Landmark XIII-D One bounded classical completion of the auxiliary pairing, or its obstruction on the unchanged sheet A parameter-local/one-retarded class, shell descent, and a classical cocycle obstruction

We freeze the classical completion route for the displayed second-order coefficient, without changing the action, the external switches, or the inherited carrier. The tested correction class has parameter-local unpropagated components and at most one new retarded metric application. A successful correction must also supply an even representation of the compact vector-field algebra preserving the complete free-shell equation ideal.

These requirements have a decisive consequence. A causal, zero-past field variation tangent to the homogeneous free shell vanishes there by Cauchy uniqueness. Every shell-descending action in the frozen class therefore induces the trivial representation on the existing classical shell quotient.

We specialize the copied XII-F witness family so that its detecting parameter is an explicit commutator of two allowed compact parameters. The auxiliary pairing is nonzero on that commutator. Its necessary cocycle condition consequently fails for every representation in the frozen class, excluding both the slot equation and an integrated compensating functional in that class.

An admitted parameter-dependent bilinear cochain cancels the copied value algebraically. Its kernel is displayed, but it is not the variation of a compensating functional under the declared action. This distinction prevents algebraic cancellation from being released as an integrated completion.

The ending is Proved obstructed for the explicitly frozen class, not for arbitrary nonlocal actions outside it. The quantum two-line extension remains Unresolved. No quantum gauge-consistency defect or interacting differential is opened.

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

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

and scope{ Route selected: Route A, with Route D as the obstruction alternative inside one class. Route B is the evaluation-only result already released in XIII-A. Route C is not opened.

Class frozen: ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}(f,g,W)$, the parameter-local/ one-retarded, shell-descending class of Definition def:class. Parameter locality of its unpropagated components is an explicit ansatz, not an inference about every possible correction.

Constructed: the class definition, its forced trivial shell representation, the necessary cocycle calculation, and one admitted cancellation cochain.

Proved obstructed: an integrated cancellation in this class for the detecting fixed data of Section sec:witness. Every member satisfying the representation and descent interfaces fails the cancellation target.

Outside the class: arbitrary nonlocal actions on Cauchy data and other completions remain undecided. The result is not a universal completion no-go.

${\mathcal{C}}_W$ remains an evaluation set. No quotient by ${\mathcal{N}}_W$ is taken. XIII-C's quantum extension and the quantum gauge-consistency gate remain closed.}

The three endings have their inherited meanings:

1. Constructed: the named correction exists, is admitted, and satisfies every declared interface.

2. Proved obstructed: an explicit incompatibility excludes every successful map in a previously frozen class.

3. Unresolved: a required map, representation, cocycle, admission, or support theorem has not been established.

A missing map is not a no-go. A sufficient wavefront criterion that fails is not, by itself, a nonexistence theorem.

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

with

The coefficientwise kernel ${\mathcal{N}}_W$ is not a two-sided ideal: compact tests give

and an admissible rank-one ordering change satisfies

Forcing $\hslash 1=0$ does not force $1=0$ over $\mathbb{C}[[\hslash ]]$. It does rule out a torsion-free formal quantum quotient with the inherited brackets. No reduced algebra was handed off XIIIA.

Neither the commutator nor the ordering counterexample is recomputed here.

The vertex

is admitted. The tree datum

is constructed, proper, and microcausal. Ordered values of $R_1$ are $0$ or $\frac{i}{\hslash }[\mathcal{O},{\mathcal{V}}_g{]}_{{\star }_H}$. The literal normalization $R_1={\mathcal{V}}_g{\star }_H\mathcal{O}$ is proved obstructed. The joint two-line extension was left unresolved XIIIB.

This tree datum is not the correction studied below.

The eleven-branch inventory, the widened class ${\mathfrak{C}}_{\mathrm{tot}}\subset {\mathfrak{C}}_{\mathrm{part}}$, and the elementary parallel-line extensions are constructed. The parallel kinetic primitive has scaling degree $8$ and codimension $4$.

Collision exclusion and the $Z_3$ bounds do not establish the joint off-collision product: distinct null-related vertices can carry a nonzero wavefront cancellation. The coefficient linear in $\hslash$ remains Unresolved. That is a failed sufficient product criterion, not a nonexistence theorem. The free differential was not applied to ${R_1}^{(1)}$. No gauge-consistency defect was opened XIIIC.

