Landmark XIV-E Joint symbol of the six-branch retained germ, or one named characteristic prescription Exact operator reduction and the non-clean product-phase stop

We continue XIV-D's unresolved characteristic composition problem for the unchanged binary pair. The first calculation concerns the complete six-branch smooth-slot expression, not the temporal component of one branch.

An exact regrouping removes the explicit second derivative of the scalar test from the combined $B_4+B_7$ expression and retains its remaining first-derivative terms. The coordinate-response branches $B_5+B_6+B_8$ combine into one divergence with all source coefficients present. The scalar-response branch $B_9$ is expressed through the complete advanced observation root. These are identities of existing tree kernels, not definitions of their joint contraction with the cross-lines.

We then identify the ambient spaces relevant to a common symbol calculation. The copied characteristic stratum has codimension four in the three-vertex space. Its intersection with the residual metric attachment diagonal has codimension eight in the common sixteen-dimensional space. The elementary sum of the three selected line phases and the attachment phase has differential rank six, rather than seven, while its critical set has codimension eight. It is neither a nondegenerate nor a clean phase. An exact quadratic identity exhibits the two transverse directions invisible to the linearized light-cone equations.

These results do not provide a different common phase, a joint symbol of the complete coefficient, or lower-order microlocal control. No complete cancellation or failure-of-cancellation theorem is obtained, and no characteristic prescription is constructed. The exact ending is Unresolved at E1, within D2/J2. J3–J5 and the quantum Ward-defect question remain unopened.

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

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

and first missing step{ Frozen: the pair, six-branch germ, Hadamard matrix, inverse coefficient $-1$, switches, width, and ${\mathfrak{C}}_{\mathrm{part}}$.

Proved here: an exact regrouping of the entire retained smooth-slot expression; its complete scalar-response root formula; the distinction between the two characteristic lifts; and failure of the elementary sum-of-line-phases construction to be clean.

Not proved: a common principal-symbol calculus for the grouped characteristic contraction, cancellation of its complete symbol, control of its lower-order terms, or a characteristic prescription.

First stop: E1, within D2/J2. The missing step is the common microlocal composition theorem. The phase calculation below excludes one proposed shortcut, not every possible construction.

No ${R_1}^{(1)}$, Ward defect, new contact class, or characteristic subtraction is released.}

The three endings apply to the retained characteristic construction:

1. Constructed: cancellation of the complete retained germ, with the necessary lower-order control, or one prescription with proved degree, normalization, ambiguity, and wavefront control. J3 may then be opened only for that distribution.

2. Proved obstructed: a demonstrated incompatibility excludes every map in a class fixed before the test.

3. Unresolved: the joint symbol, a required remainder estimate, or a prescription theorem is missing.

A nonzero $B_6$ numerator is not the second ending. Failure of a sufficient product test is not a nonexistence theorem. A successful local characteristic prescription would not, by itself, admit ${R_1}^{(1)}$ before J3–J5.

The frozen observable is

The inverse and ambiguity conventions are

Let $A,B,Y$ be the pairwise separated neighborhoods of $u_0,x_0,y_0$, and let $\overline{Z}\subset A$ be the residual metric neighborhood. In the common variables

the mixed observable kernel satisfies

The other branches remain in the global pair and vanish only on the proved germ or species channel. In particular, $K_8$ and $K_9$ retain both gravitational-response legs.

The kernels $K_6,K_8,K_9$ are supported on the observable metric diagonal $a=z$. The retained mixed germ has 24 potentially surviving labelled matchings.

The leading fiber identity

is a branch diagnostic. It does not decide the complete sum.

No joint symbol or characteristic prescription was constructed. The first missing step was D2 XIVD.

The retained triple is

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

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

It is smooth of codimension four. For

the differentials have rank two on ${\mathcal{N}}^{\circ }$ and satisfy

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

Collision-supported changes vanish on the retained collision-free neighborhood, while the original attachment variables are retained XIVC,XIVD.

