Landmark XIV-G Operator and pullback composition of the regrouped retained germ Individual symbols, residual testing, and the exact G2 relation stop

We continue XIV-F's unresolved characteristic problem for the unchanged binary pair. The selected route works from XIV-E's regrouped operator expression and uses individual conormal relations, without constructing a common product phase or reusing the excluded clean-excess-two matching.

The characteristic test is placed in the common sixteen-dimensional space, on the intersection with the observable attachment diagonal. The copied positive conormal relation specifies a diagnostic cancellation, not a joint symbol relation. In particular, its zero covector cannot serve as the base point of an ordinary punctured principal-symbol bundle.

Two operator-level distinctions are proved. First, after a smooth residual metric test, the differentiated gravitational source is a differential operator of order at most two. Its local symbol is well typed, but does not supply the symbols of the composed response legs. Second, a compact balanced kernel can have nonzero attachment-diagonal wavefront directions even though every smooth residual test produces a smooth attachment kernel.

A separate factorwise audit uses two individually defined conormal factors of the retained numerical incidence. Their own relations and orders are named before their symbols are used. The ordinary two-factor product test fails at the copied characteristic covectors, even after those factors have been legitimately grouped. Thus individual symbols do not bypass the missing pullback hypothesis.

These results do not construct the complete internal symbol relations, a common symbol bundle, a remainder theorem, or a characteristic prescription for the grouped six-branch coefficient. The ending is Unresolved at G2, within the inherited F2/E1/D2/J2 stop. J3–J5 and the quantum Ward-defect question remain unopened.

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

Figure fig:scope-XIV-G records that separation. The drawing adds no hypothesis.

and first missing gate{ Selected route: operator and pullback calculus from the complete regrouped expression, with no common product phase.

Copied boundaries: the elementary phase is non-clean; the direct clean-excess-two matching is excluded; the tangent defect is not an excess fiber; the rank-fourteen image is not a Lagrangian relation in $T^{\ast }W$.

Proved here: the local differential-operator typing of the tested source; loss of attachment information under smooth residual testing; and failure of an automatic factorwise pullback argument at the retained characteristic covectors.

First missing gate: G2. The complete internal symbol relations and the common relation and bundle for the grouped coefficient have not been constructed.

Ending: Unresolved. No joint symbol, complete cancellation, characteristic prescription, ${R_1}^{(1)}$, or Ward defect is released.}

Only XIV-F, XIV-E, XIV-D, XIV-C, XIII-C, XIII-B, XII-E, and XII-C are analytic manuscript inputs. The new proofs below are conditional on their stated handoffs. They do not verify every predecessor premise anew.

Quoted passages are copied from the supplied manuscripts. Displayed formulas are recorded with local labels. No incidence ledger, elementary phase theorem, clean-matching theorem, or branch fiber identity is reconstructed.

“A successor receives:

1. the unchanged XIV-E package and grouped target;

2. the selected route and the precise correction to its excess-two wording;

3. the regular manifold $S_6$ and exact formula $q_R=\alpha \delta |{\mathbf{n}}_1-{\mathbf{n}}_2{|}^2$ on it;

4. the non-clean intrinsic conormal matching theorem;

5. the rank-two tangent-defect bundle and its quadratic form;

6. the rank-fourteen projection and one-dimensional frequency fibers;

7. the persistence of the tangent discrepancy under smooth frequency reparameterization;

8. the exact F2 stop, with no symbol or prescription released.

”

“The successor receives no failure-of-cancellation theorem and no permission to infer one from $Q\lambda$. It receives no ${R_1}^{(1)}$” XIVF.

In the common space $W$, the regular manifold is

Its parameterization is

The exact formula is

The selected conormal matching has

It is smooth and not clean. Its tangent-defect quotient has rank two and is identified by

The copied quadratic form is

The covector-sum projection satisfies

Its image is isotropic of dimension fourteen, not Lagrangian in $T^{\ast }W$.

“The two components of $\alpha v-\beta w$ measure a failure of clean tangency. They are not, merely by that calculation, an excess-two integration fiber.”

The two-dimensional tangent discrepancy persists under the specified smooth frequency reparameterizations and local vertical product reductions.

The ending is “Unresolved at F2, within the inherited E1/D2/J2 stop” XIVF.

These statements are boundaries for the present route. The matching ${\mathcal{F}}_{\mathrm{match}}$ and the map $J$ are not reused as a composition relation.

With

the complete retained expression is

where

and

“This identity retains all six branches. It involves no characteristic composition and no regularization.”

For $F={E_m}^{\mathrm{A}}f$, the complete scalar-response term is

“Both the derivative and mass terms are retained” XIVE.

