of the coefficient, retaining its coordinate ramp and scalar switching terms, and distinguish tangency from closure. The copied XII-F witness remains a classical boundary of the claim, not a quantum anomaly. A reduced algebra and an integrated cancelling completion remain unresolved. No new quantum domain or retarded product is released.

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

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

and proof boundary{ Constructed: one fixed classical evaluation subclass and its conditional cancellation identity; the free-differential display; the $b\gamma$ commutator; compact detecting smearings; and the rank-one ordering calculation.

Proved obstructed: the coefficientwise restriction quotient with the inherited linear fields, unchanged causal brackets, and nonzero formal unit. The new vanishing space is not a two-sided ${\star }_H$-ideal and is not preserved by every admissible ordering transport.

Unresolved: a separately constructed reduced algebra or an integrated observable completion cancelling the full-shell remainder. Neither absence is a universal no-go theorem.

The inherited carrier is unchanged. No $R_1$, time ordering, quantum Ward identity, interacting differential, physical BRST class, loop, state, antibracket, master equation, or sharp-width limit is supplied.}

The endings Constructed, Proved obstructed, and Unresolved refer to a specified object and its stated requirements. They are not ordered assertions about every possible successor theory. In particular, a classical cancellation can be constructed while its naive quantum implementation is obstructed.

The principal question is whether the first reading can be implemented by simply quotienting the entire inherited carrier by coefficientwise vanishing on ${\mathcal{C}}_W$. We give a negative answer to that particular question. We do not select the observable subalgebra, equivalence relation, or product of a different reduction.

The obstruction uses regular linear fields. It does not require a singular composite product, an extension at coincident points, or a renormalization condition. Its mechanism is the existing central commutator. The ordering calculation concerns functionals, not movement of configurations by a change of product data.

Five manuscripts supply mathematical inputs. Their roles are separated below; other predecessor results are retained only as boundaries.

Use the supplied Minkowski conventions

The ramp ${\sigma }_{\epsilon }$ is smooth, zero for $t\le 0$, and one for $t\ge \epsilon$. The switch is $g\in {C_c}^{\infty }({M_T}^{\circ };[0,1])$, with $P_g=\mathrm{int}\{g=1\}$. All fields are ambient smooth configurations. No spatial decay is imposed. Tensor indices and dual tests are identified using $\eta$.

For a symmetric covariant tensor $r$ and a vector $v$,

Parentheses include $\frac{1}{2}$. The frozen identities are

The inverse remains ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{E}_0^{\mathrm{R}}$, with coefficient $-1$ XIB,XIIA.

The independent multiplet is $(\gamma ,\varphi ,c,\bar{c},b)$. Only $c,\bar{c}$ are odd; their ghost numbers are respectively $1,-1$. The other fields have ghost number zero. The shell is

Odd identities mean coefficientwise identities in every finite Grassmann algebra, as in XII-A. In particular, $b$ is an independent off-shell coordinate. Equation (eq:shell) is not an off-shell substitution rule.

With the inherited roots

put

For compact plateau parameters and a free scalar,

where

The last tensor is supported in $\mathrm{supp}\xi$. The companion has no ramp-layer contribution and vanishes if

There are admissible data satisfying (CP) for which ${\mathfrak{B}}_{\xi }$ is strictly nonzero. The corresponding admitted lift has nonzero class under $Q_H$. An additional cancelling completion was not constructed XIIF,XIID.

This is the substance of XII-F's release box: a conditional cancellation, a counterexample to unqualified invariance, and a separately unresolved completion. We do not repeat its dependence cylinder calculation.

XII-E fixes, without a sign change,

and the coefficient evaluated below is

The added term has no metric argument. Neither $\mathcal{O}$ nor ${{\Phi }^{(2)}}_{\mathrm{D}}$ has an independent $b,c,\bar{c}$ argument XIIE,XIIF.

XII-A supplies

where ${\mathcal{N}}_{\mathrm{sol}}[[\hslash ,\kappa ]]$ is a two-sided ideal and

For linear fields the causal relation is

The relevant even blocks are

with $E_0={E_0}^{\mathrm{R}}-{E_0}^{\mathrm{A}}$ the ambient causal wave operator.