Neither copied geometric proposition nor the copied $B_6$ fiber identity is re-proved.

XIV-D established the following necessary distinctions:

1. an extension with wavefront set contained only in $N^{\ast }N$ on a whole neighborhood would require its off-$N$ reference distribution to be smooth;

2. in a codimension-four extension problem, degree at least four and allowance of the full conormal bundle permit tangential multiples of ${\delta }^{(4)}(r)$ unless an additional normalization or uniqueness result removes them;

3. on a collision-free neighborhood, such nonzero characteristic-supported differences are not members of ${\mathfrak{C}}_{\mathrm{part}}$.

These constraints do not assert the existence or the actual degree of the selected characteristic distribution XIVD.

XIII-E's common-frequency family is recorded as an attempt: its admissible-set convergence does not establish convergence through this stratum XIIIE. No such limit is imported.

XIV-A and XIV-B are not analytic inputs. The value $-8I$ is not a subtraction, and a classical breaking density is not an anomaly of the missing graph distribution.

Use

All external data remain those fixed in XIV-C and XIV-D. No observation, switch, or width is chosen after a symbol calculation.

The carrier remains

The frozen time-ordered line is

Metric–scalar blocks vanish. The numerical species factors and the inverse coefficient $-1$ are unchanged.

The interaction is

where

The scalar-sourced slot of $\mathcal{O}$ is included once. It is not removed because its scalar–scalar channel vanishes on the retained output-location patch.

The complete formal two-line expression remains

On the retained mixed germ it is

The factor $i$ includes the two mixed slot orderings of XIV-D.

This difference stays grouped. Formula (eq:target) specifies the composition to be constructed; it does not assert its existence on the characteristic set.

The class ${\mathfrak{C}}_{\mathrm{part}}$ is unchanged. Its defined graphwise changes are collision-supported contacts with the inherited spectator, external-weight, normal-order, support, and normalization conditions. It controls allowed differences of constructions, not a reference extension across the separated characteristic set.

There is no cutoff of ${\mathcal{N}}^{\circ }$, triangle residue, new characteristic contact class, antifield differential, spurion, replacement Hadamard matrix, or sharp-width limit. The auxiliary neighborhoods describe kernel restrictions, not a cutoff of the characteristic configuration.

No universal class of successful characteristic prescriptions is asserted to have been constructed. Thus no universal obstruction is tested against an unspecified class.

The following calculation concerns all six retained branches. It is performed before discussing their principal symbols.

For a smooth metric tensor $r$, put

Define

Let $G$ be the residual metric direction, $h$ the contracted metric direction, and $\psi$ the contracted scalar direction. These are smooth directions of the classical functional, not values of singular line distributions. Write

The source ${\mathcal{J}}_{\mathrm{D}}$ is the unchanged source of XII-C:

The unintegrated curvature representative, and hence every coefficient of ${\mathcal{J}}_{\mathrm{D}}^{\prime }[G]h$, is fixed as in XII-C. No source coefficient is set to zero by the following notation XIIC,XIIE.

The density derivative is

The copied mixed branch derivatives are

They include all metric-slot choices, source derivatives, and both response legs. Set

Define the smooth coefficient

and the complete tensor

For all smooth directions for which the inherited slab response formulas are used,

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

Proof. Symmetry of ${\partial }_{\mu }{\partial }_{\nu }\psi$ gives

Compact integration by parts in the second term of ${\mathcal{M}}_4$ yields

The last integral cancels ${\mathcal{M}}_7$ exactly. The first term of ${\mathcal{M}}_4$ and the remaining integral are precisely

Next, the definition of $U$ expresses ${\mathcal{M}}_5$ as the retarded inverse of the divergence of the first two terms in (eq:Btensor). The third term is ${\mathcal{M}}_6$. Since

the fourth term gives ${\mathcal{M}}_8$, with its inherited sign. Linearity of ${E_0}^{\mathrm{R}}$ combines these three branches.

