Landmark XIV-D Cancellation or prescription on the retained noncoincident characteristic stratum Complete retained incidence and the unresolved joint-symbol test

This manuscript continues XIV-C's characteristic gate for the unchanged binary pair. No quantum retarded coefficient was released there, so no Ward defect is formed here.

We display the complete second-derivative ledger and the two nonzero species channels of the grouped two-line expression. A common attachment/residual-variable description makes the retained local comparison precise. In its mixed channel, $B_1,B_2,B_3$ vanish as kernel germs by their field-location supports; the scalar-sector branches vanish in that channel. The scalar–scalar channel is excluded only on the specified retained output-location patch. The complete potentially surviving mixed germ therefore contains $B_4,\dots ,B_9$, including both gravitational-response legs.

We also compute the full leading fiber-tensor numerator of $B_6$, rather than its temporal monomial alone. It is nonzero for suitable residual field jets. This is a branch diagnostic, not a nonzero symbol section of the complete six-branch germ.

The required common microlocal symbol calculation and its lower-order control are not established. Neither cancellation of the complete germ nor failure of that cancellation is proved. Consequently no characteristic prescription is entered on the strength of the branch diagnostic. Necessary degree, wavefront, and ambiguity conditions for a future prescription are separated from its existence.

The exact ending is Unresolved at D2, within J2. No characteristic extension, admitted ${R_1}^{(1)}$, quantum Ward defect, new contact class, or sharp-width limit is constructed.

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

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

and first missing step{ Frozen: XIV-C's pair, Hadamard matrix, inverse coefficient $-1$, switches, width, characteristic configuration, and ${\mathfrak{C}}_{\mathrm{part}}$.

Constructed here: the complete smooth-slot Hessian ledger; its grouped species decomposition; the common-variable local incidence comparison; and the reduction of the retained mixed germ to $B_4,\dots ,B_9$.

Additional diagnostic: the full fiber-tensor leading numerator of $B_6$ is computed. It is not the full branch sum.

First missing step: D2. A controlled joint-symbol description of the complete retained germ, including lower orders, has not been supplied.

Ending: Unresolved. No full cancellation or nonzero full symbol section is asserted. D3 is not entered; J3–J5 and the Ward-defect question remain closed.}

The three endings concern the characteristic construction:

1. Constructed: cancellation of the complete retained germ, with the requisite lower-order control, or one characteristic prescription with its degree, ambiguity, and wavefront theorem, discharges the named J2 problem.

2. Proved obstructed: an explicit incompatibility excludes every prescription in a class fixed before the test.

3. Unresolved: the complete sum is uncontrolled, or a required prescription or estimate is missing.

A failed sufficient product criterion is not the second ending. A nonzero numerator of one branch does not establish failure of cancellation of the complete pair. Even a successful J2 prescription would not by itself release ${R_1}^{(1)}$.

The pair is

with

The scalar-sourced slot is retained once.

The fixed triple is

where $\epsilon <t_{\ast }<t_{\ast }+2L<T$, $f(x_0)\ne 0$, and $g=1$ near these points.

The characteristic stratum is

It is smooth of codimension four. For

the differentials have rank two there and satisfy

No tangent vector makes all three selected oriented evaluations strictly positive.

The temporal $B_6$ monomial has nonzero coefficient and nonvanishing line differential symbols at the retained triple. That statement is not a residue of the complete branch sum.

There is a collision-free neighborhood $V$ of the retained three-vertex configuration. Graphwise collision contacts restrict to zero there. No characteristic prescription or cancellation of the complete pair was released, and J3–J5 were not reached XIVC.

These geometric statements and the temporal monomial diagnostic are not re-proved.

XIII-C supplies the eleven branches, 48 labelled matchings, the partial-diagonal class, and the elementary parallel-line extensions. XIII-B supplies the tree datum, support control, retarded convention, and corrected ordered values. XII-E supplies the completion formulas and their admitted tree kernels. XII-C supplies the fixed density representative and actual cubic source XIIIC,XIIIB,XIIE,XIIC.

XIV-A and XIV-B are recorded only. Their relative antifield complex is not an analytic input. The auxiliary value $-8I$ is not a subtraction or normalization datum.

