Landmark XII-F The gauge-fixed remainder of Φ (2) D : companion support, the auxiliary pairing, and what is not a Ward identity

XII-E assembled the scalar-linear second-order relational coefficient and left its gauge-fixed remainder explicit. We decide that remainder without changing the source, inverse, insertion sign, or parameter space.

The propagated companion has no ramp-layer contribution. Its switching display is retained, its cancellation under the copied causal plateau condition is stated with that hypothesis, and an explicit off-condition diagnostic shows why universal cancellation cannot omit a support assumption.

The Einstein remainder equals a compact auxiliary pairing on the gauge-fixed free shell. It vanishes on configurations satisfying $D^{\ast }b=0$ near the parameter support, but it does not vanish on the whole shell. We exhibit a massless free scalar, a free pure-trace metric with affine auxiliary field, a compact plateau parameter, and an observation satisfying the causal plateau condition for which the assembled variation is strictly negative. The calculation also fixes the value of the auxiliary pairing. Its compact microcausal kernel defines a nonzero class in the inherited carrier.

Unqualified classical invariance of the named coefficient is therefore Proved obstructed. A cancelling interacting co-transformation is not supplied by the free differential; its interface is recorded, not constructed. The scalar-sourced slot is included once and has zero metric variation. No quantum retarded product of the cubic source, state, loop, interacting differential, sharp-width limit, old-class repair, or Einstein-jet repair is introduced.

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

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

and proof boundary{ Constructed: the companion support statement, an explicit off-causal-plateau diagnostic, the shell rewriting of the Einstein row, and cancellation on the named local configuration subclass.

Proved obstructed: unqualified invariance of ${{\Phi }^{(2)}}_{\mathrm{D}}$ on the complete gauge-fixed free shell. A compact plateau witness satisfies the copied causal plateau condition and has a strictly nonzero auxiliary pairing.

Unresolved as a separate future problem: an additional co-transformation or observable completion cancelling that pairing. Its absence is not promoted to a general no-go theorem.

No quantum retarded product, interacting differential, state, loop, antibracket, sharp limit, or coupled-jet repair is claimed. The inherited inverse coefficient and insertion sign are unchanged.}

The three endings are Constructed, Proved obstructed, and Unresolved. They apply to the named problem, not to every possible completion. A missing map is not a no-go. A nonzero variation of a specified classical lift is not removed by subsequently reducing the parameter space.

The closed chain is

The remaining question is whether the auxiliary row of that coefficient vanishes on the gauge-fixed free shell under the copied causal plateau condition.

The scalar-linear coefficient is

where

It belongs to ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$ and has class $Q_H{{\Phi }^{(2)}}_{\mathrm{D}}\in {\mathcal{E}}_H$. The opposite-sign assembly has the doubled two-jet defect already detected in XII-E. That sign question is not reopened XIIE.

For $\xi \in {\mathfrak{g}}_U$ and $P_m\varphi =0$, define

With ${\Delta }_{\xi }F=F[\gamma +K\xi ,\varphi ]-F[\gamma ,\varphi ]$,

The second term also equals ${\mathcal{A}}_1[f,g](e_{\xi };\varphi )$. On ${\mathcal{C}}_{\mathrm{sol}}$, $A\gamma =-2D^{\ast }b$, so its auxiliary form is

These are classical identities of the specified lifts, not a quantum Ward identity or an action on the quotient algebra XIIE.

Let $K_f=\mathrm{supp}f$. If $P_m\varphi =0$ and

then ${\mathcal{A}}_1[f,g](K{\rho }_{\xi };\varphi )=0$. The companion is retarded from the compact parameter support above the ramp. Its switching display and the associated sampling supports are those in Section sec:companion. No new ramp term is attached to it XIID,XIIE.