Finally ${\mathcal{M}}_9$ is unchanged. Adding the three groups proves (eq:fullregroup).

The theorem proves an exact cancellation of the explicit second-scalar-test-derivative terms in $B_4+B_7$. It does not prove cancellation of those two branches, much less of all six.

It also does not establish that either branch is of lower microlocal order in a common characteristic calculus. That conclusion would require the calculus and order comparison still missing at E1.

Let

The inherited smooth advanced operator makes $F$ smooth.

The last term of (eq:fullregroup) is

Both the derivative and mass terms are retained.

Proof. Transposition of the inherited zero-data Green operator gives

Compactness of $g$ permits integration by parts in the divergence. This gives (eq:B9root). No equation for $\psi$ is assumed.

The mass term is not deleted merely because it has fewer explicit scalar derivatives. A prospective remainder estimate must control it, together with every other lower-order term.

For an arbitrary $G\in {C_c}^{\infty }(Z;{\mathrm{Sym}}^2{T}^{\ast }M)$, pair the residual metric slot of $K_{\mathrm{ret}}$ against $G$. Denote the resulting attachment kernel by $L_G$:

Testing the residual slot is a distributional pairing. It neither changes the observable nor sets $z=a$ in branches where that equality is absent.

Formula (eq:fullregroup), with (eq:B9root), gives the same attachment distribution $L_G$ as the sum ${{\sum }_{j=4}}^9⟨K_j,G⟩$.

For every such $G$, this kernel is compact and satisfies

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

Proof. The exact smooth-slot identity proves equality on tensor-product tests $h(a)\psi (b)$. Finite sums of these tests determine a distributional kernel, giving the asserted equality.

Each inherited $K_j$ is a compact tree kernel with the covector sum-zero bound in $(a,b,z)$. Multiplication by the smooth residual test and pushforward over its compactly supported variable set the residual covector to zero. The resulting covectors therefore satisfy $\xi +\eta =0$. Smooth testing and the finite sum preserve compactness Hormander (Section 8.2).

The sum-zero bound does not prove that multiplying $L_G$ by the two cross-lines is allowed. Their endpoint covectors can cancel the two nonzero attachment covectors. Nor does testing each fixed $G$ supply the uniform estimates needed to reconstruct a joint characteristic distribution in all four common variables.

Let

in the coordinate order $(a,b,z,y)$, with dimension sixteen. Let

Define

The overline in ${\overline{\mathcal{N}}}_{\mathrm{att}}$ is notation for this full lift, not a topological closure.

Locally at the retained point,

The intersection defining ${\tilde{\mathcal{N}}}_{\mathrm{att}}$ is transverse.

Proof. The map $P$ is a submersion, so the copied codimension-four stratum lifts to a codimension-four submanifold with $z$ free.

The equation $a-z=0$ has four independent components. Every conormal of ${\overline{\mathcal{N}}}_{\mathrm{att}}$ has zero $z$ component, whereas a conormal to ${\mathcal{D}}_{\mathrm{obs}}$ has opposite $a$ and $z$ components. A covector belonging to both conormal spaces must consequently have its $z$, then its $a$, component zero, and hence is zero. This proves transversality and the codimension-eight conclusion.

The codimension-four characteristic ambiguity statement copied from XIV-D refers to a specified codimension-four extension problem. It must not be transferred to ${\tilde{\mathcal{N}}}_{\mathrm{att}}$ with the same delta order or degree threshold without identifying the ambient space.

For example, the numerical three-vertex graph may be extended first and its observable attachment added later. That is not automatically the same operation as extending a common kernel already containing derivatives of $\delta (a-z)$. Their compatibility requires a theorem.

Consider the smooth homogeneous candidate

where

Its signs encode the copied oriented conormals. A simultaneous reversal of the Fourier convention reverses the whole expression and changes none of the rank conclusions.