For $G\in {C_c}^{\infty }(Z;{\mathrm{Sym}}^2{T}^{\ast }M)$,

The kernel $L_G$ is the tested sum of $K_4,\dots ,K_9$, is compact, and satisfies

“This is an intermediate attachment-kernel bound, not admission of the final quantum coefficient.”

“Nor does testing each fixed $G$ supply the uniform estimates needed to reconstruct a joint characteristic distribution in all four common variables” XIVE.

In

define

The copied codimensions are

The elementary phase of XIV-E is neither nondegenerate nor clean. Its auxiliary equations have rank six and its critical set has codimension eight.

On the copied affine transverse slice,

“Equation (eq:Equadratic) is a geometric identity, not a scaling degree of a distribution.”

The ending is “Unresolved at E1, within D2/J2” XIVE.

The frozen pair has

On the retained common-variable patch,

The kernels $K_6,K_8,K_9$ are supported on $a=z$.

“The complete mixed germ is therefore a six-branch sum. In particular, neither the coordinate gravitational leg $B_8$ nor the scalar gravitational leg $B_9$ may be omitted.”

The inherited mixed species factor is $i$. The scalar–scalar factor is $i/2$, and that channel vanishes only on the specified retained output-location patch.

For $p^2=0$ and $r=p\otimes k+k\otimes p$,

“The proposition computes only one branch's leading differential numerator. It supplies no sum identity or sum nonvanishing theorem for that complete expression.”

The first unresolved step is D2, within J2 XIVD.

The fixed retained points are

where

and the fixed switch is one near these points.

In the order $(b,a,y)=(x,u,y)$,

It is smooth of codimension four. With

the cone differentials have rank two and satisfy

The oriented line conormals have the copied positive relation.

“Every defined contact change in ${\mathfrak{C}}_{\mathrm{part}}$ for this graph restricts to zero on $V$.”

The first unresolved gate is J2 XIVC.

The neighborhoods $A,B,Y$ have pairwise disjoint closures, $\overline{Z}\subset A$, and $B\times A\times Y\subset V$. They specify restrictions of kernels, not removal of the retained characteristic configuration.

XIII-C supplies the eleven branches, 48 labelled matchings, the frozen class ${\mathfrak{C}}_{\mathrm{part}}$, and the primitive parallel-line extensions. In particular, the temporal kinetic primitive has degree eight in collision codimension four.

“The next extension attempt must supply:

1. a joint prescription across the noncoincident characteristic configurations, or a proved cancellation of those configurations in the complete selected pair;

2. uniform transverse scaling and wavefront estimates for the three-vertex and four-vertex collision strata and their intersections;

3. compatible extensions with the permitted partial contacts, rather than only isolated primitive extensions;

4. the full output wavefront estimate after proper pushforward.

” The complete coefficient remains Unresolved XIIIC.

The admitted tree datum is

The convention is

Its strictly ordered full-support values are zero or the scaled commutator $(i/\hslash )[\mathcal{O},{\mathcal{V}}_g{]}_{{\star }_H}$.

“Properness cannot repair an undefined distribution product, and an extension theorem cannot be substituted for properness” XIIIB.

The inherited formulas are

where

The roots satisfy

The completion and filled-slot kernels are admitted tree kernels XIIE.

For $q(\rho )=\eta +\rho \gamma$,

is the fixed unintegrated cubic representative, with

Its actual weighted source is

Here $C_2$ is the quadratic coefficient of

and

The source is density weighted before variation, at cubic order only XIIC.

No coefficient in (eq:source) is set to zero. The retained external points do not specify the locations of all internal source vertices.

XIII-E's common-frequency family is an attempt; its finite-regulator and admissible-domain conclusions are not characteristic convergence here XIIIE.

XIV-A and XIV-B are recorded classical boundaries, not analytic inputs. XIII-A, XIII-D, and XII-F are evaluation, class, and auxiliary-pairing boundaries, not graph data. The value $-8I$ is not a normalization.

Use

All external functions, widths, and propagators remain fixed. The carrier and line conventions are

Metric–scalar propagator blocks vanish.

The interaction is

The observable contains its scalar-sourced slot once:

The grouped retained target is

The cross-line difference stays grouped. No intermediate mixed $H$–${G_{\mathrm{act}}}^{\mathrm{A}}$ product is defined.

The endings, applied only to (eq:target), are:

1. Constructed: a common microlocal representation, a joint principal-symbol section, and a complete remainder estimate establish the required cancellation; or one characteristic prescription has all named domain, degree, normalization, ambiguity, and wavefront data proved.

