Landmark XIV-H Internal kernel relations on the operator route The two gravitational-response legs, the internal characteristic-diagonal transition, and the first missing amplitude theorem

We continue XIV-G's operator route for the unchanged six-branch retained germ. The residual metric variable is retained throughout. No common product phase, excess construction, factorwise cross-line retry, or characteristic prescription is introduced.

The untested local source is written as a distributional incidence on its two field slots. This gives exact internal kernel maps for the coordinate and scalar gravitational-response legs. On the retained separated-endpoint patch, the coordinate leg reduces to a derivative of the existing massless retarded convolution. An explicit calculation evaluates that convolution. Both response legs consequently admit finite conormal representations with their source coefficients and scalar-root mass term retained.

The two placements in the remaining internal branch are expressed as submersion pullbacks of tensor products of the existing retarded kernels. Their regular separated-cone and noncharacteristic-diagonal symbol charts are identified. At the internal characteristic diagonal, however, the diagonal piece and the limiting flow-out piece are distinct Lagrangians. Their intersection is clean, but their union is not one smooth Lagrangian relation. Sectorwise symbols therefore cannot be identified as a single ordinary symbol there.

A uniform internal amplitude construction across that transition is not supplied. The first missing gate is H2, rather than H3: the existence of the tree kernel does not discharge the stronger internal-symbol requirement. No common characteristic operator representation of the complete grouped coefficient, joint section, remainder theorem, or quantum output is released. The target remains Unresolved within G2/F2/E1/D2/J2.

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

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

release and first stop{ Constructed here: the untested source incidence; exact internal kernel formulas for $K_5,K_8,K_9$; regular conormal representations for $K_8,K_9$; and regular symbol charts and transition geometry for the two $K_5$ placements.

Not constructed: a uniform internal amplitude theorem for the $K_5$ characteristic-diagonal transition. Its defined tensor-product kernel is not promoted to such a theorem.

First stop: Unresolved at H2. Declaring H2 complete and then reporting the same missing theorem at H3 would misstate the dependency.

The clean intersection proved below is not called a non-clean matching. It is not the tangent defect of XIV-F.

No common characteristic representation, joint section, transverse degree, prescription, ${R_1}^{(1)}$, or Ward defect is released.}

Only XIV-G, XIV-F, XIV-E, XIV-D, XIV-C, XIII-C, XIII-B, XII-E, and XII-C are analytic manuscript inputs. Their handoffs are used with their stated scope. No incidence inventory, predecessor phase theorem, or clean-excess theorem is re-proved.

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.

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

For a smooth compact residual test $G$,

Its local polynomial differential symbol is not the symbol of either composed gravitational-response leg.

Smooth residual testing can hide attachment-diagonal wavefront directions. The compact balanced-kernel example in XIV-G proves that limitation, not a model of the characteristic singularity of $K_{\mathrm{ret}}$.

The ordinary final product test for the two individually defined numerical factors in XIV-G fails at the retained opposite covectors. Their conormal orders ${\mu }_U=-\frac{1}{2}$ and ${\mu }_V=-1$ belong only to those factors.

No complete internal symbol relations or common symbol bundle were supplied XIVG.

Neither factor order is used as a transverse degree below. The factorwise audit is not repeated.

On the regular six-constraint stage,

The tangent-defect map is

Its quotient has rank two. It is not an established excess fiber.

The selected matching has actual dimension fifteen and linearized matching dimension seventeen. The covector-sum map satisfies

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

The direct clean-excess-two identification was excluded. The grouped target remained Unresolved at F2 XIVF.

Neither $S_6$, the matching manifold, nor $J$ is used as an internal composition relation in this paper.

With

the complete retained smooth-slot expression is

where

and

All six branches are retained.

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

The derivative and mass terms are both retained.

For each fixed smooth compact $G$, the tested attachment kernel is compact and satisfies the balanced attachment bound. This does not give the untested joint symbol or its uniform estimates XIVE.

The full observable is

On the retained mixed patch,

The kernels $K_6,K_8,K_9$ are supported on the actual local source or metric attachment incidence $a=z$.

The relation

is limited to the leading differential numerator of $B_6$. It is not a section of the complete sum XIVD.

In the common space

the test locus is

where ${\ell }_{\mathbf{n}}=(1,\mathbf{n})$ and $\mathbf{n}\in S^2$. It has codimension eight in the sixteen-dimensional space.