Use

All widths, observations, switches, and propagators remain those fixed in XIV-C. The carrier is unchanged:

The time-ordered line remains

Its metric–scalar blocks vanish. No replacement Hadamard matrix is selected.

The ambiguity class remains the inherited ${\mathfrak{C}}_{\mathrm{part}}$. Its defined graphwise members have collision support and the form

with the inherited weights, spectator coefficients, two residual field slots, support closure, and normalization requirements.

No class of characteristic prescriptions is declared to have been constructed. In particular, a universal obstruction is not tested against an unspecified class.

Define the polarized stress tensor by

Thus $T(\varphi ,\varphi )=T_0[\varphi ]$, and

Put

The first-response operators are

Arguments $(f,g)$ on ${\mathcal{A}}_1$ are suppressed when fixed.

The inherited filled slots are

These are the already admitted classical slots, not new quantum products with ${\mathcal{J}}_{\mathrm{D}}$ as an argument.

For a finite intrinsic description of the fixed source, write $q(\rho )=\eta +\rho \gamma$ and

The curvature representative is the unintegrated one of XII-C, with

Coefficient extraction is finite at this order. It fixes the same local quadratic source, including its first and second ramp derivatives; it changes neither the representative nor the action.

The convention remains

The requested two-line coefficient is specified formally by

This difference stays grouped. No separate mixed product of $H$ and ${G_{\mathrm{act}}}^{\mathrm{A}}$ is assigned an existence statement.

Choose small neighborhoods $A,B,Y$ of $u_0,x_0,y_0$, respectively, with pairwise disjoint closures and

in the vertex order $(x,u,y)$. Choose a neighborhood $Z$ of $u_0$ with $\overline{Z}\subset A$.

The variables used below are:

In the retained $B_6$ graph, $(b,a,y)$ is the original triple $(x,u,y)$ and the local metric vertex enforces $a=z$.

For other branches, $z$ remains an independent field variable. It is not set equal to $a$. This is a restriction of a kernel germ to an open set, not an excision of ${\mathcal{N}}^{\circ }$, a change of switches, or a cutoff of the triangle.

There is no characteristic cutoff, triangle-residue subtraction, new contact class, antifield differential, spurion, convolution sheet, restriction quotient, new gravitational vertex, or sharp-width limit.

The first new calculation concerns the complete second-derivative sum. It precedes any further branch diagnostic.

Let $\mathcal{C}({\gamma }_1,{\gamma }_2;\psi )$ be the symmetric polarization in the metric arguments of ${{\mathcal{A}}_2}^{\mathrm{sc}}(\gamma ,0;\psi )$:

Define symmetric bilinear metric-valued maps

Then the complete observable is

This formula includes both legs of each filled slot.

For smooth metric directions $h,h_1,h_2$ and scalar directions $\psi ,{\psi }_1,{\psi }_2$,

and

The mixed blocks are symmetric under exchanging the derivative slots.

Proof. The stress tensor (eq:Tpolar) is symmetric bilinear. Differentiating $-\frac{1}{2}\int g\gamma T(\varphi ,\varphi )$ once in each indicated direction gives (eq:VHessian).

Each of $\mathcal{C}$ and $\mathcal{D}$ is symmetric bilinear in its metric arguments. Each occurrence of ${\mathcal{A}}_1$ is linear in both displayed arguments. The first two terms of (eq:polarO) are linear in $\varphi$, while its last term is cubic in $\varphi$ and independent of $\gamma$. Applying the product rule gives all three formulas in (eq:OHessian), with the stated factors of two.

All operations in this proposition concern smooth directions of existing classical functionals. No characteristic product is used in its proof.

For a block-diagonal even line $L$, define ${\mathrm{Contr}}_L^{\gamma \varphi }$ and ${\mathrm{Contr}}_L^{\varphi \varphi }$ to mean the two-slot contractions using, respectively, $L_{\gamma \gamma }\otimes L_{\varphi \varphi }$ and $L_{\varphi \varphi }\otimes L_{\varphi \varphi }$, with the indicated derivative blocks.

Where the contractions are defined, species expansion of (eq:grouped) gives

On the unresolved domain this is a formal specification only.

The factor $i$ in the mixed channel includes its two equivalent slot orderings. There is no two-metric channel because ${{\mathcal{V}}_g^{\prime \prime }}_{\gamma \gamma }=0$. No metric–scalar line has been added.