An admissible ordering difference is a smooth graded-symmetric bisolution $d$, respecting parity and ghost number. Its map is

It acts on functionals, preserves the full-shell ideal, and descends to an isomorphism ${\bar{\alpha }}_d:{\mathcal{E}}_H\to {\mathcal{E}}_{H^{\prime }}$ XIIA. No preservation of the new vanishing space is included in that input.

XII-C retains

and its named fixed-input invariant jet remains empty. Thus $S_{\mathrm{EH}}$ has not been removed XIIC. The original detected XI-G classes, including their stated covectors $\pm (k_1,k_2,0)$ with $k_1+k_2=2\mathrm{d}t$, stay excluded under the inherited qualification XIIA. Our restriction is not a new representative theorem for those classes.

XII-B, XI-C–F, X-A–H, VII–IX, and Landmark I are not additional prerequisites. In particular, no X-G reciprocity or Bogoliubov claim is used. Smooth wave existence and finite propagation are the standard analytic input for one new compact-smearing construction BGP (Chapter 3).

Fix $f,g,m,\epsilon ,T$ and a nonempty open set $W$ satisfying

This choice precedes all parameter selections. Since $g$ is compactly supported, $\overline{W}$ is compact. Define

and

The dependence on $W,f,g$ is suppressed only after those data are frozen.

The plateau handoff is used with this declared parameter region. There is no later reduction of its parameter class. Requiring one condition throughout $W$ is different from choosing a new configuration-dependent neighbourhood for each parameter.

One has

Proof. Take $\gamma =b=0$, with arbitrary free scalar and ghost solutions. These configurations satisfy the shell relation and belong to ${\mathcal{C}}_W$. For $u\in {\mathcal{C}}_W$ and $\xi \in {\mathfrak{g}}_W$, the compact tensor ${\mathcal{L}}_{\xi }{\beta }_{f,g}[\varphi ]$ is supported inside $W$. Equation (eq:auxiliary) therefore gives ${\mathfrak{B}}_{\xi }(u)=0$.

Nonemptiness does not mean that arbitrary metric data can be combined with $b=0$: the equation $b=D\gamma$ must still hold. Scalar and ghost data are unrestricted within their free smooth solution classes in the example just given. No classification of admissible metric data is needed for this proposition.

For the detecting data of XII-F, with a compact spatial realization,

The parameter is a compact realization of $a(t){\partial }_t$, with $a\ge 0$ a nonzero bump. XII-F proves

It also proves that this lift has nonzero full-shell carrier class XIIF (Theorem on the nonzero auxiliary pairing). The value and its factor $8$ are copied, not recomputed.

The witness can be included in fixed restriction data by choosing $W$ to contain its compact parameter support and choosing the switch to cover the relevant causal compact sets before evaluation. The stronger uniform plateau requirement below is compatible with that realization: the switch may be chosen one near $J^{-}(\mathrm{supp}f)\cap J^{+}(\overline{W})$ as well as near $\overline{W}$. This is still among XII-F's allowed witness switches, since it covers the smaller causal set generated by $\mathrm{supp}\xi$.

For such detecting fixed data,

The first inclusion is not asserted to be equality.

Equation (eq:strict-detection) is observation-qualified. For example, $f=0$ makes every auxiliary pairing zero and gives ${\mathcal{C}}_{\mathrm{def}}(W;0,g)={\mathcal{C}}_{\mathrm{sol}}$. A copied existential witness cannot prove strictness for every already frozen observation and switch. By contrast, Section sec:ideal proves that ${\mathcal{C}}_W$ is proper and that the naive quotient fails for every nonempty $W$, independently of whether the chosen observation detects $\mathfrak{B}$.

Let $K_f=\mathrm{supp}f$. Impose the configuration-independent hypothesis

A convenient stronger sufficient version replaces $W$ by $\overline{W}$. The stronger set is compact and lies strictly between the slab's temporal endpoints. Its use makes explicit that such support data can be selected without reference to a particular configuration.

Under {(UCP${}_W$)},

This is an identity of evaluations of the prescribed off-shell lift.

Proof. Every term of $D{s}_{\xi }[\gamma ]$ is supported in $\mathrm{supp}\xi$. The inherited support statement gives

Thus {(UCP${}_W$)} implies (CP) for every smooth $\gamma$. The companion vanishes for the free scalar in $u$.