Writing

the copied positive relation is

It is diagnostic input data, not an output symbol relation.

Defined collision contacts vanish on the retained collision-free attachment neighborhood XIVC,XIVD.

The neighborhoods $A,B,Y$ have pairwise disjoint closures, $\overline{Z}\subset A$, and their closures lie in the open slab. They are small enough that $b^0>a^0>\epsilon$ on $B\times A$. The retained configuration is not removed.

XIII-C supplies four unresolved obligations:

1. a characteristic prescription or complete-pair cancellation;

2. uniform transverse estimates at the relevant collision strata and their intersections;

3. compatible partial contacts with their singular spectators;

4. the output wavefront theorem after proper pushforward.

The parallel temporal kinetic primitive has collision scaling degree eight in codimension four XIIIC.

XIII-B supplies the admitted selected tree datum

the retarded convention

and the corrected strictly ordered values: zero or $(i/\hslash )[\mathcal{O},{\mathcal{V}}_g{]}_{{\star }_H}$.

Properness is support control for an existing distribution, not a construction of a missing product XIIIB.

The response and first-completion formulas are

The existing completion kernels are tree kernels XIIE.

The actual source is

Its fixed unintegrated cubic representative is

The selected weighting is at cubic order only XIIC.

Every source coefficient below is that of (eq:actualsource). None is replaced by a simply source-switched expression.

XIV-A and XIV-B are not analytic inputs. The value $-8I$ is not a normalization. XIII-E and a resolved paired construction remain unentered.

The carrier, pair, inverse, switches, width, and ambiguity class remain

The inverse coefficient is $-1$.

The target is still

The difference stays grouped. No mixed $H$–$G_{\mathrm{act}}$ product is declared.

The selected class uses the original common space $W$, with $z$ retained. It permits:

1. named tensor products, submersion pullbacks, differential operators, and support-proper pushforwards of the existing tree kernels;

2. individual regular conormal or canonical-relation charts, with their orders and amplitudes specified;

3. a composition across intersecting relation charts only after its matching and uniform amplitude hypotheses have been proved.

A wavefront inclusion, a union of relation charts, or an exact distributional tensor-product formula does not by itself supply the amplitude theorem in (iii).

There is no common product phase and no identification with XIV-F's excluded matching. The two gravitational-response legs are kept separate until their own kernel relations have been written.

The distinction in item (iii) does not prohibit a future generalized operator theorem. It requires that theorem to be stated and proved rather than supplied by terminology.

The endings concern only the internal relations and common operator representation:

1. Constructed: the necessary internal relation and amplitude maps exist, and one common representation is justified on a stated domain.

2. Proved obstructed: a demonstrated incompatibility excludes every representation in Definition def:class.

3. Unresolved: an internal amplitude map, matching hypothesis, or common representation remains missing.

A constructed representation would not by itself discharge the joint-section, remainder, or quantum-admission gates.

H2 is a prerequisite, not a label that may be passed by recording an existing tree kernel. If its required internal amplitude theorem is missing, the paper stops there. H3 cannot be declared reached by renaming that same gap.

No characteristic prescription is opened. There is no retry of the factorwise product of XIV-G.

Use $\eta =\mathrm{diag}(1,-1,-1,-1)$ and $\square ={{\partial }_t}^2-{\Delta }_{\mathbf{x}}$. Let $R(x-v)$ denote the kernel of ${E_0}^{\mathrm{R}}$. Define constant fiber maps by

where $I$ ranges over symmetric metric components. All sums over $I$ use the inherited tensor pairing.

The coordinates $(a,z)$ are the two metric field slots: $h_I(a)$ is the contracted direction and $G_J(z)$ the residual direction. They are not replaced by a smooth residual test in the kernel formulas.

The actual source is a finite sum of local quadratic monomials. Its full derivative can therefore be written

where all symmetrization factors are absorbed in $q$. The smooth coefficients are the fixed coefficients of (eq:actualsource), including the ramp derivatives.

Define the untested incidence kernel

The product of deltas in (eq:Qincidence) is defined. Its support is $v=a=z$, and its underlying conormal relation is

The source kernel has conormal order at most three in the twelve-dimensional $(v,a,z)$ space. Its amplitude is the finite polynomial obtained by applying the displayed derivatives to the two independent diagonal deltas.