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

3. Unresolved: the common symbol, remainder, or prescription theorem is missing.

A vanishing leading section is sufficient only when the remainder control actually supplies the required distribution. A formal order improvement does not define an otherwise undefined product.

The elementary phase failure, the excess-route failure, and $Q\lambda$ are not universal obstruction theorems for this target.

The selected route must identify:

1. a canonical or characteristic relation in a named ambient space;

2. amplitude orders and a common symbol bundle;

3. internal-vertex maps for $K_5,K_8,K_9$;

4. the attachment diagonal and its derivatives;

5. the complete tensor coefficients and grouped cross-line difference;

6. a remainder estimate controlling every unrepresented term.

Neither inherited wavefront inclusions nor the isotropic image of $J$ supplies this list.

The construction starts from (eq:regroup). Already defined conormal or Lagrangian factors may be described on their own relations. Their subsequent operator compositions and pullbacks must satisfy their own hypotheses.

There is no common sum of line phases. No integration over ${\mathcal{D}}_{\tan }$ is introduced. Melrose–Uhlmann composition is not reapplied to ${\mathcal{F}}_{\mathrm{match}}$.

Before assigning a joint symbol to (eq:target), the route requires:

1. a named common ambient space and characteristic test locus;

2. valid individual relations, orders, and bundles for each factor used in the proposed composition;

3. proved internal composition maps for $K_5,K_8,K_9$;

4. a justified pullback or other operator composition for the complete grouped coefficient;

5. a common symbol identification and a uniform remainder theorem.

The ordinary pullback theorem is used only when its normal-set condition has been checked. If that condition fails, it supplies no automatic reference distribution.

The standard pullback test asks that the wavefront set of the distribution being pulled back avoid the normal set of the map. For the diagonal embedding, this becomes the familiar exclusion of opposite product covectors Hormander (Theorems 8.2.4 and 8.2.10).

Removing a phase construction does not remove this condition.

The following remain unentered:

1. a resolved or blown-up paired representation, with a new ambient representation and its own clean-matching theorem;

2. a continuation of XIII-E's common-frequency family, with a new convergence theorem on tests meeting the stratum.

Neither is substituted after the stop below.

There is no cutoff of ${\mathcal{N}}^{\circ }$, restoring limit, characteristic subtraction, new contact class, or change of propagator.

The tangent defect is not renamed as an excess fiber. The fourteen-dimensional image of $J$ is not used as a Lagrangian relation in $T^{\ast }W$. No branch numerator is promoted to a joint section.

All common-kernel statements take place in

in the order $(a,b,z,y)$.

For the characteristic test in this paper, choose

It has codimension eight in $W$. The residual metric variable is not projected out.

The numerical stratum ${\mathcal{N}}^{\circ }$ has codimension four in the three-vertex space. Its full lift ${\overline{\mathcal{N}}}_{\mathrm{att}}$ also has codimension four, with $z$ free. Neither is substituted for (eq:chosenN) in a degree statement. No characteristic degree is assigned to any of them.

Let

At $w\in N$, define the already copied oriented conormals

They satisfy

Define the diagnostic selection set

Its entries are four separate cotangent selections: the three line covectors and a zero attachment covector. It is not a canonical relation for the complete coefficient.

The natural candidate for a characteristic conormal component, if such a component were constructed, would be

This merely names a possible target bundle. It does not assert that the grouped coefficient belongs to a conormal class on it.

Cotangent addition sends every tuple in (eq:diagnostic) to the zero covector of $T_w^{\ast }W$.

Accordingly, reducing the diagnostic tuple by its copied linear dependence does not produce a point of ${\Lambda }_{\mathrm{char}}$. The diagnostic alone cannot define an ordinary principal-symbol section of the grouped coefficient.

Proof. The sum is

By definition, the zero covector is removed from ${\Lambda }_{\mathrm{char}}$. An ordinary conormal or Lagrangian principal symbol is a section over its punctured cotangent relation, not a value assigned at this zero covector.

The calculation specifies a cancellation of input covectors. It specifies neither membership of an output distribution in a conormal class nor an amplitude on an output relation.

This does not exclude a generalized characteristic representation. It identifies the additional theorem such a representation would need. The isotropic image from XIV-F cannot fill the missing symbol relation.

The following first operator calculation concerns the full source, not the $B_6$ numerator.

For a fixed smooth residual metric test

define

This is a metric-density-valued expression.

The map ${\mathsf{Q}}_G$ is a matrix-valued differential operator of order at most two:

Its coefficients are smooth and supported in $\mathrm{supp}G$.