The auxiliary term vanishes by Proposition prop:inclusion. Finally, the additional scalar-sourced term in (eq:full-object) is unchanged by a metric-only shift. Equation (eq:copied-remainder) proves the result.

No carrier quotient is needed for this argument. Its uniform quantifiers concern evaluations on the stated class; they do not turn evaluation into a homomorphism for ${\star }_H$.

The finite shift does not preserve the shell. If one tries to maintain the auxiliary relation after $\gamma \mapsto \gamma +K\xi$, the required auxiliary change is

This does not generally preserve the homogeneous free equations. The compact residual intersection is zero XIB. Therefore the theorem compares a prescribed shifted off-shell evaluation with its value at an initial shell configuration. It is not a nontrivial residual symmetry of ${\mathcal{C}}_W$.

The frozen odd differential is

Its ghost number is one. The even finite parameter difference ${\Delta }_{\xi }$ has ghost number zero. Matching a differentiated vector field in the two expressions does not identify these operations.

For $u\in {\mathcal{C}}_W$, the free variation is an odd tangent vector to ${\mathcal{C}}_W$. Consequently $s_0{\mathcal{N}}_W\subseteq {\mathcal{N}}_W$ for the classical vanishing space defined in Section sec:ideal.

Proof. On the shell,

Also $\square b=D\square \gamma =0$, so $s_0(\square \bar{c})=i\square b=0$. The other free equations are unchanged. The additional restriction satisfies $s_0(D^{\ast }b)=0$. These statements are interpreted with an external odd infinitesimal parameter. Differentiating a polynomial identity on this linear restriction space proves preservation of its vanishing space.

Tangency is not the assertion $s_0\mathcal{O}=0$. We now give an exact display rather than replacing the compact parameter by a free ghost.

For a symmetric tensor $r$, set

Write

The inherited formulas are

Here $V_{1,g}[r]v$ is the divergence-form scalar operator appearing in the formula for $p_1$, with $\varphi$ replaced by $v$. By XII-E's affine slot identity,

No lower coefficient or additional metric slot has been inserted.

Put

The last equality uses the polarization of the already fixed quadratic response. It is not a newly selected response or co-transformation; $s_0{r}_g[\varphi ]=0$.

Define the explicit quadratic derivatives

Suppose now that the multiplet is on the free shell. Then $Dz=0$ and ${\partial }_{\mu }{a_1}^{\mu \nu }[\gamma ]=-b^{\nu }$. The first coordinate variation is

Thus the free ghost equation removes the ${\sigma }_{\epsilon }Dz$ term, not the coordinate layer term.

For $v=c^{\mu }{\partial }_{\mu }\varphi$, direct differentiation gives $V_{1,1}[z]\varphi =-\frac{1}{2}{P}_{m}v$ on the free scalar shell. Because $gv$ is compactly supported in the slab interior,

This formula retains the local switched term and both parts of $[P_m,g]v$. It requires no compact support assumption on $c$.

For completeness, write $Y=s_0{X}_2$ and $p_{2,z}=s_0{p}_2$. Expanding all external weight derivatives in the completion gives

All source-owned ramp terms remain inside the fixed derivative $\dot{k}$; none is set to zero. This notation does not require a new expansion of the gravitational Euler source. Equivalently, if $t_{\mathrm{D}}$ is XII-E's inherited polarized insertion kernel, its entire contribution is fixed by

This specifies that contribution without discarding any of its smooth source coefficients.

With the preceding notation,

It is an admitted odd, ghost-number-one polynomial. No vanishing on ${\mathcal{C}}_W$ is asserted.

Proof. Differentiate each of the five terms of (eq:O-expanded). All undifferentiated coefficient fields are even, so the displayed first variations acquire no additional Koszul signs. Equations (eq:ghost-X2)–(eq:ghost-p2) are the product rule applied to (eq:completion-display). The shell equations justify only the simplifications stated above. Admission follows from the inherited preservation of ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ by $s_0$ XIIA.

The coordinate ${\sigma }_{\epsilon }^{\prime }$ terms and the scalar $\partial g,\square g$ terms are visible. Homogeneous ghosts need not vanish in their sampling regions. The uniform plateau hypothesis on compact parameters consequently does not delete these terms. Furthermore, $\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$ belongs to a finite two-jet transformation; it is absent from the infinitesimal rule $s_0\gamma =\mathcal{R}c$. Theorem thm:classical is therefore not a proof of free BRST closure.