No quantum extension premise is used in the classical proof.

On ${\mathcal{C}}_{\mathrm{sol}}$, under the copied causal-plateau condition,

The companion vanishes under (CP). Local vanishing holds if $D^{\ast }b=0$ near $\mathrm{supp}\xi$.

The compact free-shell witness satisfying (CP) has

Its carrier class is nonzero. A cancelling co-transformation was not supplied. The scalar-sourced slot has zero metric variation and is included once XIIF.

For the kernel calculation only, we also use XII-F's admission of the auxiliary row: for fixed $\xi ,f,g$ its bilinear kernel in $(b,\varphi )$ is compact and has wavefront set contained in $Z_2$. This is the kernel used in its nonzero-class proof, not the cubic kernel of ${\mathcal{C}}_{3,\mathrm{D}}$.

The displayed lift is

with

The opposite insertion sign is already detected and is not reopened XIIE.

The action is XII-C's cubic-only density weighting,

Its fixed-input jet ${{\mathcal{J}}_{g,\sigma }}^{\mathrm{D}}({\varphi }_m)=\varnothing$ is retained as a boundary, not an interaction entry XIIC. The inverse coefficient remains $-1$. The free rule $s_0\gamma =\mathcal{R}c$ is odd and has ghost number one; it is not the even rule $Y_{\xi }$ studied here XIIA,XIB.

Use

The backgrounds $g$ and ${\sigma }_{\epsilon }$ are fixed; neither is a spurion. The independent fields are $(\gamma ,\varphi ,c,\bar{c},b)$, with no antifields. The complete shell is

Fields are ambient smooth configurations. No spatial decay is required. Grassmann-valued identities are interpreted coefficientwise.

Fix $f,g,m,\epsilon ,T,W$ before selecting correction maps, with

and

Its bracket is the ordinary vector-field bracket

This is not the free ghost solution space.

Impose the uniform hypothesis

Then the copied remainder, including the scalar-sourced slot, is

The domain of the cancellation target is the complete shell. ${\mathcal{C}}_W$ is only a subsequent evaluation set.

The order is classical, ${\hslash }^0$, at the displayed ${\kappa }^2$ coefficient. One additional even correction is allowed. No insertion counted by $\lambda$ is added, and no coefficient of a quantum retarded product is constructed.

Route B is the conditional evaluation identity of XIII-A. Its naive quantum quotient is already obstructed. This paper does not construct a reduction.

Route C would change the problem to an interacting or quantum construction. It is not selected. In particular, the unresolved two-line coefficient is neither used nor differentiated.

The production envelope leaves the unpropagated parts of a rule unspecified. We make one precise choice: those parts are parameter-local, and the only new propagation is at most one application of ${G_{\mathrm{gr}}}^{\mathrm{R}}=-\mathcal{P}{E}_0^{\mathrm{R}}$.

This is an explicit restriction defining the tested class. It is made before the witness is used in a calculation. The theorem below does not exclude arbitrary nonlocal transformations of Cauchy data outside this realization.

For $\xi \in {\mathfrak{g}}_W$, write $S_{\xi }=\mathrm{supp}\xi$. A parameter-local expression has the form

where the sum is finite and the coefficients are smooth polynomial expressions in field jets and the fixed backgrounds. Its support is contained in $S_{\xi }$.

A structural proposal in ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}(f,g,W)$ consists of an even, ghost-number-zero, parameter-linear field rule

possibly accompanied by one fixed compensating functional $F$ of the metric and scalar slots, with the following restrictions.

1. The rule has parameter-local parts and, at most, one new retarded output

   $$r_{\xi }[\Phi ]={G_{\mathrm{gr}}}^{\mathrm{R}}{j}_{\xi }[\Phi ],\mathrm{supp}{j}_{\xi }[\Phi ]\subseteq S_{\xi }.$$

   The source is smooth, compact, polynomial in the fields and linear in $\xi$.