Formula (eq:phase) is the elementary candidate obtained by adding the selected three line phases and the metric attachment phase. It is not asserted to represent the full six-branch sum. Its validity must be tested before amplitudes can be compared using it.

For a nondegenerate phase, the differentials of its auxiliary critical equations are independent on the critical set. This is the standard local criterion for the principal-symbol construction for Lagrangian distributions Wunsch (Section 8.2).

For the present calculation, call a candidate clean if its auxiliary critical set is smooth and its tangent space equals the kernel of the differential of those critical equations. This is the tangent-space requirement needed for clean critical-set reduction. It is weaker than full independence but still fails below.

Shrink the original neighborhoods, if necessary, without deleting the retained configuration, so that

throughout them.

On these neighborhoods,

Consequently the auxiliary critical set of (eq:phase) is

Proof. Set $\xi =b-a$ and ${\eta }^{\prime }=y-a$. The equations imply

The time-order inequalities make $\xi$ and ${\eta }^{\prime }$ nonzero future null vectors. Write

Their scalar product is $\alpha \beta (1-\mathbf{n}\cdot \mathbf{m})$, so it vanishes exactly when $\mathbf{n}=\mathbf{m}$. The second time inequality gives $\alpha <\beta$. This is the copied parameterization of ${\mathcal{N}}^{\circ }$.

The auxiliary derivatives of $\Phi$ are

Their simultaneous vanishing gives the stated critical set.

This identifies the zero locus; it does not re-prove the copied smoothness, dimension, or rank of ${\mathcal{N}}^{\circ }$.

At the retained characteristic configuration, the seven auxiliary critical equations of (eq:phase) have differential rank six. Their critical set has codimension eight in the twenty-three-dimensional base/frequency space.

Thus:

1. the candidate is not nondegenerate;

2. it is not clean under the stated tangent-space criterion;

3. on the ray

   $$t_R=\frac{c}{2\beta },t_1=\frac{c}{2\alpha },t_2=\frac{c}{2(\beta -\alpha )},\kappa =0,c>0,$$

   its entire differential vanishes.

It cannot be used as the ordinary common phase defining a principal symbol of (eq:target).

Proof. The three cone differentials have copied rank two. The four components of $\mathrm{d}(a-z)$ are independent of them: looking at their $z$ components forces every coefficient in a proposed linear relation to vanish. The total rank is therefore six.

By Proposition prop:lifts and Lemma lem:zerolocus, the critical set has dimension

inside a space of dimension $16+7=23$. Its codimension is eight.

The kernel of the differential of the critical equations has dimension $23-6=17$, whereas the tangent space of $C_{\Phi }$ has dimension fifteen. They are unequal. Independence and the clean tangent-space condition both fail.

On the displayed frequency ray, the base differential is

by the copied positive relation. The auxiliary differential also vanishes on $C_{\Phi }$. Thus $\mathrm{d}\Phi =0$ there even though the frequency variables are nonzero.

The theorem excludes this elementary phase shortcut. It does not prove that every multiphase, resolved, boundary-value, or other characteristic construction fails.

In particular, an amplitude cancellation could change the relevant representation. Establishing that cancellation still requires the complete amplitude calculation and its remainder estimates; it cannot be inferred from the geometric rank defect.

The rank calculation has a concrete second-order form.

Fix one point of ${\mathcal{N}}^{\circ }$ with parameters $\alpha ,\beta ,\mathbf{n}$. Choose an orthonormal spatial basis transverse to $\mathbf{n}$. Let $v,w\in {\mathbb{R}}^2$ be transverse spatial displacements, and set

Here $(0,v)$ and $(0,w)$ have no time or longitudinal spatial component.

On this affine transverse slice,

There are two independent relative transverse components in $\alpha v-\beta w$. They are not detected by the first differentials of the three cone equations at the base point.