Equation (eq:species), together with (eq:VHessian)–(eq:OHessian), is the complete tensor-valued algebraic sum. It does not replace the requested microlocal composition theorem.

Write

and

Set

Let ${\int }_f$ denote integration against $f$ over the observation variable. The mixed branch derivatives ${\mathcal{M}}_j[\gamma ;h,\psi ]=(B_j{)}_{\gamma \varphi }^{\prime \prime }[h,\psi ]$ are

Thus

For the scalar–scalar block, put ${\mathcal{N}}_j=(B_j{)}_{\varphi \varphi }^{\prime \prime }$. Its complete ledger is

Equations (eq:allmixed) and (eq:allscalar) are the ordinary second derivatives of all eleven inherited branches. No tensor component, source coefficient, filled-slot leg, or scalar-slot bijection has been discarded.

Proof. For $B_1,\dots ,B_7$, vary each of the two metric occurrences in turn, and replace the single scalar by $\psi$. This gives ${\mathcal{M}}_1,\dots ,{\mathcal{M}}_7$, including the symmetric derivative of $a_2$.

The two metric occurrences in $B_8,B_9$ are inside $r_{\mathrm{D}}$. Their complete variation is $r_h$, giving ${\mathcal{M}}_8,{\mathcal{M}}_9$. The definition of ${\mathcal{J}}_{\mathrm{D}}$ fixes every coefficient entering $r_h$; no layer term is separately set to zero.

The last two branches have no metric argument. Their scalar slot derivatives give (eq:allscalar). The first nine branches have only one scalar occurrence and therefore have zero second scalar derivative.

For each mixed branch the two observable metric choices and two interaction scalar choices reproduce its four labelled matchings. For each scalar-sector branch the three observable scalar pairs and two bijections reproduce its six matchings. The count remains $9\cdot 4+2\cdot 6=48$.

Because ${\mathcal{M}}_j$ is trilinear in $(\gamma ,h,\psi )$, let $K_j$ denote its compact distributional kernel:

All field and line derivatives are included in this kernel. Existence of these individual kernels follows from the inherited tree constructions, not from a two-cross-line composition.

The complete mixed kernel is

It lives on the common variables $(a,b,z)$ even when a branch was originally written with additional internal vertices. Those vertices have been integrated by its established tree kernel map.

This comparison does not pull a four-vertex distribution back to an arbitrarily imposed three-vertex diagonal. Such a pullback would require its own wavefront test FR (Appendix A).

On $A\times B\times Z$,

Moreover $K_6,K_8,K_9$ have support in $\{a=z\}$. The mixed derivatives of $B_{10},B_{11}$ are identically zero.

These are distributional support statements, not evaluations of singular kernels at $(u_0,x_0,u_0)$.

Proof. In $B_1$, the scalar and one metric slot share the inner vertex. Selecting the other metric slot for contraction forces $b=z$; selecting the metric at the scalar vertex forces $a=b$. Thus

The same slot-location argument applies to $B_3$, giving the same support inclusion.

All three observable slots in $B_2$ share one vertex, so

Since $\overline{A}\cap \overline{B}=\varnothing$ and $\overline{Z}\subset A$, these supports miss $A\times B\times Z$. Differentiating a distribution does not enlarge its support, so the argument covers all derivative placements.

Both metric slots of $B_6$ share their local vertex. Both metric slots of $B_8,B_9$ share the source vertex, including when the source differentiates either slot. Hence their mixed kernels are supported on $a=z$.

The last assertion follows either directly from the absence of a metric argument or from (eq:allmixed).

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.

In the scalar–scalar channel of (eq:species), the interaction's unused field is its metric at $y$. In the mixed channel, it is a scalar at $y$ and the residual metric comes from the observable at $z$.

Retain the interaction vertex in $Y$ and the output metric variable in $Z$. The scalar–scalar channel has zero kernel germ on this patch.

Proof. Its residual metric variable equals the interaction vertex. Its kernel consequently has support in $z=y$, including any field derivatives. Since $\overline{Y}\cap \overline{Z}=\varnothing$, the restriction is zero.

This does not delete $B_{10},B_{11}$ from the frozen pair. Their scalar channel remains part of (eq:species) outside the retained output-location patch.