In the Fourier convention where ${\partial }^{\nu }$ has symbol $(i\xi {)}^{\nu }$, its degree-two differential symbol is

This is a fiber-valued polynomial on $T^{\ast }M$. It is allowed to vanish and is not asserted to be elliptic.

Its Schwartz kernel is supported on the source diagonal $s=a$ and is a finite sum

These statements retain every coefficient of the actual source, including its ramp derivatives.

Proof. The copied quadratic source is a finite sum of local monomials

with constant fiber contractions and the fixed external ramp coefficients. Its directional derivative at $G$ is the sum of the two variations of every monomial.

Collecting the derivatives of $h$ gives (eq:QG). Each coefficient contains a derivative of $G$ of order at most two, multiplied by fixed smooth external data. It is therefore smooth and supported in $\mathrm{supp}G$.

The symbol (eq:Qsymbol) is the algebraic highest-degree differential polynomial of that operator. Lower-degree coefficients remain in (eq:QG); none is omitted.

The kernel identity follows from the distributional formula for differentiating $h(s)$ through the diagonal delta. It has exactly the stated support. No propagator or characteristic multiplication is used.

Equation (eq:Qsymbol) is a symbol of the local operator ${\mathsf{Q}}_G$, on the diagonal operator relation. It is not the symbol of

and still less the common symbol of the coordinate and scalar response legs after their cross-line composition.

In the untested kernel, the residual metric variable $z$ remains present in the local source incidence. Testing it against $G$ is not an internal-symbol theorem in all variables.

The complete tested-kernel identity is retained as an input. The next proposition proves why its regularity conclusions cannot be strengthened without a uniform joint theorem.

There exists a compact distribution $T(a,b,z)$ on $A\times B\times Z$ such that

and $T$ has nonzero wavefront directions on $a=z$, while

is smooth for every $G\in {C_c}^{\infty }(Z)$.

Proof. Choose smooth compact functions ${\chi }_A,{\chi }_B,{\chi }_Z$, nonzero near the retained $a,b,z$ coordinates, with ${\chi }_A{\chi }_Z$ not identically zero. Set

Its support is compact. The conormal directions of the attachment diagonal are

They have zero covector sum. Wherever the smooth weight is nonzero, the diagonal delta has these nonzero wavefront directions; multiplication by that nonvanishing weight does not remove them.

Pairing the residual variable gives

which is smooth for every smooth compact test $G$.

This is a counterexample to a proposed inference from tested regularity, not a replacement for $K_{\mathrm{ret}}$. It satisfies the same relevant compactness and balanced-covector condition.

Smooth tests determine a distributional kernel uniquely. The proposition does not deny that fact. It shows that the regularity of every tested kernel can conceal attachment wavefront directions of the untested kernel. Those directions are precisely among the data a later joint composition must control.

Thus the tested equality of XIV-E remains valid, but it cannot supply the missing common symbol bundle or uniform characteristic composition estimates by itself.

The source diagonal in (eq:Qkernel) is a defined local operator map. The internal response constructions require more.

For example, the source response and coordinate response have the exact operator structure

The internal variables $s,v$ are governed by these kernels and the prescribed local source incidences. They are not set equal to $u_0,y_0,x_0$.

The two placements in the $B_5$ part of (eq:Btensor), and the coordinate and scalar response maps in $B_8,B_9$, must be composed on their own relations. Their existing tree-kernel pushforwards prove existence of those classical kernels, not a uniform Lagrangian-symbol description of their later characteristic composition.

The present paper does not supply:

1. the complete internal canonical relations with $z$ retained;

2. their microlocal orders and common amplitude identifications;

3. the necessary internal composition and remainder estimates.

These are the first missing data of G2.

The factorwise audit below identifies an additional limitation of an automatic pullback argument. It does not bypass this internal stop or enter G3.

The selected route permits individual symbols only after their relations, orders, and bundles have been named. The following audit does that for two elementary factors of the retained numerical incidence.

It does not assign a symbol to the complete regrouped coefficient. It does not evaluate a singular graph at a point and does not add separate branch numerators.

We retain the original temporal numerical factors only for this pullback diagnostic. The full fiber coefficient and every other retained branch remain outside its conclusion.

Write $F_{\mu }$ for the frozen scalar time-ordered numerical line of mass $\mu$, with its fixed nonzero fiber constants separated. Consider, on $W$,

Smooth inherited weights may be included. The statements about ellipticity below concern the retained open region where those weights are nonzero.

The two products defining $U$ and $V$ will be justified individually. Their further multiplication is not assumed.

Define

Near the retained point,

The first assertion follows because $\mathrm{d}{q}_R$ has no $z$ component and is nonzero. The second follows from independence of $\mathrm{d}{q}_1,\mathrm{d}{q}_2$.