The insertion is ${\mathcal{C}}_{3,\mathrm{D}}={\mathcal{A}}_1[u_{\mathrm{D}};\varphi ]$, with both legs retained. Its compact collar root and compact polarized kernel are constructed, $\mathrm{WF}(t_{\mathrm{D}})\subset Z_3$, and $Q_H{\mathcal{C}}_{3,\mathrm{D}}\in {\mathcal{E}}_H$. The sign $L{u}_{\mathrm{D}}={\mathcal{J}}_{\mathrm{D}}$ differs from the action-coefficient sign $L{r}_{\mathrm{D}}=-{\mathcal{J}}_{\mathrm{D}}$ XIID.

XII-C fixes

Its pure-trace layer contraction is $-6({\partial }_{\mu }{\sigma }_{\epsilon })\psi {\partial }^{\mu }\psi$. The named invariant jet ${{\mathcal{J}}^{\mathrm{D}}}_{g,{\sigma }_{\epsilon }}({\varphi }_m)$ remains empty by a late-switch compatibility defect, not by failure of a Green operator XIIC.

XII-A supplies

with the complete free shell

The equation and action operators have this same kernel. The original detected XI-G classes remain excluded XIIA,XIG.

XI-B fixes

The plateau residual intersection is $\{0\}$, and propagated support is not layer support. Its ghost rule $s_0\gamma =\mathcal{R}c$ is not a nonlinear metric co-transformation XIB.

X-A's scalar-completion covariance has already been consumed by XII-E. We do not repeat it or import any quantum or sharp-limit claim. The new calculations use smooth wave uniqueness and finite propagation to realize compact local witnesses BGP.

F0 is Recorded. In particular, the auxiliary row is not assumed to vanish.

Use

The inherited ramp ${\sigma }_{\epsilon }$ is smooth, zero for $t\le 0$, and one for $t\ge \epsilon$, with $0<\epsilon <T$. The switch and parameter space are

Fields are ambient smooth configurations; no decay is required. All tensor pairings use the flat Lorentz identifications. Parentheses include $\frac{1}{2}$.

For a symmetric covariant tensor $r$,

Define

The inherited first completion is

In particular, $\partial g$ remains inside the scalar divergence.

The root notation is fixed by

These are XII-D's roots, not new choices. The raw advanced root can extend below $t=0$; multiplication by the compact parameter in $U$ makes all pairings used here compact without a new boundary prescription.

For a contravariant density $v$,

The Lie derivative of $\beta$ is the ordinary covariant tensor Lie derivative.

For fixed $\xi ,f,g$ set

On ${\mathcal{C}}_{\mathrm{sol}}$,

Proof. $A\gamma =-\square \mathcal{P}\gamma -2D^{\ast }D\gamma =-2D^{\ast }b$. For the compact tensor $q={\mathcal{L}}_{\xi }{\beta }_{f,g}[\varphi ]$, $\int q\cdot D^{\ast }b=\int (Dq{)}_{\nu }{b}^{\nu }$ by the flat adjoint identity.

Thus the copied variation is

Neither $K{\rho }_{\xi }$ nor ${\mathcal{L}}_{\xi }\gamma$ is replaced by a ghost variation. No new Noether identity is derived.

Let

Since every term of $s_{\xi }$ contains $\xi$ or a derivative of it,

Finite propagation gives

The minimum time of $S_{\xi }$ is strictly greater than $\epsilon$. Consequently ${\rho }_{\xi }$ and all its derivatives vanish before that time.

For the handed companion,

Proof. The tensor $K{\rho }_{\xi }$ is supported above the ramp, so ${\sigma }_{\epsilon }K{\rho }_{\xi }=K{\rho }_{\xi }$. Using $DK=\square$ gives

where the last equality follows from zero past data.

This removes a ramp factor from this companion calculation. It does not assert that the propagated field is supported in the parameter region or in the ramp layer.