Pairing its $z$ slot against a smooth $G$ recovers ${\mathsf{Q}}_G$, but this testing is not used to define the untested relation.

Proof. The map

is a submersion onto ${\mathbb{R}}^8$: variations of $z$ and $a$ can prescribe its two outputs independently. The two-delta product is its pullback of the delta at the origin in ${\mathbb{R}}^8$.

Its conormal covectors have the stated zero-sum form. The defining submanifold has codimension eight in dimension twelve. With normal amplitude order at most two, the conormal order is

Lower derivative monomials belong to the corresponding lower filtered orders.

The pairing formula follows directly from the distributional evaluation of the derivatives of the deltas. It reproduces (eq:sourcepolynomial).

Only this actual incidence forces $v=a=z$. No internal vertex is assigned $u_0$, $y_0$, or $x_0$ by hand.

For operator notation, the incoming $(a,z)$ covectors may be sign-reversed in the usual primed relation. Below we use unprimed kernel covectors in $T^{\ast }W$.

Before composing either gravitational leg, set

with the contracted metric index understood in the source kernel.

This is the untested kernel of $r_h(b)=\mathcal{P}{E}_0^{\mathrm{R}}({\mathcal{J}}_{\mathrm{D}}^{\prime }[G]h)(b)$. It retains the residual variable.

On the separated patch $b\in B$, $a\in A$, the endpoints $b,a$ never coincide. Moving the external derivatives in (eq:Qincidence) outside the integral writes (eq:responsekernel) as a finite sum of derivatives, of total order at most two, of

The differentiated coefficients are included in this identity; they are not omitted remainder terms.

The coordinate leg has two internal variables: $v$ is the actual source vertex and $s$ the intermediate coordinate-response vertex. Its scalar attachment is $b$. Its untested kernel is

This is the kernel of

after the scalar-slot derivative is transferred to the kernel.

The original maps are therefore:

The first is the regular local incidence of Proposition prop:sourceincidence. The other two are the existing retarded maps.

In (eq:K8map), replacing the intermediate factor ${\sigma }_{\epsilon }(s)$ by one leaves the restricted kernel unchanged.

Proof. The source incidence enforces $v=a=z$, with $a^0>\epsilon$. The support of the two retarded kernels enforces

Hence $s^0\ge a^0>\epsilon$, where ${\sigma }_{\epsilon }(s)=1$ and all its derivatives vanish.

Differentiation does not enlarge distributional support. Thus every term containing ${\sigma }_{\epsilon }(s)-1$ or a derivative of that factor has empty support on this patch.

This lemma uses an identified internal vertex and proved causal support. It does not use the locations of the retained external triple as substitutes for internal variables.

The source coefficients $q$ remain those of the actual source. No simply weighted source is substituted.

Define $B=R\ast R$. For fixed compact endpoint sets the intermediate integration is support-proper, since its support lies in compact causal diamonds.

With the normalization $\square {E_0}^{\mathrm{R}}=\mathrm{id}$,

The second equality is an equality of the already defined retarded convolution distributions.

Proof. The first formula is the retarded massless wave kernel in the declared normalization.

Let $\rho =|\mathbf{\xi }|$ and take the Fourier transform only in the spatial variables. The first kernel has transform

Its causal time convolution is

At $\rho =0$ the continuous value is $\Theta (t)t^3/6$.

The spatial Fourier transform of $\Theta (t)1_{\{|\mathbf{x}|\le t\}}/(8\pi )$ is the same expression, by radial integration over the ball of radius $t$. The formulas define locally tempered distributions, so equality of their spatial Fourier transforms proves the result.

No frequency regulator or characteristic limit is introduced by this calculation. It concerns the existing retarded internal propagators only.

On the retained patch, (eq:K8map) equals

Since $b\ne a$ and $b^0>a^0$, it is a finite sum of derivatives of total order at most three of smooth tensor coefficients times

\delta((b-a)^2)\delta^{(4)}(a-z).
 \label{eq:K8seed}
\end{proposition}

\begin{proof}
After Lemma~\ref{lem:weight}, constant fiber maps
commute with the scalar propagation.
Causal convolution and differentiation give
\[
 \int R(b-s)
          \partial_{s^\alpha}R(s-v)\,\,\mathrm d^4s
       =\partial_{b^\alpha}B(b-v),

which proves (eq:K8reduced).