Use the conormal order convention in which a normal Fourier amplitude of order $d$ on a codimension-$k$ submanifold of an $n$-dimensional space gives conormal order

Principal symbols are taken in the usual conormal symbol quotient, with half-density and Maslov factors and the fixed numerical fiber factors. Local defining functions and the fixed coordinate density trivialize those factors for the calculations here Wunsch (Section 8.1).

The individual relations are:

and the selected positive sector

When a joint normal order for $V$ is used, it is understood microlocally in a sector where both positive frequencies are nonzero and comparable. No statement about the axes of that normal sector is needed.

These are relations of the two factors in (eq:UV), not the relation of (eq:target).

Both $U$ and $V$ in (eq:UV) are defined distributions on the retained neighborhood.

The relevant conormal directions of $U$ have the form

Those of $V$ in the selected sector have the form

Proof. For $U$, a forbidden product cancellation between the retarded factor and the attachment delta would have zero $z$ covector. The attachment covector would therefore be zero. The retarded covector would then also be zero. Thus the product criterion passes.

For $V$, the first line depends on $(y,a)$ and the second on $(y,b)$. In a forbidden cancellation, the $a$ component forces the first nonzero line selection to vanish, and the $b$ component forces the second to vanish. The criterion again passes.

Differentiation preserves the relevant wavefront inclusions. The pullback and product rules therefore give the stated directions Hormander (Theorems 8.2.4 and 8.2.10).

These preliminary products use no matching with the third line. They do not invoke the excluded fiber product.

At a nonzero future null separation, the massless retarded line has its nonzero delta-type normal singularity. The frozen scalar boundary-value lines have nonzero simple-pole normal parts on the selected conormal sides. The massive line has the same principal normal pole; its less singular terms do not change that individual principal symbol. These are properties of the already frozen numerical lines, not choices of replacement propagators.

For the present Fourier convention, denote their nonzero individual leading normal amplitudes by

The selected time-ordered orientation is the copied one. A simultaneous reversal of Fourier convention changes the signs of all selected normal frequencies, not the argument.

On ${\Lambda }_U$, near the selected retarded direction, $U$ has normal amplitude order one and hence conormal order

Its leading normal amplitude is a nonzero conventional factor times

On ${\Lambda }_V^{+}$, in the stated positive comparable-frequency sector, $V$ has normal amplitude order two and hence conormal order

Its leading normal amplitude is a nonzero conventional factor times

Both individual symbols are nonzero at the copied characteristic selections. They are not a joint symbol of their product.

Proof. The normal amplitude of a delta distribution has order zero. A derivative acting on its normal singularity multiplies its leading amplitude by $i{\theta }_R{\partial }_{a^0}{q}_R$. Multiplication by the attachment delta adds four independent normal variables with order-zero amplitude. This gives (eq:Usymbol) and its order.

For $V$, the two normal defining functions are independent. In their separate adapted normal coordinates the simple-pole amplitudes have order zero on their selected half-rays. The two temporal derivatives of the second line give the leading degree-two factor in (eq:Vsymbol). Derivatives of amplitudes and the massive lower normal terms have lower order in this sector. The usual individual conormal symbol calculus applies because these normal coordinates are regular Wunsch (Section 8).

On $N$, with $\delta =\beta -\alpha >0$,

The selected normal frequencies are nonzero, as are $c_R,c_1,c_2$. Thus neither leading amplitude vanishes there.

No integration over the two tangent-defect directions occurs in this proof. Each factor is examined on its own regular conormal relation.

At every retained point $w\in N$ where the diagnostic weights are nonzero, the two defined factors in (eq:UV) admit opposite nonzero wavefront covectors:

and

Therefore the ordinary two-factor product criterion does not define $UV$ at these configurations.

The individually assigned relations and symbols do not justify this final product step.

Proof. Choose the individual normal frequencies

and choose zero attachment covector.

The factor symbols in Proposition prop:factorsymbol are elliptic at these conormal selections. Hence the selected directions are actual individual wavefront directions, not merely members of an upper bound.

The copied positive relation gives their zero sum. Thus the wavefront sets of $U$ and $V$ contain opposite nonzero covectors at the same point. The sufficient product hypothesis fails Hormander (Theorem 8.2.10).

The conclusion is failure of an automatic product construction, not nonexistence of every possible product prescription.

The following shortcut is excluded for the retained numerical incidence:

Assign symbols to the already defined retarded/attachment factor and cross-line factor, and then invoke the ordinary diagonal-pullback theorem without any further characteristic analysis.

This exclusion does not establish failure of cancellation of the complete grouped six-branch coefficient.