For free $\varphi$, put $q_{\xi }={{\rho }_{\xi }}^{\mu }{\partial }_{\mu }\varphi$ and $v_f={E_m}^{\mathrm{A}}f$. The display from XII-E is

where

It follows from the free-scalar identity $V_{1,1}[K\rho ]\varphi =-P_m(\rho \cdot \partial \varphi )$, the zero-data relation ${E_m}^{\mathrm{R}}{P}_m(g{q}_{\xi })=g{q}_{\xi }$, and Lemma lem:no-ramp. We use it as the handed switching formula, not as a new cubic response.

After transposition onto $v_f$, the terms in (eq:companion-display) sample ${\rho }_{\xi }$ and its derivatives only in

This set is above the ramp. Under (CP), ${\mathfrak{C}}_{\xi }=0$.

Proof. The local term is supported in $\mathrm{supp}((1-g)f)\cap \mathrm{supp}{\rho }_{\xi }$. The advanced test $v_f$ is supported in $J^{-}(K_f)$. Every coefficient of the second term contains a derivative of $g$, whose support lies in $\mathrm{supp}\mathrm{d}g$; derivatives of ${\rho }_{\xi }$ have support contained in $\mathrm{supp}{\rho }_{\xi }$. Apply (eq:rho-support). Cancellation under (CP) is Input statement input:CP.

The set (eq:sampling-support) is not a claim that the full functional dependence on $\gamma$ is localized there: ${\rho }_{\xi }$ was itself propagated from $S_{\xi }$. Nor is it a support bound for the scalar response before smearing.

The condition (CP) uses $\mathrm{supp}(K{\rho }_{\xi })$, not $\mathrm{supp}{\rho }_{\xi }$. Thus its cancellation need not arise by deleting each term of (eq:companion-display) separately. For example, a locally constant tail of ${\rho }_{\xi }$ can make the local term and a propagated switching term cancel.

A useful configuration-independent sufficient condition is

It implies (CP) for every $\gamma$ for this fixed parameter and observation. We use (UCP) in the decisive witness.

There are admissible compact data, a free scalar, and a free metric for which

The example violates (CP), not the cancellation theorem under (CP).

Proof. Work first in a dependence cylinder where the spatial cutoffs are one. Let $m=0$, $\varphi =t$, and $\gamma =t\eta$. Choose $a\ge 0$, $a\not\equiv 0$, supported in a time interval $(t_a,t_b)$ above the ramp. Let

A compact spatial cutoff will be inserted below.

Write $r={\mathcal{L}}_v(t\eta )$. Its nonzero local components are

Direct differentiation gives

Hence the linear-in-$\lambda$ companion parameter is

with zero initial data. Since $({{\partial }_t}^2+k^2)(ta)=t(a^{\prime \prime }+k^{2}a)+2a^{\prime }$,

Choose a temporal switch $g_0$ equal to one on the support of $a$, falling smoothly and nontrivially from one to zero in $(t_c,t_d)$ with $t_b<t_c$, and zero near an observation time $t_o>t_d$. Its earlier rise occurs before $\mathrm{supp}a$. Choose $k$ sufficiently small that

whenever $t\in [t_c,t_o]$ and $s\in \mathrm{supp}a$. Above $\mathrm{supp}a$,

For ${\rho }^{(1)}$, the free scalar $\varphi =t$ gives

The coordinate leg is ${\rho }^{(1)}$ by Lemma lem:no-ramp. Because $g_0=1$ on the support of $R^{\prime \prime }+k^{2}R$, the linear companion integrand is

For $t$ just after $t_d$, every nonzero source term in this integral has negative sign: $g_0^{\prime }\le 0$, $R^{\prime }>0$, and the sine kernel is positive. Thus $d(t)<0$.

Choose $f\ge 0$, nonzero, supported there and near $x^1=0$, where $\cos (k{x}^1)>0$. Its linear companion value $J$ is strictly negative. The exact finite parameter has

Linearity of the propagation and of ${\mathcal{A}}_1$ therefore gives

for a finite constant $J_2$ independent of $\lambda$. It is nonzero for all sufficiently small positive $\lambda$.