2. Every nonlocal component of $Y_{\xi }$ is a finite differential expression in this same $r_{\xi }$, multiplied by smooth polynomial field coefficients. Thus

   $${Y_{\xi }}^A={{\ell }_{\xi }}^A+{\mathsf{M}}^A[\Phi ]r_{\xi },A\in \{\gamma ,\varphi ,b\},$$

   with zero retarded output permitted.

3. The output in (eq:one-retarded) has zero past data and, on the slab, support in $J^{+}(S_{\xi })\cap M_T$. There is no second new retarded application.

4. A proposed cancellation effect must be an even, ghost-number-zero homogeneous quadratic polynomial with a compact polarized kernel bounded by $Z_2$. For a direct slot correction, that effect is

   $${K_{\xi }}^Y=⟨{{\mathcal{O}}^{\prime }}_{\gamma },{Y_{\xi }}^{\gamma }⟩+⟨{{\mathcal{O}}^{\prime }}_{\varphi },{Y_{\xi }}^{\varphi }⟩.$$

   For an integrated compensating-functional proposal, the effect is ${\delta }_{\xi }F$, where $\delta$ is the declared action below. The functional $F$ is fixed independently of $\xi$.

5. No new Einstein–Hilbert vertex, witness-dependent contact, additional field, or modification of $r_{\mathrm{D}}$ is permitted.

For a successful member, the directional rule must define an even derivation of the ordinary polynomial carrier,

preserve ${\mathcal{N}}_{\mathrm{sol}}$, and induce a representation with

Bars denote the inherited classical shell quotient, not a new quotient by ${\mathcal{N}}_W$.

The auxiliary component is allowed only to make the multiplet rule well defined and to test equation preservation. Since $\mathcal{O}$ and the proposed $F$ have only metric and scalar arguments, it does not independently change those functionals.

The class is nonempty as a structural class: the zero rule and zero functional satisfy its bounds and define a representation. They do not satisfy the nonzero cancellation target.

A parameter-indexed family $C_{\xi }$ is not automatically the variation of one functional $F$. Calling $C_{\xi }=-{\mathfrak{B}}_{\xi }$ a “counterterm” does not establish ${\bar{\delta }}_{\xi }\bar{F}=-{\bar{\mathfrak{B}}}_{\xi }$. The latter is the integrated interface tested here.

The only direct slot target is

An integrated compensating functional must satisfy the corresponding equation ${\bar{\delta }}_{\xi }\bar{F}=-{\bar{\mathfrak{B}}}_{\xi }$. No map satisfying only a smooth pointwise assignment is released.

For comparison, define

These constant directional derivatives commute: $[T_{\xi },T_{\zeta }]=0$. They are not the representation in (eq:representation).

They also fail equation-ideal preservation. For a compact test $q$,

but

can be a nonzero constant.

Indeed, if a nonzero compact $\xi$ had $\square K\xi =0$, Cauchy uniqueness would give $K\xi =0$. The Killing equation in Minkowski space then gives ${\partial }_{\mu }{\partial }_{\nu }{\xi }_{\rho }=0$, so $\xi$ is affine and cannot be nonzero with compact support. An appropriate compact component test $q$ therefore detects $\square K\xi$ BGP (Chapter 3).

The natural tensor pullback rule is not a remedy on the fixed free sheet. With the convention

it has the ordinary vector-field representation sign, but fails shell descent. At the free configuration $\gamma =\eta$, $b=0$, its metric variation is $-K\xi$, and the same equation test is nonzero. The fixed flat wave operator is not preserved by an arbitrary compact diffeomorphism.

These comparisons do not enter the cocycle calculation.

For every structural proposal in ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$ and every smooth input,

Every component vanishes on an open time band preceding $S_{\xi }$.

Proof. The parameter-local terms are supported in $S_{\xi }$. The retarded output is supported in its causal future. Differentiation and multiplication by smooth coefficients do not enlarge distributional support. Since $S_{\xi }$ is compact inside $\{t>\epsilon \}$, it has a minimum time strictly above $\epsilon$.