Proof. The theorem supplying the proposed final pullback requires the absent normal-set condition, as Theorem thm:pullbackfailure shows.

The calculation concerns two elementary numerical factors of a retained incidence. It neither constructs nor evaluates the complete tensor and branch sum in (eq:target). It also does not analyze that sum's lower-order terms. No full cancellation conclusion follows.

The cross-line factor in the actual target remains grouped as ${H_T}^{\otimes 2}-H^{\otimes 2}$. The preceding audit does not expand it into mixed $H$–${G_{\mathrm{act}}}^{\mathrm{A}}$ products or replace it by $V$.

In particular, the audit is not a proof that the complete grouped coefficient has the displayed wavefront direction. A cancellation theorem for that coefficient would have to address the complete sum before claiming such a direction is present or absent.

For the route fixed in Definition def:route:

1. the common ambient space and test locus are named: $W$ and $N={\tilde{\mathcal{N}}}_{\mathrm{att}}$;

2. the copied positive relation supplies diagnostic input covectors, not a common symbol relation;

3. the complete tested local source has a valid differential operator symbol;

4. smooth residual testing alone does not preserve the attachment wavefront information needed by joint composition;

5. the ordinary factorwise pullback shortcut fails on the specified retained numerical incidence;

6. the complete internal symbol relations and the common relation and bundle for (eq:target) have not been constructed.

The first unresolved gate is G2, within the inherited F2/E1/D2/J2 stop.

Proof. The choice (eq:chosenN) gives (i). Proposition prop:zerosum gives (ii). Proposition prop:sourceoperator gives (iii). Proposition prop:testing gives (iv). Theorem thm:pullbackfailure and Corollary cor:shortcut give (v).

The source polynomial and the individual factor symbols supply none of the missing internal composition and uniform common-kernel symbol theorems listed in Section sec:G2. No alternative representation or characteristic prescription has been constructed. This is exactly (vi), and it prevents entry into G3.

The selected no-common-phase route is not universally impossible. What fails is the automatic passage from individual symbols and inherited wavefront bounds to the ordinary final pullback.

A representation using cancellations of the full sum, or a separately proved generalized operator composition, could in principle advance this route. Neither is supplied here. No class of all such constructions has been proved incompatible.

G3 is not entered as a construction. The individual symbols above do not populate a symbol bundle for (eq:target).

The complete regrouped pieces retain the following status:

A loss of an explicit test derivative is not an assigned microlocal order improvement. A vanishing numerator in one presentation would have to be recorded with its remaining terms; it would not delete a branch.

G4 is not entered. Its remainder theorem would have to control:

1. lower-order terms of every numerical line;

2. contributions from the internal vertices of $K_5,K_8,K_9$;

3. derivatives of the attachment diagonal;

4. the two relative transverse directions identified by the copied defect;

5. every retained source and root term.

The tangent defect is a geometric constraint, not an integration fiber or a characteristic scaling degree.

Neither a zero joint section with adequate remainder control nor a nonzero full section on a named open relation is asserted.

The results of this paper establish neither cancellation nor failure of cancellation of the complete grouped coefficient (eq:target).

Proof. The source symbol belongs to a local differential operator. The testing example establishes a limitation of a regularity inference. The factorwise audit establishes failure of the ordinary final pullback hypothesis for one retained numerical incidence.

None constructs a common principal-symbol section of the complete coefficient or its controlled remainder. The copied $Q\lambda$ identity and the rank of $J$ do not supply those missing data.

No nonzero full joint section has been established. G5 is therefore not entered as a construction.

A future prescription must name all seven data:

1. its ambient space;

2. a reference distribution on a proved domain;

3. the exact characteristic locus across which it extends;

4. agreement on that domain with the grouped frozen prescription;

5. a finite uniform transverse degree;

6. a closed conic wavefront bound retaining off-locus singularities;

7. a normalization selecting one map, or a uniqueness theorem.

The present choice $N={\tilde{\mathcal{N}}}_{\mathrm{att}}$ is a codimension-eight common-kernel test stage. It is not silently converted into a codimension-four numerical extension problem.

The canonical product is known on the inherited admissible domain $U_{\mathrm{can}}$. No equality

is asserted. A proved reference distribution on the required complement remains part of the construction, not a consequence of the existence of the individual lines.

The copied extension warning remains:

on a whole neighborhood would require the reference to be smooth away from $N$. It cannot erase the already existing off-$N$ singularities. A prospective bound

requires its own microlocal extension hypotheses.