Now replace $v$ by a compact spatial realization equal to the printed expression throughout the observation dependence region. Take $g=g_0(t){\zeta }_g(\mathbf{x})$, with ${\zeta }_g=1$ on the same larger cylinder and compactly supported beyond it. Choose $U$ to contain $\mathrm{supp}{\xi }_{\lambda }$, with closure in the early plateau where $g_0=1$. The cutoffs are chosen before the calculation and large enough that their transition regions cannot influence $\mathrm{supp}f$. Finite propagation preserves (eq:R-diagnostic)–(eq:d-diagnostic) on the observation region BGP.

At the observation $g=0$, and it lies in the causal future of the parameter support. During the falling switch, $(K{\rho }_{{\xi }_{\lambda }}{)}_{00}$ has leading term $2\lambda R^{\prime }(t)\cos (k{x}^1)$, nonzero in the relevant causal past. Thus (CP) fails. The fields $\gamma =t\eta$ and $\varphi =t$ are free; one may set $b=D\gamma$ and the ghosts to zero.

This proves that a universal companion cancellation cannot simply discard its support hypothesis. It does not prove that (CP) is necessary for every individual zero value: accidental cancellations and special fields are possible. F2 ends Constructed as a support statement and diagnostic, not Proved obstructed for the on-(CP) identity.

Suppose $\Phi \in {\mathcal{C}}_{\mathrm{sol}}$, (CP) holds, and

Then ${\Delta }_{\xi }{{\Phi }^{(2)}}_{\mathrm{D}}=0$. For a free-shell configuration, (eq:local-closed) is equivalent to $A\gamma =0$ on that neighbourhood. In particular, constant $b$ gives zero auxiliary pairing.

Proof. Under (CP), the companion vanishes. By compact integration by parts,

The compact tensor ${\mathcal{L}}_{\xi }\beta$ is supported in $S_{\xi }$. The result follows from (eq:local-closed) and Lemma lem:shell. A constant vector field has $D^{\ast }b=0$.

This is a restriction of configurations, not of ${\mathfrak{g}}_U$. It is smaller than the complete gauge-fixed shell. It is not a theorem about representatives of XI-G's old cubic class on its global Einstein subshell.

Fix times

Choose a nonzero nonnegative bump $a\in {C_c}^{\infty }((t_a,t_b))$. Choose a nonzero nonnegative observation

where $B_r$ is a spatial ball centered at the origin.

Choose a spatial cutoff $\zeta$ equal to one on $B_R$, where

and supported in a larger ball. Set

Choose an open relatively compact $U$ containing $\mathrm{supp}\xi$ with $\overline{U}\subset \{\epsilon <t<T\}$. Choose $g\in {C_c}^{\infty }({M_T}^{\circ };[0,1])$ equal to one on a neighbourhood of both $\overline{U}$ and

These sets are compact and separated from the temporal endpoints, so such a switch exists. This freezes an admissible tuple $(f,g,U,\xi )$ before evaluation. It satisfies (UCP), hence (CP).

Put

The configuration (eq:main-fields) belongs to ${\mathcal{C}}_{\mathrm{sol}}$. Its auxiliary field and its Einstein tensor coefficient are

In particular $D^{\ast }b\ne 0$.

Proof. $\square \psi =2-2=0$, hence $\square \gamma =0$ and $\square \varphi =0$. Trace reversal gives $\mathcal{P}(\psi \eta )=-\psi \eta$, so $b^{\nu }=-{\partial }^{\nu }\psi$, which is (eq:main-Ab)'s first formula. Its divergence is zero, and

with the other components zero. Lemma lem:shell gives the second formula. The ghost equations hold because the ghosts vanish.

The affine $b$ is permitted: no decay or compact-support requirement was imposed on free configurations. All test pairings remain compact. The example is deliberately not a constant auxiliary field.