Proof. Orthogonality to $\mathbf{n}$ and nullity of ${\ell }_{\mathbf{n}}$ give

Therefore the left side of (eq:quadratic) equals

Expanding the last square gives exactly the right side.

All three expressions have zero first derivative at $v=w=0$. The linear map $(v,w)\mapsto \alpha v-\beta w$ has rank two, which identifies the two relative components invisible to that linearization.

Equation (eq:quadratic) is a geometric identity, not a scaling degree of a distribution. Its quadratic behavior does not determine the transverse degree, convergence, or wavefront set of the six-branch product.

To use the notation

for (eq:target), a successor must supply a common microlocal representation and composition theorem for the grouped coefficient. In particular, it must identify:

1. the relevant canonical or characteristic relation in the chosen ambient space;

2. the amplitude orders and the common symbol bundle;

3. the maps accounting for additional internal vertices in $K_5,K_8,K_9$;

4. the observable attachment diagonal and its derivatives;

5. the complete tensor coefficients and the prescribed grouping of the cross-line difference;

6. a remainder estimate that controls every term not represented by the proposed leading section.

The inherited wavefront inclusions do not themselves supply these data. The ordinary principal-symbol construction requires more than a wavefront-set inclusion Wunsch (Section 8).

Theorem thm:regroup supplies an exact common operator expression before characteristic composition. Theorem thm:nonclean proves that simply adding the selected line phases does not supply the missing calculus.

Neither theorem assigns the requested joint principal symbol.

A vanishing numerator in a particular presentation would have to be recorded with its remaining terms. It would not authorize deleting a branch from this table.

Let $U_{\mathrm{can}}\subset W$ denote an open domain on which the complete specified graph products satisfy the inherited product criteria, with their smooth weights and attachment maps. The grouped expression there is the canonical reference $t^{\mathrm{can}}$.

No equality

is asserted. Establishing a reference on the whole complement of a proposed extension locus is itself part of the missing composition problem.

Consequently an extension of “the off-characteristic product” must not be invoked before that product and its precise domain have been named.

The proved results establish neither of the decisive E1 conclusions:

1. a zero joint-symbol section for the complete six-branch coefficient with controlled remainder;

2. a nonzero joint-symbol section of that complete coefficient.

They also do not demonstrate an incompatibility excluding every prescription under the frozen contract.

Proof. Theorem thm:regroup is an exact identity for the already defined smooth-slot tree kernels. It does not compose them with the cross-lines on the characteristic set.

Theorem thm:nonclean rules out the elementary phase candidate, not all possible representations. Proposition prop:quadratic identifies its geometric degeneracy but supplies no distributional amplitude estimate.

The copied $Q\lambda$ identity concerns only $B_6$. None of these statements supplies a common joint-symbol section or a remainder theorem for (eq:target). Nor does any statement quantify over all prescriptions in a previously fixed candidate class.

The first unresolved step is therefore E1, within the inherited D2/J2 gate. It is not E2 on the premise that cancellation has failed: failure of complete cancellation has not been proved.

No prescription is constructed in this section. Any future map must name:

1. its ambient graph or common-kernel space;

2. a distribution $t^{\circ }$ on a proved reference domain;

3. the exact closed characteristic locus $N$ across which it is extended;

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

5. a finite uniform transverse degree in specified normal coordinates;

6. a closed conic wavefront bound retaining the inherited off-characteristic singularities;

7. a normalization selecting a map, or a uniqueness theorem under the proved degree and wavefront hypotheses.

The numerical stratum ${\mathcal{N}}^{\circ }$, the full lift ${\overline{\mathcal{N}}}_{\mathrm{att}}$, and the attachment intersection ${\tilde{\mathcal{N}}}_{\mathrm{att}}$ are not interchangeable choices of $N$. The chosen stage must also account for other internal compositions present in the complete sum.

For a stated codimension-$q$ locus, use normal coordinates $(s,r)$ with $r\in {\mathbb{R}}^q$. A finite transverse degree must be proved uniformly over compact tangential sets with the necessary microlocal seminorm control.