Set

For a formal series, vanishing means that each coefficient belongs to ${\mathcal{N}}_W$. Thus its formal vanishing space is ${\mathcal{N}}_W[[\hslash ,\kappa ]]$, not a condition allowing cancellation between different powers of the formal parameters.

One has ${\mathcal{N}}_{\mathrm{sol}}\subseteq {\mathcal{N}}_W$. The space ${\mathcal{N}}_W$ is an ideal for ordinary pointwise polynomial multiplication, and the vector-space restriction quotient exists. These facts do not establish its compatibility with ${\star }_H$. We test that compatibility using two regular fields.

Let

and define

Both are even, ghost-number-zero regular linear functionals. Every $L_q$ belongs to ${\mathcal{N}}_W$.

Compact integration by parts gives

The metric dual smearing is

These are the actual off-shell derivative kernels. In particular, $L_q$ has no metric derivative kernel off shell. Replacing its argument by $D\gamma$ before computing the product would change the lift and obscure which block performs the contraction.

Put $\chi =E_{0}h$. Then

The formula holds for every admissible replacement $H^{\prime }$ with the same causal part.

Proof. There is at most one contraction between linear functionals. Their commutator is therefore exactly

In covariant flat identification, $v_h$ is $\eta h$. Applying only the prescribed block gives

Indeed $D$ already contains trace reversal, so ${\mathcal{P}}^2=\mathrm{id}$ leaves the divergence of $\eta \chi$ with the displayed minus sign.

Using (eq:uq),

The causal solution satisfies $\square \chi =0$, proving (eq:commutator). The calculation keeps the coefficient $-1$ in ${\Delta }_{b\gamma }$ throughout. Finally, admissible changes leave the graded antisymmetric part unchanged.

No ${\Delta }_{\gamma \gamma }$ contraction was used. The positive sign of the final Hessian pairing results from integration by parts and the action of $D^{\ast }$ on $-\partial \chi$; it is not obtained by changing the sign of the causal metric block.

For every nonempty $W$ satisfying (eq:W), there are real compact $q,h$ as above such that $C(q,h)>0$.

Proof. Choose $(t_0,{\mathbf{x}}_0)\in W$ and a small open dependence diamond $V$ whose closure is contained in $W$. Consider the polynomial

It solves $\square {\chi }_{\mathrm{loc}}=0$ and has ${{\partial }_t}^2{\chi }_{\mathrm{loc}}=2$.

At $t=t_0$, multiply its spatial Cauchy value by a smooth compact cutoff equal to one on a ball large enough to contain the Cauchy domain of dependence of $\overline{V}$; use zero time derivative. The smooth wave Cauchy problem produces a real solution $\chi$ with compact Cauchy data. By finite propagation and uniqueness, it agrees with ${\chi }_{\mathrm{loc}}$ on $V$. On every bounded time interval its spatial support is bounded.

Choose a smooth temporal function $\theta$ which is zero before one time and one after a later time, with its transition compactly inside $(0,T)$. Set

This is a smooth compact test in ${M_T}^{\circ }$. The functions $\theta \chi$ and $(\theta -1)\chi$ solve the sourced equation with respectively zero past and zero future data. Uniqueness gives

This is the compact time-slice construction in the supplied wave conventions XIIA,BGP.

Choose a nonzero nonnegative $\omega \in {C_c}^{\infty }(V)$ and put $q^{00}=\omega$, with all other components zero. Then

This construction is independent of the observation $f$ and of the scalar mass.

This is a new compact-smearing construction, not a recalculation of XII-F's cylinder integral. Its local polynomial is used only to detect the Hessian in the existing linear causal bracket.

For every nonempty $W$, ${\mathcal{C}}_W⊊{\mathcal{C}}_{\mathrm{sol}}$. Moreover the constraint smearings used above are not elements of ${\mathcal{N}}_{\mathrm{sol}}$.