Define the already handed compact-source response

This evaluates XII-E's Einstein row. It is not a new application of a propagator to ${\mathcal{J}}_{\mathrm{D}}$.

In the observation dependence region the spatial cutoff is one. By (eq:main-Ab),

Indeed, for a constant diagonal contravariant density and a temporal parameter, its $00$ component acquires $-a^{\prime }{v}^{00}$ and its spatial components acquire $+a^{\prime }{v}^{ij}$.

Set

On the observation region,

and

Proof. The trace of (eq:main-w) is $8a^{\prime }$. Consequently $\mathcal{P}{w}_{\xi }$ has only its $22$ and $33$ components nonzero, both equal to $4a^{\prime }$. The spatially homogeneous zero-data inverse satisfies ${E_0}^{\mathrm{R}}{a}^{\prime }=B$, giving (eq:main-e).

This local calculation equals the compact-source calculation: all sources capable of influencing the observation have spatial coordinates in $B_R$, where $\zeta =1$. The transitions of $\zeta$ are outside that causal past. Nested retarded operations do not enlarge the overall causal dependence region. Finite propagation therefore justifies the homogeneous formulas BGP.

For (eq:main-e), the relevant density coefficients are

The entire response is supported after $t_a>\epsilon$. Thus the coordinate ramp is one and

On the causal region where $e_{\xi }$ contributes, $g=1$ by (eq:witness-diamond). Since ${\partial }_t\varphi =2t$ and ${\partial }_1\varphi =2x^1$,

Hence

Subtracting proves (eq:main-integrand).

For the compact admissible tuple (eq:main-parameter)–(eq:main-fields), (CP) holds and

Equivalently,

Thus unqualified invariance of ${{\Phi }^{(2)}}_{\mathrm{D}}$ on the complete gauge-fixed free shell is false.

Proof. The switch satisfies (UCP), so the companion is zero. The shell membership is Lemma lem:main-shell. By the copied root identity, ${\mathfrak{E}}_{\xi }={\mathcal{A}}_1[f,g](e_{\xi };\varphi )$. Lemma lem:main-evaluation computes this value. For every observation time, $t>t_b$ and $a$ is a nonzero nonnegative bump, so the inner integral is strictly positive. The nonzero nonnegative $f$ makes the displayed value strictly negative. Lemma lem:shell then gives (eq:pairing-value).

The theorem is an existential counterexample in the frozen admissible class, already at $m=0$. It does not assert a nonzero value for every mass, observation, switch, or parameter. One counterexample suffices to exclude the unrestricted invariance theorem.

The parameter space has not been shrunk. The field $D^{\ast }b$ is nonzero exactly where a nonconstant auxiliary field can be detected. The constant-$b$ zero check in Proposition prop:subclass remains consistent with the integration by parts.

The witness proves a classical nonzero value. We now identify the carrier statement it supports, without defining a gauge action on the quotient.

For fixed $\xi ,f,g$, ${\mathfrak{B}}_{\xi }[b,\varphi ]$ is an even, ghost-number-zero compact microcausal polynomial of degree $(1,1)$ in $(b,\varphi )$. For the tuple of Theorem thm:obstruction,

Also the named lift ${\Delta }_{\xi }{{\Phi }^{(2)}}_{\mathrm{D}}$ belongs to ${\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$, and

for that tuple. These conclusions persist at classical-symbol order under every admissible smooth change of ordering.

Proof. XII-D's compact root kernel $B_{f,g}(y,x)$ satisfies $\mathrm{WF}(B_{f,g})\subset Z_2$. On $U$, its collar factor is one, so ${\mathcal{L}}_{\xi }{\beta }_{f,g}$ is obtained from that kernel by differential operators and multiplication by fixed smooth compact parameter coefficients. Applying $D$ does not enlarge its wavefront set. Its $y$ support is in $S_{\xi }$ and its scalar projection is in the compact set

The resulting bilinear kernel is compact with wavefront set in $Z_2$, which excludes the two all-causal cones. Thus it belongs to XII-A's domain.