Suppose a rule in ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$ defines (eq:delta) and preserves ${\mathcal{N}}_{\mathrm{sol}}$. Then, for every $u\in {\mathcal{C}}_{\mathrm{sol}}$,

Consequently its induced action is

Proof. Apply ideal preservation to all compactly smeared linear equation functionals. The metric and scalar tests give

The auxiliary relation gives

These identities follow because the compact tests separate smooth functions and ${\delta }_{\xi }E$ vanishes on the shell whenever $E\in {\mathcal{N}}_{\mathrm{sol}}$.

By Lemma lem:causal, the metric and scalar variations vanish, with their first time derivatives, on a Cauchy surface preceding $S_{\xi }$. Smooth homogeneous wave uniqueness therefore makes them zero. The auxiliary identity then gives ${Y_{\xi }}^b[u]=0$. For Grassmann-valued fields the same argument applies to every coefficient BGP (Chapter 3).

Finally, evaluation of (eq:delta) on $u$ pairs each derivative with a zero variation. Thus ${\delta }_{\xi }G$ vanishes on ${\mathcal{C}}_{\mathrm{sol}}$ for every admitted $G$, proving (eq:trivial-rep).

This specifies the representation before any cocycle test: every successful shell-descending proposal in the frozen class has the trivial module action. The zero off-shell rule realizes this representation, so the conclusion is not based on an unconstructed action.

The theorem uses the simultaneous requirements of causal zero-past output and free-shell tangency. It does not state that every retarded response vanishes. A sourced response generally fails the homogeneous tangent equations.

Nor does it state that every action on free Cauchy data is trivial. An action with nonzero past data or an unpropagated nonlocal component lies outside the frozen class.

Write ${\bar{\mathfrak{B}}}_{\xi }$ for the class of the admitted auxiliary row in ${{\mathcal{F}}^{\mathrm{sol}}}_{\mathrm{mc},\mathrm{pol}}$. It is an even linear map from ${\mathfrak{g}}_W$ to that vector space. Parameters and the external functions are not varied.

For a representation satisfying (eq:representation), an equation

for one fixed functional $G$ requires

For a direct slot proposal, take $G=\mathcal{O}$. For a compensating-functional proposal, take $G=F$.

Proof. Apply ${\bar{\delta }}_{\xi }$ to the equation with parameter $\zeta$, subtract the reversed equation, and use (eq:representation). The result is

This is the degree-one Chevalley–Eilenberg condition with the specified representation convention CE.

For the forced action (eq:trivial-rep), the condition becomes

There is no substitution of compact translations into this equation.

If there are $\xi ,\zeta \in {\mathfrak{g}}_W$ with ${\bar{\mathfrak{B}}}_{[\xi ,\zeta ]}\ne 0$, then no member of ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}$ satisfies all the representation, descent, and cancellation interfaces.

More directly, any parameter $\theta$ with ${\bar{\mathfrak{B}}}_{\theta }\ne 0$ excludes the target (eq:target) for every shell-descending rule in the class.

Proof. The first assertion combines Theorem thm:triviality with Lemma lem:cocycle. For the second, the left side of (eq:target) is zero on the shell by (eq:Y-zero), whereas its required right side is nonzero.

Likewise, ${\bar{\delta }}_{\theta }\bar{F}=0$ for every fixed compensating functional $F$. Adding one such functional cannot change the conclusion.

The criterion excludes successful maps in the whole frozen class, not just an enumerated ansatz. A proposal whose directional operator or composed kernel is undefined already fails an interface; any proposal for which those interfaces are defined is subject to the same theorem.

The nonzero commutator required by Proposition prop:criterion is established in Section sec:witness using the copied witness, not a new cylinder integration.

A kernel calculation is useful because it isolates what the obstruction does not say.

Let the inherited auxiliary kernel be denoted $b_{\xi }$, with the normalization

Fiber components are understood. Define the new metric–scalar cochain

This is a functional of the allowed observable slots. It does not make $b=D\gamma$ an off-shell relation in the inherited carrier.

The family (eq:C-definition) is even, has ghost number zero, is linear in $\xi$, and has field degree $(1,1)$ in $(\gamma ,\varphi )$. Its unsymmetrized kernel is

where $y$ is the auxiliary slot replaced by the metric slot. After normalized species symmetrization it is compact and satisfies $\mathrm{WF}(c_{\xi })\subset Z_2$. Hence ${\mathcal{C}}_{\xi }\in {\mathcal{F}}_{\mathrm{mc},\mathrm{pol}}$.