There are 24 potentially retained labelled mixed matchings on this germ: four for each of the six branches. This is not a count of nonzero tensor monomials.

For each inventory graph, retain its original two attachment variables $a,b$ and the interaction vertex $y$. Every connected collision contact involving $y$ is supported in

Thus every defined inherited contact vanishes on $Y\times A\times B\times Z$ while these attachment variables are retained.

Proof. The only edges incident to $y$ are its two cross-lines, whose observable endpoints are $a$ and $b$. A connected vertex subset containing $y$ and at least one further vertex must therefore contain at least one of those endpoints. Collapsing the subset forces $y=a$ or $y=b$.

Intersections and the corresponding contracted loci retain that equality when expressed in the original attachment variables. Differential operators do not enlarge support. Projection over other internal variables preserves the displayed equality because $y,a,b$ are not projected out.

The three neighborhoods have disjoint closures, proving the vanishing.

After projecting out the attachment variables themselves, a contact can produce a nonlocal-looking output through its spectator. No assertion of vanishing after that different projection is made.

The full formal coefficient on the retained mixed attachment/output germ is obtained by contracting ${{\mathcal{V}}_g^{\prime \prime }}_{\gamma \varphi }$ with the complete kernel

against the grouped line difference in the first line of (eq:species).

All other branch contributions are zero on this germ for the reasons proved above. No collision contact defines or changes this germ while the attachments are retained.

Proof. Use Proposition prop:Hessian, Lemma lem:ledger, Proposition prop:reduction, Lemma lem:channel, and Lemma lem:contact.

This is a complete incidence statement. It is not yet a statement that the displayed contraction exists on the characteristic set.

On the retained germ, the required object is

This is the complete sum to be tested, not its $B_6$ summand.

The copied relation is used with $(x,u,y)=(b,a,y)$. For $\ell =(1,1,0,0)$ and $p=\eta (\ell ,\cdot )$, its line covectors in the order $(b,a,y)$ are

Their sum is zero. The residual metric variable has zero covector in this copied product-test selection.

On the microlocally admissible open sets, each contraction in (eq:retainedfull) is an ordinary defined graph composition. At the retained characteristic configurations, the inherited theorem does not define it.

The kernels $K_j$ are defined distributions, but a wavefront inclusion alone does not assign them a common scalar principal symbol for the composition (eq:retainedfull).

Some kernels include the observable metric diagonal $a=z$. Others arise from tree compositions with additional internal vertices. To use a common microlocal symbol calculation one must specify the corresponding conormal or multiphase representations, their composition maps, and their orders. One must then control the complete grouped amplitudes and the remaining terms. The distribution product and pushforward criteria do not supply this information merely from an incidence count Hormander.

In particular, the following replacements are not made:

1. evaluating $K_j$ at a singular point;

2. setting every hidden internal vertex equal to one of the retained three vertices;

3. adding differential numerators from inequivalent unidentified phases as if they were one wavefront section;

4. calling a cancellation of highest differential factors a proof that all lower-order singularities disappear;

5. expanding the grouped line difference into intermediate products whose existence is unknown.

The first missing analytic step is a controlled common symbol/composition description of the complete germ (eq:retainedfull). No equality to zero and no nonzero full symbol section is proved here.

The following calculation sharpens the inherited monomial diagnostic. It does not replace the complete six-branch test.

For a covector $p$, put $n=p^{♯}$ and suppose $p^2=0$. Let $G_{\mu \nu }$ be the residual metric value at its source vertex, let $k_{\mu }$ be the residual scalar derivative at the interaction, and set

The complete density variation satisfies

After the metric propagator fiber and the principal kinetic stress variation are contracted, the corresponding $B_6$ leading differential numerator is a fixed nonzero conventional factor times

It is not identically zero as a fiber-tensor polynomial.

Proof. The principal kinetic variation of the stress tensor has fiber coefficient

In four dimensions ${\mathrm{tr}}_{\eta }S=-\lambda$, so trace reversal gives

The signs and factors from ${{\mathcal{V}}_g^{\prime \prime }}_{\gamma \varphi }$ and the fixed metric line are nonzero. The mass term has lower scalar-line differential order and is not part of this principal differential numerator.