The initial configuration in Theorem thm:obstruction is on ${\mathcal{C}}_{\mathrm{sol}}$ and gives a nonzero value. Therefore ${\mathfrak{B}}_{\xi }\notin {\mathcal{N}}_{\mathrm{sol}}$.

The functional ${{\Phi }^{(2)}}_{\mathrm{D}}$ is already admitted. Substitution of the smooth configuration $K\xi$ into some of its metric slots, followed by subtraction, preserves compactness and microcausality. Hence its named finite-difference lift is admitted. Its nonzero shell evaluation proves it also lies outside ${\mathcal{N}}_{\mathrm{sol}}$.

For the present switch, (UCP) holds independently of $\gamma$. Thus the companion vanishes for every free scalar and metric, and the two lifts ${\Delta }_{\xi }{{\Phi }^{(2)}}_{\mathrm{D}}$ and ${\mathfrak{B}}_{\xi }$ have the same restriction to ${\mathcal{C}}_{\mathrm{sol}}$.

Finally, the inherited ordering maps are the identity modulo $\hslash$. A nonzero classical-symbol class cannot be annihilated by such transport XIIA.

This assertion does not require a new admission theorem for the cubic source. The only new kernel observation concerns the already displayed auxiliary remainder.

For a merely configuration-dependent instance of (CP), cancellation holds on that configuration. It must not automatically be asserted as an identity on the entire shell quotient. The stronger (UCP) used above supplies the uniform hypothesis needed for that quotient identity.

The compact shift $K\xi$ generally does not preserve ${\mathcal{C}}_{\mathrm{sol}}$. Consequently ${\Delta }_{\xi }$ here is the difference of two prescribed off-shell evaluations, tested on an initial shell configuration. We do not claim it is an automorphism or a derivation of ${\mathcal{E}}_H$. The nonzero result excludes invariance of the named classical lift on all such initial free configurations.

Theorem thm:obstruction supplies the nonzero premise for this section. It obstructs ${{\Phi }^{(2)}}_{\mathrm{D}}$ alone. It does not obstruct every possible further observable completion.

A proposed rule on the existing multiplet would have to specify

and its dependence on the parameter, fields, coupling order, and Cauchy data. If it is intended to act on the carrier, preservation of the equation ideal and compatibility with the inherited product must also be supplied. Merely naming its auxiliary component does not meet these requirements.

At the level of the named lift, a cancelling interface under a uniform causal plateau hypothesis is

For a finite rule, the corresponding finite coefficient law must replace this infinitesimal display. If the proposed correction acts at a lower coefficient in the full relational expansion, that coefficient and its coupling order must be printed separately. It cannot be added to the already assembled second metric slot a second time.

There is an elementary slot restriction: the off-shell formula ${{\Phi }^{(2)}}_{\mathrm{D}}[\gamma ,\varphi ]$ has no independent $b,c,\bar{c}$ argument. Therefore

Changing only auxiliary or ghost variables cannot cancel (eq:main-value) in this formula. A more general proposal would need an induced action on $\gamma$ or $\varphi$, or a separately defined observable term involving the additional slots.

XII-A's free rule

does not supply (eq:cancelling-interface). It has ghost number one and is not the even finite parameter transformation used in the calculation. Substituting a compact plateau parameter for a free ghost also changes the equation domain; the nonzero compact residual intersection is absent by XI-B.

No cancelling map is selected here. In particular, $b$ is not replaced by an antifield, no contact term is chosen after the witness is known, and no symbol ${\hat{s}}_{\mathrm{int}}$ is introduced. The existence of a suitable additional completion is Unresolved. This separate status does not weaken the proved counterexample to invariance of ${{\Phi }^{(2)}}_{\mathrm{D}}$ alone.