On the separated chronological patch,

The source incidence contributes at most two external derivatives, and the scalar attachment contributes one more. Derivatives of the source coefficients are smooth coefficients of lower terms. This gives the finite representation (eq:K8seed).

Set

and define

The $y$ variable is passive, with zero covector. The five differentials $\mathrm{d}q,\mathrm{d}{r}^0,\dots ,\mathrm{d}{r}^3$ are independent because $b\ne a$ and the $z$ components isolate the four attachment differentials.

Consequently $S_{\mathrm{resp}}$ has codimension five, and ${\Lambda }_{\mathrm{resp}}$ is a sixteen-dimensional Lagrangian kernel relation. It is not the characteristic test locus $N$, which has codimension eight.

For the remainder of the paper, the order convention is: a normal Fourier amplitude of order $d$ along a codimension-$k$ submanifold of an $n$-dimensional space has conormal order

This is the individual conormal convention Wunsch (Section 8.1).

On the retained separated-endpoint patch,

The order is an upper filtered order and may decrease when its complete tensor coefficients cancel.

A full finite amplitude representative is obtained from (eq:K8reduced): apply its finite differential operators to the normal amplitude of $\delta (q){\delta }^{(4)}(r)$. No symbol of the grouped quantum coefficient is assigned.

Proof. The seed has normal amplitude order zero. The representation in Proposition prop:K8exact has at most three derivatives. Its normal amplitude is therefore a finite polynomial of degree at most three in the five normal frequencies, with smooth coefficients. Equation (eq:order), with $n=16,k=5$, gives $3-(16-10)/4=3/2$.

For clarity, the interior propagation matching in (eq:K8map) can be examined without the source diagonal. In primed covectors, a separated massless retarded flow relation is locally

on a fixed sign sheet with the separation future directed.

Two such propagations match by equality of the intermediate point and covector. On an interior segment, the matching is parameterized by

The matching equations have exactly these linearized tangent solutions. The projection retains $v,\xi ,{\tau }_1+{\tau }_2$ and has a one-dimensional local fiber.

This is a genuine clean internal propagation matching on its separated interior chart. Its one-dimensional fiber comes from splitting an already fixed null segment. It is unrelated to XIV-F's rank-two defect. No endpoint symbol integral is inferred from this interior chart: the exact convolution proof above already includes those endpoints distributionally.

Let $F={E_m}^{\mathrm{A}}f$, the inherited smooth advanced observation root. The scalar leg uses the response kernel (eq:responsekernel), with its output at the scalar attachment $b$. Its exact untested kernel is

Formula (eq:K9) is the complete untested kernel of the inherited $B_9$ root. It retains the derivative and mass terms.

On the separated chronological patch,

It has a finite conormal amplitude obtained from the source incidence and the displayed differential operators. The derivative term has normal amplitude degree at most three; the mass term has degree at most two.

Proof. Substitute the untested response kernel into (eq:root). Transferring the derivative of $\psi (b)$ to the kernel gives the first line of (eq:K9); the second line is its mass term.

On the retained patch,

because $b^0>a^0$ and the endpoints are separated. Equation (eq:responseseed) therefore expresses the response as at most two derivatives of smooth coefficients times $\delta (q){\delta }^{(4)}(r)$.

The first line of (eq:K9) adds at most one derivative. The second adds none. The same regular conormal coordinates as in (eq:Sresp) give the stated orders and amplitudes.

The equality of the two response-leg relation supports does not identify their amplitudes. Equations (eq:K8reduced) and (eq:K9) remain separate.

No assertion that the principal amplitude of either leg is nonzero is needed or made.

Define the coordinate-response line

Thus

The two terms of $K_5$ have the actual internal maps

and

respectively. The equalities $s=a$ and $s=z$ come from the displayed local metric factor in that placement. They are not assignments of a singular internal point to a retained configuration.

The untested mixed kernel of $B_5$ is

Each placement is a finite differential expression, of total kernel derivative order at most four, in the products

These products are defined, including at $a=z$.

Proof. Start with the two terms

Transfer the divergence to the incoming retarded kernel and the scalar derivative to the final kernel. The two signs cancel. Insert (eq:L), and use the actual local incidences $s=a$ and $s=z$. This gives (eq:K5).