Because $p^2=0$, the $b_2{\eta }^{\alpha \mu }$ term makes no contribution after contraction with $p_{\alpha }{p}_{\mu }$. Write $\tau ={\mathrm{tr}}_{\eta }G$. The remaining quadratic expression is

For variation $r$,

Its directional derivative is therefore

proving (eq:B6identity).

The retarded divergence and observation scalar derivative supply the two factors of $p$ in (eq:B6identity). The frozen scalar line normalizations and the nonzero kinetic coefficient supply only fixed nonzero factors.

For example, $G_{00}=1$ with all other components zero, $p=(1,-1,0,0)$, and $k=(1,0,0,0)$ give $Q=\lambda =1$. The original weights are nonzero at the retained configuration.

This proof performs the fiber sum for this leading $B_6$ numerator. It does not evaluate an undefined graph product. It does not determine the contributions of $K_4,K_5,K_7,K_8,K_9$, their common microlocal orders, or their lower-order terms.

Proposition prop:B6tensor proves neither cancellation nor failure of cancellation of (eq:retainedfull). It does not justify advancing to D3 on the premise that complete-pair cancellation has failed.

Proof. The complete retained germ is the six-branch expression of Theorem thm:incidence. The proposition computes only one branch's leading differential numerator. It supplies no sum identity or sum nonvanishing theorem for that complete expression.

The complete retained incidence is named by (eq:retainedfull), but its joint-symbol cancellation test and lower-order control have not been established. Neither legal decisive D2 result—a complete cancellation theorem or a nonzero full symbol section—is released.

The first unresolved step is D2, within J2.

Proof. The algebraic and local-support identification is Theorem thm:incidence. The inherited characteristic relation does not satisfy the sufficient product criterion. No common microlocal composition theorem for (eq:retainedfull) has been proved.

Proposition prop:B6tensor cannot fill that gap by Corollary cor:branch. No independent prescription is constructed below. The missing conclusion is therefore reported as Unresolved, not as a universal incompatibility.

D3 is not entered as a construction, because no complete failure-of-cancellation result or independent prescription has been supplied. This section corrects two possible misstatements of its requirements.

A future prescription must specify:

1. a reference distribution $t^{\circ }$ on a stated off-characteristic domain;

2. the exact characteristic locus and graph or common kernel space on which it is extended;

3. agreement with $t^{\circ }$ and with the grouped frozen line prescription;

4. a finite uniform transverse degree;

5. a closed conic wavefront bound and its extension theorem;

6. the normalization selecting a permissible map under the unchanged ambiguity contract.

It is not enough to say that every separate line is defined. The reference joint distribution must be defined on the domain on which agreement is required.

Let $N$ be a submanifold of an open set $U$, and let $t\in {\mathcal{D}}^{\prime }(U)$ extend $t^{\circ }\in {\mathcal{D}}^{\prime }(U\setminus N)$. Then

Consequently a bound $\mathrm{WF}(t)\subset N^{\ast }N$ on all of $U$ would require $t^{\circ }$ to be smooth on $U\setminus N$.

Proof. Wavefront sets are local and commute with restriction to an open set. The conormal bundle $N^{\ast }N$ has base points only in $N$. Hence the proposed bound leaves no wavefront point outside $N$, which is equivalent to smoothness there.

Accordingly, “wavefront contained in the named conormal” must not be used to erase the inherited off-characteristic line singularities or observable diagonal singularities.

A minimal-extension target may have the form

but its validity requires additional microlocal hypotheses. In particular, the conormal landing condition is a hypothesis, not an automatic consequence of finite scaling degree Dang (Sections 5–7).

In the common-variable description, pre-existing observable diagonals such as $a=z$ are also retained. They are not interaction collision counterterms and are not removed by calling the new characteristic locus $N$.

No such wavefront theorem is proved for (eq:retainedfull).

The following conditional statement identifies a genuine ambiguity issue without adding an ambiguity to ${\mathfrak{C}}_{\mathrm{part}}$.

Let $N$ be a smooth codimension-four submanifold and choose local coordinates $(s,r)$ with $N=\{r=0\}$, $r\in {\mathbb{R}}^4$. Suppose an extension $t$ of a fixed $t^{\circ }$ has transverse scaling degree at most $d$, where $d\ge 4$.