When their hypotheses hold, submanifold extension theorems distinguish the degree-below-codimension uniqueness case from the case requiring normal-jet normalization BF (Section 6). Those hypotheses are not supplied by a superficial derivative count.

A prospective minimal wavefront estimate has the form

in the stated ambient space. Such an estimate requires the appropriate microlocal extension hypotheses, including control of how the off-$N$ wavefront approaches $N$ Dang (Sections 5–6).

The copied restriction warning remains in force: $\mathrm{WF}(t)\subset N^{\ast }N$ alone cannot retain a nonsmooth reference outside $N$.

In a common kernel already containing $\delta (a-z)$ or its derivatives, the inherited attachment singularities must also be accounted for. No estimate for adding those attachments after characteristic extension is asserted here.

For a genuine codimension-four problem, the copied ambiguity diagnostic applies conditionally when the degree is at least four and the full characteristic conormal is allowed:

can agree with the reference off the stratum while preserving the stated degree allowance and conormal bound.

On the retained collision-free neighborhood, a nonzero such difference is not in ${\mathfrak{C}}_{\mathrm{part}}$. Hence a future prescription must not identify it as an inherited collision contact.

For a different extension stage, the relevant codimension and supported-distribution structure must be recomputed. For example, ${\tilde{\mathcal{N}}}_{\mathrm{att}}$ has codimension eight, while an amplitude on the three-vertex numerical space still faces codimension four. No actual degree is assigned to either unresolved distribution.

Under suitable conormal wavefront hypotheses, supported extension differences have finite normal delta-derivative expansions with smooth tangential coefficients Dang (Section 8). This is an ambiguity theorem for an existing extension problem, not a construction of the selected reference or extension.

No characteristic-supported term is added here. No normalization constant is selected, and ${\mathfrak{C}}_{\mathrm{part}}$ is not enlarged.

XIII-E's family can be continued only as that named attempt. To advance it on the present germ, one would need a new convergence theorem on tests meeting the characteristic set, with the complete branch sum and uniform wavefront estimates retained.

Finite-regulator admission, convergence on canonical subdomains, and independence of auxiliary compact localization do not give that theorem XIIIE. None is claimed here.

No cutoff removing the characteristic stratum and no restoring limit is introduced as an alternative proof.

J3 opens only for a distribution supplied by a successful E1 or E2 construction. The copied parallel-line degree eight is a collision degree, not a degree along ${\mathcal{N}}^{\circ }$.

The exact integrations by parts in Theorem thm:regroup do not supply uniform transverse bounds for the unresolved characteristic product, larger collision strata, or their intersections.

J4 remains closed: no characteristic prescription has been composed with the singular spectators and compatible partial contacts.

J5 remains closed: inherited properness applies to an existing distribution. It does not construct (eq:target) or prove the required output wavefront bound.

The intermediate attachment-kernel bound of Proposition prop:kernels is not the J5 theorem for the degree-two residual-field output.

No $s_{\mathrm{ext}}$, $s_{\Psi }$, or $s_0$ is applied to ${R_1}^{(1)}$. No binary Ward defect is formed. The already constructed tree coefficient is not substituted for the missing quantum coefficient.

The classical breaking density from XIV-A and the relative complex of XIV-B remain separate from this analytic problem. Neither the value $-8I$ nor a classical evaluation identity normalizes the graph.

For the frozen contract:

1. The complete six-branch smooth-slot expression equals (eq:fullregroup), with its full scalar-response term given by (eq:B9root).

2. The characteristic lift with the residual metric free has codimension four, while its intersection with $a=z$ has codimension eight.

3. The elementary candidate phase (eq:phase) has rank-six auxiliary equations and a codimension-eight critical set. It is neither nondegenerate nor clean.

4. The transverse identity (eq:quadratic) exhibits the two relative directions missed by the first-order cone equations.

These conclusions establish an exact operator reduction and a specific phase-construction failure. They do not establish the joint characteristic distribution.