On the complete free shell,

Proof. The source functional is bilinear and parameter-linear. Replacing its first smooth argument by $D\gamma$ gives the transpose formula (eq:C-kernel). A differential operator does not enlarge support or wavefront set. The finite species symmetrization preserves the sum-zero bound.

A nonzero tuple in either all-future or all-past closed causal cone cannot have zero covector sum. The bound $Z_2$ therefore implies the inherited microcausal condition Hormander,XIIA. The field species are even and have ghost number zero.

On ${\mathcal{C}}_{\mathrm{sol}}$, the equation $b=D\gamma$ holds. Substitution gives (eq:C-cancel).

This proof uses the actual auxiliary-row kernel. It does not borrow the wavefront estimate of ${\mathcal{C}}_{3,\mathrm{D}}$ for a different functional. No new retarded application is needed; zero such applications is within the structural bound “at most one.”

Under (eq:UCP), the copied remainder and (eq:C-cancel) give the algebraic identity

That identity is not released as an integrated completion. The family ${\mathcal{C}}_{\xi }$ depends on the parameter. No single functional whose variation is this family has been constructed.

In fact, for the representation forced by the frozen class,

for every $F$. Thus a nonzero ${\bar{\mathcal{C}}}_{\xi }$ cannot be such a variation. Its cocycle is

which the next section detects.

If arbitrary parameter-indexed additions were declared to be completions without an action or integrability requirement, (eq:C-definition) would provide a cancellation prescription. That is a different, weaker interface. The present class requires a slot action or the variation of one fixed compensating functional. Its obstruction is not non-admission of the bilinear cochain.

We now specialize the detecting family, after the class and its representation have been frozen.

Choose the admissible XII-F data with a temporal bump that is a square. More explicitly, let

and choose a compact spatial cutoff $\zeta \in {C_c}^{\infty }({\mathbb{R}}^3;[0,1])$ equal to one on the dependence region used by XII-F. Take its temporal bump and spatial realization to be

These are among XII-F's permitted choices.

Choose $W$ to contain the compact parameter support and choose $g$ to satisfy (eq:UCP), as in XIII-A's embedding of the detecting tuple into fixed restriction data. The switch, observation, region, and cutoffs are fixed for the entire test. No parameter subclass is removed afterward.

Define two allowed parameters

where $t_{\ast }$ is any fixed real number. Both lie in ${\mathfrak{g}}_W$.

The parameters (eq:bracket-parameters) satisfy

Proof. Both vector fields have only a temporal component. Their spatial cutoff is independent of time. Thus the temporal component of their bracket is

There are no spatial components.

Use the copied free-shell configuration

denoted $u_{\ast }$. Its shell membership, compact realization, causal-plateau hypothesis, and evaluated sign are those of XII-F.

Set

The copied witness gives $I>0$ and

No retarded cylinder calculation is repeated.

For these detecting fixed data,

Consequently every shell-descending representation in ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}(f,g,W)$ fails the necessary cocycle condition for the cancellation target. No member satisfies all declared interfaces.

Proof. The nonzero shell value (eq:B-theta) proves that ${\mathfrak{B}}_{\theta }\notin {\mathcal{N}}_{\mathrm{sol}}$. By Theorem thm:triviality, every eligible action induces $\bar{\delta }=0$. Therefore

At $u_{\ast }$ its representative value is $8I>0$. Apply Proposition prop:criterion.

For every shell-descending slot rule in the class,

For the admitted cochain of Section sec:kernel,

Nevertheless that cochain is not an integrated compensating functional in the frozen class.

Proof. The first assertion is (eq:Y-zero). The second is (eq:C-cancel). The lack of an integrated realization follows from Theorem thm:obstruction, or directly from ${\bar{\delta }}_{\theta }\bar{F}=0$.

The result is observation-qualified. For example, $f=0$ makes the pairing vanish and removes this detection argument. We do not assert obstruction for every frozen observation, every mass, or every conceivable correction class. For the detecting fixed data, however, the exclusion applies to every member of the class, not merely to one candidate.