For the first product in (eq:K5products), the coordinate map

is a diffeomorphism onto its image. The product is therefore the pullback of $1\otimes R\otimes R\otimes 1$. The second placement uses the analogous diffeomorphism

Tensor products of distributions exist, and these pullbacks are defined. Differentiation and multiplication by the fixed smooth weights preserve existence.

This proves the distributional kernel formula. It does not by itself assign one ordinary symbol class through the internal cone vertex.

The common $W$ variables are never replaced by the tested kernel $L_G$ of XIV-E.

On the first placement, define

When both separations are nonzero and future null, their defining differentials are independent: the $b$ and $z$ endpoint components isolate the two coefficients in a proposed relation.

The regular kernel relation is

on that separated chronological sector.

For the second placement use

and

Each placement of $K_5$ has a conormal representation of upper order one on its corresponding separated two-cone relation.

The normal amplitude is the finite polynomial obtained by differentiating

and multiplying by the full coefficients in (eq:K5). This statement includes all lower differential terms on these regular charts.

Proof. On these sectors both retarded time factors are constant one. The two defining functions are regular and independent. The undifferentiated normal amplitude has degree zero, and the displayed kernel has at most four derivatives.

The ambient dimension is sixteen and the regular base codimension is two. Equation (eq:order) gives

The exact polynomial amplitude description follows by applying the finite differential expression to the two normal deltas.

The matching used here is a submersion pullback of a tensor product. It has no failed product hypothesis. This is an internal tree calculation, not the factorwise cross-line audit of XIV-G.

Consider the first placement near $a=z$, with $b-a$ a nonzero future null separation. In the relative internal variable $r=a-z$, the retarded kernel has a diagonal operator part. Away from the internal characteristic covectors

its leading Fourier multiplier is $-1/p(\kappa )$, because the symbol of $\square$ is $-p(\kappa )$.

This describes the ordinary elliptic diagonal chart of the fixed retarded kernel. It is not an ellipticity assertion on its characteristic set.

The corresponding diagonal kernel relation is

The second placement has the analogous formula with $b-z$ and $z-a$.

On a conic chart where the outer cone frequency and the internal diagonal frequency are nonzero and comparable, and where

each $K_5$ placement has an ordinary conormal representation of upper order $\frac{1}{2}$.

Its amplitude is obtained from the finite differential formula (eq:K5), the outer delta amplitude, and the internal classical multiplier $-1/p(\kappa )$, together with their lower terms.

Proof. The internal multiplier has order $-2$ on the stated noncharacteristic chart. The complete kernel differentiates its two elementary factors at most four times, giving total normal amplitude order at most two.

The regular diagonal-plus-outer-cone submanifold has codimension five in dimension sixteen. Thus

The separation from $p(\kappa )=0$ ensures that the multiplier and its differentiated coefficients are classical symbols on this chart. The finite kernel formula retains their lower differential terms.

The frequency comparability is a chart condition for this order statement, not a cutoff introduced into the observable. No uniform assertion is inferred as that chart approaches $p(\kappa )=0$.

We now keep both the diagonal and the limiting flow-out of the internal line in the first $K_5$ placement. This is an internal relation calculation. It is not the matching excluded in XIV-F.

Work where the outer cone frequency $\theta$ is nonzero and the internal covector $\kappa$ is nonzero on one fixed time-oriented null sheet. Write

The diagonal Lagrangian piece is

Here $\kappa$ is unrestricted near the chosen nonzero null covector.

The separated internal flow-out piece can be parameterized by

Choose the covector sheet so that the separation $a-z$ is future directed. Its closure has $\tau =0$ as its boundary.

All covectors in these formulas have zero $y$ component. Unlike the balanced testing example, their outer $b$ covector is nonzero when $\theta \ne 0$.

The local diagonal piece ${\Lambda }_{\mathrm{d}}$ and the closure of the flow-out piece (eq:Lflow) each have dimension sixteen. They meet at

in dimension fifteen.

Their tangent intersection is exactly the tangent space of that intersection. Thus the two pieces intersect cleanly.

Their tangent sum has dimension seventeen. Consequently no single smooth sixteen-dimensional Lagrangian submanifold can contain both full local pieces at their intersection.