Proof. Using the solution $\chi$ in Lemma lem:compact-pair, take

The metric is a wave solution and $D(\eta \chi )=-\partial \chi$, so $u\in {\mathcal{C}}_{\mathrm{sol}}$. Since $\square \chi =0$,

Thus $u\notin {\mathcal{C}}_W$ and $L_q\notin {\mathcal{N}}_{\mathrm{sol}}$.

For every nonempty $W$, the space ${\mathcal{N}}_W[[\hslash ,\kappa ]]$ is not a two-sided ideal of the inherited product. Its image

is not a two-sided ideal in ${\mathcal{E}}_H$ either. These assertions hold for every admissible ordering.

Proof. Choose $q,h$ from Lemma lem:compact-pair. If the formal vanishing space were a two-sided ideal, both $L_q{\star }_H{G}_h$ and $G_h{\star }_H{L}_q$ would belong to it. Their difference would then belong to it. But (eq:commutator) is the formal scalar $i\hslash C(q,h)1$.

The zero configuration belongs to ${\mathcal{C}}_W$, so a nonzero constant cannot vanish on ${\mathcal{C}}_W$. Its $\hslash$ coefficient is therefore outside ${\mathcal{N}}_W$. This is a contradiction. The same commutator descends through the full-shell ideal, and constants survive that quotient, proving the on-shell assertion. The causal scalar is ordering-independent.

There is no associative product on

for which coefficientwise restriction is a unital homomorphism from the inherited ${\star }_H$-algebra.

More generally, a unital $\mathbb{C}[[\hslash ,\kappa ]]$-algebra homomorphism which kills every $L_q$ while preserving the inherited brackets must satisfy $\hslash 1=0$ in its target. It cannot have a nonzero unit with faithful formal scalar inclusion, or be a nonzero torsion-free formal quantum algebra.

Proof. The kernel of a homomorphism is a two-sided ideal, contradicting Theorem thm:nonideal. Alternatively, apply a proposed homomorphism to (eq:commutator). Its left side is zero because the image of $L_q$ is zero. Since $C(q,h)$ is a nonzero scalar, the result forces $\hslash 1=0$. In the coefficientwise restriction module the class of $1$ is nonzero, and multiplication by $\hslash$ merely shifts coefficients, so this relation is impossible.

Over a formal power-series ring, forcing $\hslash 1=0$ does not by itself force $1=0$. It may produce an $\hslash$-torsion quotient, including a classical specialization. After inverting $\hslash$, however, it forces the zero algebra. Neither option is the intended coefficientwise quantum restriction.

The inherited equation ideal remains valid. The obstruction is caused by adding a new condition which the causal contractions do not annihilate. Thus it is not a contradiction of XII-A's full-shell descent theorem. A smaller observable algebra, different constraint implementation, or separately proved reduction is not excluded by this argument.

Use the same $q,h,\chi ,u$ as in (eq:ordering-solution). Write $C=C(q,h)>0$. The multiplet $u$ is an ordinary real even solution with only ghost-number-zero components.

The matrix

is an admissible smooth ordering difference in XII-A's class.

Proof. It is smooth and symmetric under simultaneous exchange of species and spacetime slots. Every nonzero species is even and has ghost number zero. Since $u$ satisfies the complete shell equations, the matrix satisfies those equations in each slot, including the auxiliary relation $u_b=D{u}_{\gamma }$. Its graded antisymmetric part is zero, so $H+d$ has the same causal part as $H$. These are precisely the required conditions. No positivity or compact support condition on $d$ is imposed XIIA.

The nonzero $bb$ component of this difference is permitted. The vanishing frozen $H_{bb}$ block is not a requirement on every admissible replacement. Nor is a separate condition making the unchanged symbol rule $s_0$ a derivation for $H+d$ required: XII-A transports that differential by conjugation when necessary.

For the admissible difference (eq:rankone),

Consequently

The corresponding descended vanishing subspaces are not preserved by ${\bar{\alpha }}_d$ either.

Proof. Let $\ell =L_q^{\prime }$ be its field-independent linear derivative kernel. Then

The factor $\hslash /2$ in (eq:ordering-map) cancels the factor $2$. A second self-contraction would require a fourth derivative, which is zero. This proves (eq:ordering-square) exactly, not merely to first order.