For a genuine codimension-four problem with degree allowance at least four, smooth tangential multiples of ${\delta }^{(4)}(r)$ can remain possible unless normalization or uniqueness excludes them. On the retained collision-free neighborhood, such nonzero characteristic-supported changes are not members of ${\mathfrak{C}}_{\mathrm{part}}$.

The codimension-eight attachment intersection is a separate extension stage. No degree of the unresolved target is assigned here.

Brunetti–Fredenhagen's submanifold extension framework and Dang's microlocal extension framework are recorded only BF (Section 6)Dang (Sections 5–6). Their required reference and uniform estimates have not been proved for this target, so no extension theorem from them is applied.

XIII-E's family is not imported as a prescription. There is no cutoff-and-restore limit. The excluded excess construction is not reused under a prescription label.

J3 opens only for a distribution supplied by a successful G4 or G5 construction. None is supplied.

The parallel-line degree eight remains a collision degree, not a degree along ${\mathcal{N}}^{\circ }$, ${\overline{\mathcal{N}}}_{\mathrm{att}}$, or ${\tilde{\mathcal{N}}}_{\mathrm{att}}$. The copied defect Hessian does not alter that distinction. The individual conormal orders in Proposition prop:factorsymbol are likewise not transverse scaling degrees of the unresolved full product.

J4 remains closed: no characteristic distribution is glued to singular spectators or compatible partial contacts.

J5 remains closed: inherited properness is support control for an existing distribution, not a construction of (eq:target) or its output wavefront theorem.

No ${R_1}^{(1)}$ is released. The tree coefficient is not substituted for the missing quantum coefficient.

No $s_{\mathrm{ext}}$, $s_{\Psi }$, or $s_0$ is applied. No binary Ward defect is formed. The value $-8I$ and the classical evaluation identities do not normalize the graph.

There is no new gravitational interaction vertex, restriction quotient, state, loop value, characteristic contact class, triangle residue, or sharp-width limit.

For the frozen retained instance:

1. The chosen common space and characteristic test locus are $W$ and $N={\tilde{\mathcal{N}}}_{\mathrm{att}}$, of dimensions sixteen and eight, respectively.

2. The positive conormal cancellation is diagnostic input data, not a joint symbol section.

3. For a smooth residual test $G$, the full differentiated source ${\mathcal{J}}_{\mathrm{D}}^{\prime }[G]$ is a differential operator of order at most two with the local symbol (eq:Qsymbol).

4. Smooth residual testing can hide nonzero attachment-diagonal wavefront directions even for compact balanced kernels.

5. Individually defined conormal factors of the retained numerical incidence still fail the ordinary final product test at the selected characteristic covectors.

6. No complete internal symbol relation, joint section, controlled remainder, or characteristic prescription for (eq:target) is constructed.

The ending for the grouped target is Unresolved at G2, within F2/E1/D2/J2.

No universal incompatibility, admitted ${R_1}^{(1)}$, or Ward defect is asserted.

Proof. The copied lift geometry and (eq:chosenN) give (i). Proposition prop:zerosum gives (ii). Proposition prop:sourceoperator gives (iii). Proposition prop:testing gives (iv). Lemma lem:UVexist, Proposition prop:factorsymbol, and Theorem thm:pullbackfailure give (v).

Theorem thm:G2 identifies the first missing common composition data. Proposition prop:nocancellation separates the proved diagnostics from a cancellation verdict. No prescription is supplied in Section sec:G5, proving (vi) and the stated ending.

The selected no-common-phase route has an explicit hypothesis audit. The local source admits an ordinary differential symbol, but its composed response symbols have not been identified.

The information lost under smooth residual testing is also explicit: attachment-diagonal covectors may disappear from every tested kernel. Pointwise-in-$G$ regularity does not replace a joint microlocal theorem.

Finally, two valid preliminary factor products do not make their final product valid. The retained characteristic cancellation persists in the ordinary operator/pullback test without invoking a common phase or an excess bundle.

These results exclude an automatic factorwise shortcut. They do not exclude every possible no-common-phase construction of the complete coefficient.

A successor receives:

1. the unchanged XIV-F and XIV-E packages;

2. the selected operator/pullback route and its unentered alternatives;

3. the chosen common ambient space and codimension-eight test locus;

4. the distinction between diagnostic zero-sum covectors and an output symbol relation;

5. the complete tested-source differential typing and local symbol;

6. the compact balanced-kernel testing counterexample;

7. the individual-factor relations and the failed final pullback test, explicitly limited to the numerical incidence;

8. the exact G2 stop and the still missing internal composition and common-symbol data.

To advance this route, the successor must construct the internal relations and a justified common operator representation of the complete grouped coefficient. It must then supply the joint section and remainder, or one fully specified characteristic prescription.