Proof. The parameters for ${\Lambda }_{\mathrm{d}}$ are: $a$ with four components, the nonzero null separation $b-a$ with three components, $y$ with four components, $\theta$ with one component, and $\kappa$ with four components. Their total is sixteen.

For the flow-out closure, replace the unrestricted four-dimensional $\kappa$ by a three-dimensional nonzero null covector and add the parameter $\tau$. The dimension is again sixteen.

At $\tau =0$ the base incidence is $a=z$. The further equation $p(\kappa )=0$ gives a fifteen-dimensional intersection in the diagonal piece.

A tangent vector to the diagonal piece has $\delta r=0$, with no constraint on $\delta \kappa$. A tangent vector to the flow-out closure at $\tau =0$ satisfies

To lie in both tangent spaces it must therefore have $\delta \tau =0$ and satisfy the linearized null condition on $\delta \kappa$. These are precisely the tangent conditions of the fifteen-dimensional intersection.

The tangent sum has dimension $16+16-15=17$. A smooth sixteen-dimensional submanifold containing both pieces would have to contain this seventeen-dimensional tangent sum, which is impossible.

The use of the boundary tangent in this proposition means the tangent of the smooth extension of the flow parameterization across $\tau =0$. The one-sided flow-out has that same tangent space.

This is not a non-clean matching. The two pieces intersect cleanly.

The difficulty is that their clean intersection does not make their union one ordinary Lagrangian relation with one classical amplitude chart. The diagonal symbol chart and the flow-out symbol chart are different charts with a nontrivial transition problem.

The number seventeen in the tangent-sum calculation has no connection to XIV-F's seventeen-dimensional linearized matching space.

The second $K_5$ placement has the corresponding transition with the internal separation reversed. Neither placement is deleted, and no cancellation between them is presumed.

Proposition prop:K5exact supplies a defined kernel at the transition. It also supplies a valid distributional pullback representation there.

It does not, without a further theorem, supply:

1. a uniform amplitude space describing the diagonal and flow-out pieces together;

2. the estimates relating the noncharacteristic diagonal symbols to the separated flow-out symbols;

3. the uniform parameter dependence needed when the residual variable $z$ is retained in a subsequent joint composition.

An intersecting-relation calculus can, in other problems, encode more than one ordinary Lagrangian piece. The present paper does not invoke such a calculus to assert these three conclusions for the full tensor-valued $K_5$ kernel.

Nor is a resolved paired construction substituted for the selected route.

The noncharacteristic amplitude contains the factor $1/p(\kappa )$. Its classical symbol bounds are not uniform as $p(\kappa )\to 0$. The separated-cone representation uses a different base incidence, with $\tau >0$. Proposition prop:branching proves that these incidences cannot be identified as one smooth ordinary relation at $\tau =0$.

Thus there is an actual additional amplitude problem to solve. Merely taking the union of (eq:L5plus) and (eq:Ldiag), or quoting a wavefront bound for the existing tree kernel, does not solve it.

This observation is not a nonexistence theorem for a suitable amplitude calculus. No universal claim about all representations in Definition def:class is made.

The following internal data have been constructed:

1. the untested regular source incidence;

2. exact kernel maps for $K_8$ and $K_9$, with regular conormal amplitudes on the retained separated-endpoint patch;

3. exact tensor-product pullback formulas for both $K_5$ placements;

4. their regular separated-cone and noncharacteristic-diagonal symbol charts;

5. the clean diagonal/flow-out intersection and the failure of a single ordinary Lagrangian identification there.

A uniform internal amplitude construction across the $K_5$ characteristic-diagonal transition has not been supplied. Hence the complete H2 internal-symbol requirement is Unresolved.

The first missing gate is H2, within G2/F2/E1/D2/J2. H3 is not entered as a characteristic representation construction.

Proof. Proposition prop:sourceincidence proves (i). Proposition prop:K8exact, Corollary cor:K8order, and Proposition prop:K9 prove (ii). Proposition prop:K5exact proves (iii). Propositions prop:K5flow and prop:K5diag prove (iv). Proposition prop:branching proves (v).

None of these results constructs the uniform amplitude map across the identified internal transition. An exact distributional pullback formula is not the required symbol-map theorem. The stated first-missing-gate rule therefore places the stop at H2.

gate assignment{ A defined $K_5$ tree kernel is not a missing distribution. The missing object is its uniform internal amplitude description for the later joint composition.