For every nonzero smooth compactly supported $a(s)$,

has transverse scaling degree four and wavefront set contained in $N^{\ast }N$. Hence $t+A$:

1. agrees with $t^{\circ }$ off $N$;

2. has transverse degree at most $d$;

3. preserves any wavefront bound for $t$ that already allows all of $N^{\ast }N$.

If this characteristic neighborhood is disjoint from the permitted interaction collision supports, the nonzero difference $A$ is not a member of ${\mathfrak{C}}_{\mathrm{part}}$.

Proof. Normal dilation gives ${\delta }^{(4)}(\rho r)={\rho }^{-4}{\delta }^{(4)}(r)$. The smooth tangential coefficient adds no wavefront covector. Thus the stated degree and conormal bound hold.

The distribution $A$ vanishes off $N$. Scaling degree and wavefront set of a sum are bounded by the corresponding maximum and union, proving (i)–(iii).

Every permitted collision change vanishes in the specified collision-free neighborhood. The nonzero $A$ does not, so it cannot belong to that class.

This is a conditional ambiguity diagnostic. The actual characteristic degree of the retained full coefficient has not been computed. In particular, the proposition does not assert that it is at least four.

Under established uniform submanifold extension hypotheses, degree below codimension can give uniqueness. At or above codimension, differences are controlled by finite normal delta derivatives; microlocal conormal control makes their tangential coefficients smooth BF (Theorem 6.9)Dang (Section 8). These conclusions presuppose the relevant distribution and estimates.

No $A$ from Proposition prop:ambiguity is added here. No characteristic ambiguity class is enlarged. A future prescription must either establish uniqueness under its proved bounds or state a normalization that selects one map without disguising characteristic differences as members of ${\mathfrak{C}}_{\mathrm{part}}$.

A cutoff excluding $N$, followed by a limit restoring it, would require its own convergence, transverse-degree, and wavefront theorem. Such a construction is not introduced.

No finite-part formula, analytic continuation, subtraction constant, triangle residue, or characteristic contact is chosen. Collision contacts are not a substitute for these missing data.

Failure to supply a prescription does not prove that every prescription fails. Conversely, specifying only an admissible difference class does not construct a reference prescription. Neither inference is made here.

J3 can be opened only for a distribution actually supplied by D2 or D3. The copied parallel kinetic primitive has scaling degree eight in collision codimension four. That is not a degree along the noncoincident characteristic stratum.

No uniform transverse estimate for the complete retained germ, its three- or four-vertex collision strata, or their intersections is proved here.

J4 remains closed: no characteristic distribution has been glued to the primitive partial contacts, and no overlap theorem is supplied.

J5 remains closed: inherited properness is support control for an existing distribution. It does not define (eq:retainedfull), nor does it establish a wavefront bound for its output.

No admitted degree-two coefficient and no conclusion about ${R_1}^{(1)}$ in the carrier are written.

There is no quantum retarded coefficient on which to form the proposed binary Ward defect. The already constructed tree datum is not substituted for the missing quantum coefficient.

No $s_{\mathrm{ext}}$, $s_{\Psi }$, or $s_0$ is applied to ${R_1}^{(1)}$. No quantum product having ${\mathcal{J}}_{\mathrm{D}}$ as a separate argument is constructed.

The classical breaking density from XIV-A is not an anomaly of this unnamed graph. XIV-B's relative complex supplies neither the joint product nor a subtraction. The evaluation identity on ${\mathcal{C}}_{W,\mathrm{bos}}$ and the value $-8I$ remain in the non-identification ledger.

For the unchanged contract of D0:

1. The full smooth-slot second-derivative sum is (eq:VHessian)–(eq:OHessian), and the complete grouped species expression is (eq:species).

2. The retained mixed kernel germ is exactly the six-branch sum (eq:six); the local zero statements and channel separation are proved without evaluating a singular kernel at a point.

3. The full leading fiber-tensor numerator of $B_6$ obeys (eq:B6identity), but this does not decide cancellation of (eq:retainedfull).

4. No complete cancellation theorem, nonzero full symbol section, or characteristic prescription is constructed.

5. No incompatibility excluding every possible prescription under the unchanged contract is proved.

The exact ending is Unresolved at D2, within J2. No ${R_1}^{(1)}$ or Ward defect is released.