It receives no full cancellation theorem, no nonzero full symbol section, and no permission to call the tangent defect an excess. It receives no ${R_1}^{(1)}$.

An individual symbol is not a joint symbol. A residual test is not a uniform composition theorem. Avoiding a common phase does not avoid a pullback hypothesis. An undefined quantum coefficient is not an anomaly.

scope. The new proofs concern local source typing, the relation audit, loss of wavefront information under residual testing, and the ordinary factorwise pullback test. No joint characteristic distribution is asserted.

scope. Only XIV-F, XIV-E, XIV-D, XIV-C, XIII-C, XIII-B, XII-E, and XII-C are analytic manuscript inputs. Their handoffs retain their predecessor scope. No incidence ledger or excluded phase or matching is re-proved.

-symbol scope. The conormal orders and symbols assigned in the factorwise audit belong only to its two individually defined factors. They are not orders, degrees, or symbols of the complete grouped coefficient. The audit does not expand the grouped cross-line difference into undefined mixed products.

scope. The local symbol of ${\mathcal{J}}_{\mathrm{D}}^{\prime }[G]$ retains the full actual source. No layer coefficient is discarded. Internal vertices are governed by their existing kernel maps, not identified with the retained external points.

scope. The class ${\mathfrak{C}}_{\mathrm{part}}$ is unchanged. No characteristic-supported difference is reclassified as a collision contact, and no reference prescription is selected.

calculus. Individual regular conormal and differential symbols are used only after their local relations and orders have been named. Preliminary products pass the stated product tests. The failed final test is reported as failure of a sufficient construction, not nonexistence. No submanifold extension theorem is applied to the missing target.

scope. No renormalized quantum coefficient, Ward defect, interacting differential, state, loop value, triangle residue, or sharp-width limit is supplied.

. No successful pdflatex execution, proof-assistant verification, independently executed symbolic computation, independent referee review, or journal acceptance is asserted. The release consists of the displayed analytic proofs and the explicit unresolved gate.

XIVF

N. Cheung, Landmark XIV-F: Paired composition on a proposed excess bundle: The clean-fiber-product audit, the excess correction, and the exact F2 stop. Supplied manuscript XIV-F.tex. Immediate predecessor and unchanged geometric boundaries.

XIVE

N. Cheung, Landmark XIV-E: Joint symbol of the six-branch retained germ, or one named characteristic prescription. Supplied manuscript XIV-E.tex. Regrouping, root, tested-kernel bound, characteristic lifts, elementary phase exclusion, and E1 stop.

XIVD

N. Cheung, Landmark XIV-D: Cancellation or prescription on the retained noncoincident characteristic stratum. Supplied manuscript XIV-D.tex. Pair, six-branch germ, species split, branch fiber identity, and D2 stop.

XIVC

N. Cheung, Landmark XIV-C: The joint two-line extension of one selected binary pair: noncoincident characteristic configurations, transverse estimates, and the output wavefront gate. Supplied manuscript XIV-C.tex. Characteristic stratum, positive conormals, contact rigidity, and J2 stop.

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. Inventory, partial contacts, primitive extensions, and successor obligations.

XIIIB

N. Cheung, Landmark XIII-B: One binary causal datum for the displayed coefficient, on the unchanged carrier. Supplied manuscript XIII-B.tex. Tree datum, convention, ordered values, and properness boundary.

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. Completion formulas, slot sign, roots, and tree kernels.

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. Fixed cubic representative and actual source.

XIIIE

N. Cheung, Landmark XIII-E: One joint off-collision prescription for the selected binary pair, or its obstruction in the frozen partial-diagonal class. Supplied manuscript XIII-E.tex. Recorded common-frequency attempt only.

Hormander

L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, second edition, Springer, 1990. Section 8.2: pullbacks, tensor products, products, and wavefront bounds.

Wunsch

J. Wunsch, Microlocal analysis and evolution equations: Lecture notes from the 2008 CMI/ETH Summer School, arXiv:0812.3181, revision v3, 2023. Section 8: conormal and Lagrangian distributions, orders, individual symbols, and operator calculus.

BF

R. Brunetti and K. Fredenhagen, “Microlocal analysis and interacting quantum field theories: Renormalization on physical backgrounds,” Communications in Mathematical Physics 208 (2000), 623–661, arXiv:math-ph/9903028. Section 6: recorded submanifold extension framework; not applied to the unresolved target.

Dang

N. V. Dang, “The extension of distributions on manifolds, a microlocal approach,” Annales Henri Poincaré 17 (2016), 819–859, arXiv:1412.2808. Recorded microlocal extension and ambiguity framework; no characteristic prescription is imported.