Proof. Theorem thm:regroup and Proposition prop:root give (i). Proposition prop:lifts gives (ii). Lemma lem:zerolocus and Theorem thm:nonclean give (iii). Proposition prop:quadratic gives (iv). The limitation follows from Proposition prop:limits.

ending{ Unresolved at E1, within D2/J2.

The first missing theorem is a controlled common microlocal composition for the grouped six-branch expression. Neither full cancellation nor failure of full cancellation has been established. No characteristic prescription is constructed, and no universal obstruction is asserted.}

The full retained sum has a shorter exact precomposition form: $B_4+B_7$ retain only first scalar-test derivatives after the proved cancellation, $B_5+B_6+B_8$ appear in one complete coordinate-response divergence, and $B_9$ has an explicit complete root formula.

The geometric obstruction to the simplest symbol shortcut is also precise. The sum of the selected line phases is not an ordinary clean common phase. Two missing linear constraints appear quadratically, and the observable attachment changes the ambient codimension.

These results identify work a common characteristic calculus must perform. They are not that calculus.

A successor receives:

1. the unchanged pair and six-branch germ;

2. the exact regrouping (eq:fullregroup);

3. the complete root formula (eq:B9root);

4. the intermediate tested-kernel equality and balanced attachment bound;

5. the two characteristic lifts and their codimensions;

6. the non-clean phase theorem and quadratic defect identity;

7. the unchanged degree, ambiguity, and wavefront requirements;

8. the exact E1 stop.

To discharge the retained D2/J2 problem, it must construct a common representation with a full symbol and controlled remainder, proving cancellation of the complete grouped coefficient, or supply a characteristic prescription with its actual degree, normalization, ambiguity, and wavefront theorem.

It receives neither a nonzero full symbol section nor permission to infer one from $Q\lambda$. It receives no ${R_1}^{(1)}$.

An exact smooth-slot cancellation is not the full characteristic cancellation theorem. A branch numerator is not a joint symbol. A collision contact is not a characteristic extension. An undefined quantum coefficient is not an anomaly.

scope. The new proofs concern the entire retained smooth-slot regrouping, its root representation, the characteristic ambient-space distinction, and the elementary phase audit. No common characteristic principal symbol is asserted.

scope. XIV-D supplies the six-branch germ and derivative ledger. XIV-C supplies the rank-two characteristic geometry. XII-C fixes the actual source representative. XII-E supplies the classical completion and response formulas. XIII-E is a recorded regulator attempt, not a completed characteristic prescription.

branch deletion. The exact cancellation of explicit second scalar-test derivatives does not delete $B_4$ or $B_7$. Both gravitational-response legs remain present. Terms with fewer explicit derivatives are retained and are not declared microlocally negligible.

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

scope. No renormalized quantum coefficient, Ward defect, antifield construction, interacting differential, state, loop value, 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 mathematical calculations claimed in this release are the displayed analytic proofs.

XIVD

N. Cheung, Landmark XIV-D: Cancellation or prescription on the retained noncoincident characteristic stratum. Supplied manuscript XIV-D.tex. Complete derivative ledger, six-branch retained germ, fiber diagnostic, and unresolved D2 composition problem.

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 characteristic geometry, conormal relation, collision-contact rigidity, and the J2 stop.

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. Common-frequency attempt, admissible-set convergence, localization independence, and unresolved characteristic limit.

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, response roots, and admitted classical 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 unintegrated cubic density representative and actual weighted Euler 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. Section 6: uniform microlocal scaling 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 estimates, and supported extension ambiguities.

Wunsch

J. Wunsch, Microlocal analysis and evolution equations: Lecture notes from the 2008 CMI/ETH Summer School, arXiv:0812.3181. Section 8: conormal and Lagrangian distributions, nondegenerate phases, and principal-symbol calculus.

Hormander

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