Although ${L_q}^2$ vanishes on ${\mathcal{C}}_W$, the $\hslash$ coefficient of its transport is the nonzero constant $C^2$. Coefficientwise vanishing therefore fails. On-shell descent gives the same conclusion because that constant is not in ${\mathcal{N}}_{\mathrm{sol}}$.

The hypothesis $L_q(u)\ne 0$ is essential. Choosing a rank-one bisolution built from $u\in {\mathcal{C}}_W$ would give zero in this particular test. The theorem proves failure for an explicitly allowed ordering change; it does not assert failure for every smooth difference.

No point of ${\mathcal{C}}_W$ has been transformed by ${\alpha }_d$. The failed statement is preservation of a space of functional symbols. In contrast, preservation of the complete free-shell ideal by the same ordering maps remains an inherited theorem. The rank-one solution lies outside the newly imposed subclass, exactly where its constraint contraction can detect it.

The quotient obstruction does not answer whether an additional classical term can cancel ${\mathfrak{B}}_{\xi }$ on the full shell. Conversely, a cancellation of one evaluated pairing would not construct a reduced quantum algebra. We keep those problems separate.

Fix the ordinary vector-field bracket convention

For a proposed action on functionals, choose the representation convention

If another pullback convention is used, its corresponding sign must be changed consistently.

A necessary test for a correction $F$ satisfying ${\delta }_{\xi }F=-{\mathfrak{B}}_{\xi }$ is

Indeed, under (eq:representation), the difference of the first two terms is $-[{\delta }_{\xi },{\delta }_{\zeta }]F=-{\delta }_{[\xi ,\zeta ]}F$.

This test is well posed on the shell quotient only after the candidate $\delta$ preserves the equation ideal. That action has not been supplied here. In particular, the constant off-shell metric-shift derivatives

commute with each other. They do not furnish the nonabelian representation (eq:representation), and they generally do not preserve ${\mathcal{N}}_{\mathrm{sol}}$. They therefore cannot be inserted into (eq:cocycle) as if they were a declared action of the vector-field Lie algebra on the shell.

For a specified action and allowed correction class, failure of (eq:cocycle) excludes a correction of the stated affine type. Success supplies only a necessary compatibility condition, not solvability. Without the action, interpreting a formal failure as a universal completion obstruction would be invalid.

A possible interface is

For a degree-matched candidate, $r_{\xi }$ should be even, have ghost number zero, be linear in the even parameter and linear in the chosen even field slots, and have no scalar dependence. A retarded candidate may be required to have zero past data and support in $J^{+}(\mathrm{supp}\xi )\cap M_T$ for each fixed smooth field input.

Those conditions on smooth outputs are not sufficient for admission. The composed functional must also have a compact polarized degree-two kernel, with its wavefront set disjoint from the two all-causal cones. For example, a covector-sum-zero bound would be a sufficient kernel condition, but no such new kernel is constructed here. The existence of a source pairing on smooth fields is not substituted for this obligation.

Equation (eq:slot-target) records a range and compatibility problem; it does not assume an inverse of ${\mathcal{A}}_1$ on arbitrary auxiliary pairings. Even a pointwise assignment satisfying this slot equation would have to be realized as the variation of one declared correction, with consistent support, field degree, and parameter composition. It is that observable-completion problem which remains unresolved.

An infinitesimal solution must also be integrated into the finite coefficient law. The finite term $\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$ cannot be recovered by merely naming a linear ghost substitution. If a correction operates on a lower coefficient of the relational expansion, its coupling order must be printed; it is not added to the already filled second metric slot a second time.

For a proposed multiplet rule

the full-coefficient infinitesimal interface is

The lift in (eq:full-object) has no independent auxiliary or ghost argument. Hence a rule with ${Y_{\xi }}^{\gamma }={Y_{\xi }}^{\varphi }=0$ changes it by zero. A pure auxiliary or ghost variation cannot cancel the copied nonzero value.

A scalar variation is different: it also differentiates the scalar-sourced slot. Explicitly,

Thus the zero metric variation of that slot does not license dropping it from a scalar co-transformation calculation.

A correction containing $b$ explicitly requires a declared action on $b$, matching parity, and its own admitted kernel. The displayed shell relation alone supplies none of those transformation laws. No correction is absorbed into $Q_H$, the inherited equation ideal, or an unlabelled modification of $r_{\mathrm{D}}$.