The witness is a classical test of the correction interface. It is not a binary quantum anomaly.

The field $\varphi$ is allowed to move in the class definition. Its contribution to the full target must retain the scalar-sourced slot. Explicitly,

where the inherited derivative is

This is differentiation of the existing slot, not a new interaction entry. Its zero metric variation does not authorize deleting either term in (eq:scalar-variation).

For a shell-descending member of the frozen class all three terms vanish on the shell because ${Y_{\xi }}^{\varphi }[u]=0$. They were not omitted before applying that theorem.

The candidate cochain (eq:C-definition) has degree two, whereas $\mathcal{O}$ has degree three. It must not be identified with $⟨{{\mathcal{O}}^{\prime }}_{\gamma },r_{\xi }⟩$ merely because $r_{\xi }$ has a plausible retarded support. For instance, a field-linear $r_{\xi }$ makes that slot variation degree three. Homogeneous field scaling distinguishes it from the bilinear target.

The order of a prospective compensating effect here is ${\kappa }^2{\hslash }^0$. No term is silently added at the first coefficient. If a future proposal acts on the lower term $\kappa {\mathcal{A}}_1$, it must print the additional coupling power producing a ${\kappa }^2$ effect and supply its own transformation law.

The filled gravitational slot remains $r_{\mathrm{D}}=-u_{\mathrm{D}}$. It is neither modified nor counted a second time. The admitted parameter-dependent cochain is not inserted into that slot.

Finally, a finite transformation contains $\frac{1}{2}{\mathcal{L}}_{\xi }K\xi$ in its inherited two-jet law. A linear parameter substitution is not a construction of that finite law. The present obstruction already excludes a necessary infinitesimal integrated interface in the class; it therefore requires no assertion about an unconstructed finite action.

On the unchanged cubic-only sheet, for the detecting fixed data of Section sec:witness:

1. Every shell-descending action in ${\mathfrak{K}}_{\mathrm{loc},\mathrm{R}}(f,g,W)$ induces the trivial representation on ${{\mathcal{F}}^{\mathrm{sol}}}_{\mathrm{mc},\mathrm{pol}}$.

2. The copied auxiliary pairing is not a cocycle for that representation.

3. No slot action or integrated compensating functional in the frozen class satisfies the cancellation target on the complete gauge-fixed free shell.

4. The admitted cochain ${\mathcal{C}}_{\xi }=-{\mathfrak{B}}_{\xi }[D\gamma ,\varphi ]$ provides an algebraic parameter-dependent cancellation, not an integrated completion in the class.

The ending for the named integrated class is Proved obstructed. The result claims nothing about completions outside the parameter-local/one-retarded, shell-descending class.

Proof. Theorem thm:triviality, Proposition prop:C-admission, Lemma lem:bracket, and Theorem thm:obstruction.

This obstruction is caused by an explicit incompatibility: zero-past causal tangency forces zero field variation, whereas the target pairing has a nonzero class. It is not caused by the absence of a proposed formula, by a failed wavefront estimate, or by XIII-C's unconstructed quantum coefficient.

A successor receives:

1. the exact frozen class, including parameter locality of its unpropagated components;

2. the theorem forcing its induced shell action to vanish;

3. the explicit commutator parameters and the nonzero Chevalley–Eilenberg test;

4. the admitted cancellation cochain and its kernel formula, with the warning that it is not an integrated functional;

5. the observation-qualified obstruction and the unchanged scalar-slot and coupling bookkeeping.

A proposed successor can leave the class by allowing a nonlocal action on Cauchy data, nonzero past data, a different domain, or a different integration interface. It must state that change before claiming that the present obstruction has been avoided. No such construction is supplied here.

It receives no reduced algebra, completed quantum binary datum, interacting differential, or physical cohomology class. The class exclusion supplies no permission to differentiate the unconstructed ${R_1}^{(1)}$.

The next quantum paper still begins at XIII-C's four obligations:

1. a prescription across the noncoincident null-related configurations;

2. transverse estimates on the three- and four-vertex strata;

3. compatible partial-diagonal contacts;

4. the output wavefront estimate after pushforward.

Those obligations remain Unresolved. The present classical cocycle calculation is not the binary quantum gauge-consistency defect.