Proof. Proposition prop:Hessian and Lemma lem:ledger prove the full algebraic ledger. Proposition prop:reduction, Lemma lem:channel, and Theorem thm:incidence prove the retained-germ statement.

Proposition prop:B6tensor proves the branch-level fiber calculation, with its limitation established in Corollary cor:branch. Theorem thm:D2 identifies the first missing analytic step. Section sec:D3 supplies necessary conditions but no extension map or universal exclusion theorem.

The complete retained incidence is no longer represented by a temporal monomial or by an unspecified eleven-term symbol sum. It is a precise common-kernel germ with six potentially surviving mixed branches and both gravitational-response legs.

The remaining problem is correspondingly exact: control the grouped composition of that six-branch kernel with the two frozen cross-lines on the retained characteristic relation, including the lower-order terms.

The source and scalar-response branches have not been declared to cancel or to vanish. The $B_6$ numerator has not been promoted to a residue. No additional contact has been chosen to force a value.

A successor receives:

1. the unchanged pair and ambiguity contract;

2. the full polarized Hessian formulas and species factors;

3. every mixed and scalar branch derivative;

4. the common attachment/residual-variable kernel description;

5. the exact six-branch retained germ and its local zero proofs;

6. the full $B_6$ fiber identity, explicitly limited to its branch;

7. the necessary wavefront and characteristic-ambiguity checks;

8. the unresolved complete joint-symbol test.

To discharge J2, the successor must prove cancellation of the complete expression (eq:retainedfull) with its lower-order control, or construct a prescription for that expression with its actual transverse degree, normalization, ambiguity, and wavefront theorem.

It receives neither failure of full cancellation nor permission to infer it from Proposition prop:B6tensor. It receives no admitted quantum coefficient.

A full incidence ledger is not a joint distribution. A nonzero branch numerator is not a nonzero full symbol section. A collision contact is not a characteristic prescription. An undefined quantum coefficient is not an anomaly.

scope. This manuscript continues XIV-C's J2 problem. Its new constructions are the complete smooth-slot derivative ledger, common-variable support comparison, retained-germ reduction, and branch-level fiber identity. It does not claim to solve the joint characteristic extension problem.

scope. ${{\Phi }^{(2)}}_{\mathrm{D}}$ retains its scalar-linear predecessor meaning. The full observable $\mathcal{O}$ contains the scalar-sourced slot once. No entry is changed after the retained test.

of zero statements. The branches proved zero on the chosen kernel germ remain in the global pair. The proofs use support and species, not undefined pointwise evaluations of distributions.

scope. ${\mathfrak{C}}_{\mathrm{part}}$ is unchanged. The conditional characteristic ambiguity proposition adds no member to it and selects no reference extension.

scope. XIV-C supplies the characteristic geometry and its retained monomial diagnostic. XIII-C supplies the inventory, partial contacts, and primitive extensions. XIII-B supplies the tree and retarded convention. XII-E supplies the classical completion and tree kernels. XII-C supplies the source representative. XIV-A and XIV-B are not analytic inputs.

. No successful pdflatex execution, proof-assistant verification, independently executed symbolic computation, independent referee review, or journal acceptance is asserted. All new algebraic calculations are displayed. No numerical residue has been computed.

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. Retained configuration, characteristic geometry, contact rigidity, and the exact 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. Eleven branches, 48 labelled matchings, partial contacts, primitive extensions, and the unresolved joint problem.

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, properness, retarded convention, and ordered-support correction.

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, affine slot identity, source-slot sign, and admitted 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 density representative and actual source.

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. Sections 5–6: scaling degree and extension along submanifolds.

Dang

N. V. Dang, “The extension of distributions on manifolds, a microlocal approach,” Annales Henri Poincaré 17 (2016), 819–859, arXiv:1412.2808. Conormal landing conditions, extension wavefront bounds, and characteristic counterterm structure.

FR

K. Fredenhagen and K. Rejzner, Perturbative algebraic quantum field theory, arXiv:1208.1428v2. Distribution products, pullbacks, and functional domains.

Hormander

L. Hörmander, The Analysis of Linear Partial Differential Operators I, second edition, Springer, 1990. Wavefront sets, distribution products, pullbacks, and pushforward criteria.