XII-E's admitted scalar-sourced coefficient is

The full displayed gauge-fixed second-order coefficient is

The metric variation of (eq:full-coefficient), at fixed $\varphi$, equals ${\Delta }_{\xi }{{\Phi }^{(2)}}_{\mathrm{D}}$. In particular the witness (eq:main-value) also detects that variation of the full displayed coefficient.

Proof. XII-E's scalar-sourced-slot corollary states its admission and its absence of a metric argument XIIE. Its difference under $\gamma \mapsto \gamma +K\xi$ is therefore zero.

The tree is not recomputed. The slot enters once and is not an additional auxiliary cancellation row.

A future identity omitting $r_g$ is a metric-sector identity, not the full displayed second-order coefficient identity. A rule that varies $r_g$ under a metric-only shift has introduced a metric argument that this slot does not possess. A transformation of the scalar would be a different, explicitly specified operation.

For the inherited coefficient and parameter space:

1. The companion has the switching display (eq:companion-display), sampling support (eq:sampling-support), and no ramp-layer contribution. It vanishes under the copied condition (CP).

2. On the complete gauge-fixed free shell, the remaining Einstein row is the compact auxiliary pairing ${\mathfrak{B}}_{\xi }$. It vanishes when $D^{\ast }b=0$ near the parameter support.

3. It does not vanish universally on that shell under (CP). The admissible witness of Theorem thm:obstruction has the strictly negative value (eq:main-value). Its carrier class is nonzero.

4. Adding the scalar-sourced slot once leaves this metric variation unchanged.

Thus unqualified invariance of the named coefficient is Proved obstructed. Existence of a further cancelling completion is not decided.

Proof. Proposition prop:companion-support, Lemma lem:shell, Proposition prop:subclass, Theorem thm:obstruction, Proposition prop:aux-class, and Section sec:scalar-slot.

The worst ending among F1–F3 is Proved obstructed. F2 does not turn an off-condition diagnostic into an on-condition obstruction. F4 does not turn absence of a map into a general no-go. The actual obstruction is the explicit nonzero value of the named coefficient on a permitted initial free configuration.

• $Q_H{{\Phi }^{(2)}}_{\mathrm{D}}$ is not a quantum retarded product of ${\mathcal{J}}_{\mathrm{D}}$. No Hadamard line is contracted against ${\mathcal{J}}_{\mathrm{D}}$.

• The original detected XI-G classes with $\pm (k_1,k_2,0)$ and $k_1+k_2=2\mathrm{d}t$ remain excluded. ${{\Phi }^{(2)}}_{\mathrm{D}}$ is not a representative of those classes.

• ${{\mathcal{J}}^{\mathrm{D}}}_{g,{\sigma }_{\epsilon }}({\varphi }_m)=\varnothing$ stands. The slot $r_{\mathrm{D}}+r_g$ solves a gauge-fixed forcing equation, not the invariant coupled Einstein equation.

• The inverse coefficient is $-1$, not $-4$. No X-track mixed map, reciprocity test, or triangle residue is imported.

• No state, loop, antibracket, master identity, interacting differential, or $\epsilon \to 0$ limit is constructed.

• The local pairing hypothesis $A\gamma =0$ near $S_{\xi }$ does not decide XI-G's global Einstein-subshell representative question.

• No cone, token, conical readout, or Enter record is used.

A successor receives:

1. the two remainder functionals and their shell identification;

2. the companion's switching display, its sampling support, and the distinction between (CP) and a uniform carrier hypothesis;

3. the local cancellation subclass $D^{\ast }b=0$ near $S_{\xi }$;

4. the explicit compact plateau witness, with auxiliary pairing value (eq:pairing-value);

5. the nonzero carrier class and the qualification that the finite parameter shift is not a quotient automorphism;

6. the prospective cancelling interface, without an asserted solution, and the scalar-sourced slot recorded once.

It receives no Ward identity, quantum retarded product, interacting BRST class, or completed auxiliary co-transformation.