Compactly supported free plateau ghosts cannot replace the parameters in this proof. A homogeneous ghost with compact time support has zero Cauchy data in an earlier band and therefore vanishes by uniqueness. Pairing against that zero is not a nontrivial cancellation identity.

${\mathcal{C}}_W$ remains an evaluation set. The quantum two-line coefficient remains unresolved. Exactly one bounded integrated classical class has been excluded, with its support and representation assumptions visible.

scope. The tested class is a precise parameter-local/one-retarded realization of the production envelope. Parameter locality of unpropagated components is an additional explicit ansatz, frozen before the witness test. No universal exclusion of nonlocal Cauchy-data actions is claimed.

scope. The theorem concerns integrated cancellations admitting a shell-descending representation in this class. It does not deny existence of parameter-dependent bilinear cancellation cochains; one such cochain is constructed explicitly.

scope. The predecessor commutator, ordering counterexample, tree datum, quantum stop, assembly sign, and witness value are copied. The causal tangency theorem, representation identification, commutator realization of the detecting parameter, and cochain kernel calculation are proved here.

scope. All new actions and cocycles are even and classical. No application of $s_0$ to a quantum binary coefficient, interacting differential, master equation, state, antifield, Einstein–Hilbert interaction insertion, or sharp-width limit is supplied.

. No successful pdflatex execution, proof-assistant check, independently executed symbolic computation, independent referee review, or journal acceptance is asserted.

XIIIA

N. Cheung, Landmark XIII-A: Classical cancellation on a fixed subclass, and the obstruction to a coefficientwise restriction quotient. Supplied manuscript XIII-A.tex. Evaluation-only route, restriction-ideal obstruction, ordering counterexample, and conditional completion interface.

XIIIB

N. Cheung, Landmark XIII-B: One binary causal datum for the displayed coefficient, on the unchanged carrier. Supplied manuscript XIII-B.tex. Constructed tree datum, corrected ordered values, and unresolved two-line extension.

XIIIC

N. Cheung, Landmark XIII-C: The joint two-line extension of one selected binary pair, on the unchanged carrier. Supplied manuscript XIII-C.tex. Partial-diagonal class, primitive extensions, characteristic cancellation, and retained quantum stop.

XIIF

N. Cheung, {Landmark XII-F: The gauge-fixed remainder of ${{\Phi }^{(2)}}_{\mathrm{D}}$: companion support, the auxiliary pairing, and what is not a Ward identity}. Supplied manuscript XII-F.tex. Auxiliary-row kernel, compact witness family, copied evaluated value, and nonzero carrier class.

XIIE

N. Cheung, Landmark XII-E: Assembly of the second-order relational coefficient after admission of the density-weighted cubic insertion. Supplied manuscript XII-E.tex. Assembly sign, affine slot identity, and scalar-sourced slot.

XIIC

N. Cheung, Landmark XII-C: One declared cubic weighting: actual Euler source, changed Noether identity, and the coupled-jet question. Supplied manuscript XII-C.tex. Unchanged selected action and fixed-input jet boundary.

XIIA

N. Cheung, Landmark XII-A: One graded polynomial carrier for the finite-ramp multiplet. Supplied manuscript XII-A-v2.tex. Polynomial carrier, complete shell, null ideal, and frozen free differential.

XIB

N. Cheung, Landmark XI-B: The free de Donder multiplet for the ramp Einstein–scalar sheet. Predecessor normalization input, as stated in the supplied XII-A, XII-E, and XIII-A manuscripts. Inverse coefficient and residual-gauge distinction.

CE

C. Chevalley and S. Eilenberg, “Cohomology theory of Lie groups and Lie algebras,” Transactions of the American Mathematical Society 63 (1948), 85–124. Representation-dependent Lie algebra cocycles.

BGP

C. Bär, N. Ginoux, and F. Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society, 2007, arXiv:0806.1036. Chapter 3: smooth Cauchy uniqueness and causal Green operators.

Hormander

L. Hörmander, The Analysis of Linear Partial Differential Operators I, second edition, Springer, 1990. Differential-operator wavefront bounds used in the new bilinear kernel calculation.