It would be incorrect to mark H2 Constructed solely from that kernel's existence and then report the same amplitude gap as an H3 failure.

The legal ending is Unresolved at H2. The narrower single-Lagrangian identification is excluded, but the full class of allowed representations is not.}

The exact regrouping (eq:regroup) continues to hold for the already defined smooth-slot tree functionals. The new kernel formulas refine three pieces of that identity with $z$ retained.

They do not construct one characteristic operator representation of (eq:target).

The pieces remain:

1. $C_G$ after the exact cancellation of the explicit second derivative of the scalar test;

2. one divergence of the complete tensor $B^{\alpha \mu }[G;h]$;

3. the complete scalar root (eq:root).

The $B_6$ numerator is not substituted for the second piece or for the complete sum.

No ordinary product theorem is asserted for multiplying these kernels by the grouped cross-line difference at $N$. The copied XIV-G audit already prevents an automatic factorwise argument. It is not retried.

A generalized composition would have to be proved for its own actual relation and amplitude hypotheses. No such theorem is supplied here.

No equality

is asserted. The pre-existing canonical domain remains only the domain on which the complete specified products have already been proved.

H4 is recorded only. H3 has not been constructed.

The individual conormal orders

are internal kernel orders in their stated charts. They are not transverse scaling degrees of ${\mathcal{T}}_{\mathrm{form}}$.

If a later paper supplies the common representation, its joint-section and remainder analysis must control:

1. lower-order terms of every numerical line;

2. all internal vertices and their actual kernel maps;

3. attachment-diagonal derivatives;

4. the relative transverse directions as geometric constraints, not integration fibers;

5. every actual source coefficient;

6. both terms of the scalar root;

7. the complete grouped cross-line difference.

No zero or nonzero joint section is assumed. A common representation, even if constructed, would not by itself prove cancellation.

The results of this paper prove neither cancellation nor failure of cancellation of ${\mathcal{T}}_{\mathrm{form}}$.

Proof. The constructed formulas concern internal tree kernels and regular internal symbol charts. No joint characteristic composition of the complete six-branch sum with the cross-lines is constructed.

The transition calculation concerns the geometry of two internal relation pieces, not the summed quantum amplitude. No controlled remainder for that amplitude is supplied.

No prescription is entered. A future prescription must name:

1. its ambient space;

2. a reference distribution on a proved domain;

3. the exact locus across which it extends;

4. agreement with the grouped frozen prescription;

5. a finite uniform transverse degree;

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

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

The test locus remains

It is not converted into a codimension-four numerical extension problem.

The internal response relation ${\Lambda }_{\mathrm{resp}}$ does not change that test locus. It is a relation of a precomposition kernel, not a characteristic extension locus for the grouped quantum target.

The prospective bound

requires its own hypotheses. A bound by $N^{\ast }N$ alone cannot erase nonsmooth reference values away from $N$.

The Brunetti–Fredenhagen and Dang extension frameworks remain comparison frameworks only BF,Dang. No theorem from them is applied to the absent joint characteristic distribution.

XIII-E's family is not imported. No cutoff-and-restore limit, finite part, characteristic contact, or new ambiguity class is introduced.

J3 opens only for a distribution produced by a successful joint-section/remainder construction, or by a later fully specified prescription. Neither is supplied.

The parallel-line degree eight remains a collision degree. It is not a degree along ${\mathcal{N}}^{\circ }$, ${\overline{\mathcal{N}}}_{\mathrm{att}}$, or ${\tilde{\mathcal{N}}}_{\mathrm{att}}$, and none of the internal conormal orders above changes that statement.

J4 remains closed: no characteristic distribution has been glued to the permitted partial contacts and singular spectators.

J5 remains closed: properness remains support control for existing kernels. The internal proper convolution in this paper does not establish the proper pushforward or output wavefront theorem for the missing quantum coefficient.

No ${R_1}^{(1)}$ is released. The tree datum is not substituted for it.

No $s_{\mathrm{ext}}$, $s_{\Psi }$, or $s_0$ is applied. No Ward defect is formed. The classical breaking density and the value $-8I$ neither define nor normalize this graph.

There is no new vertex, state, numerical loop value, residue subtraction, or sharp-width limit. The inverse remains $-1$; neither the X-track inverse nor a spurion law is introduced.

For the unchanged retained instance:

1. the local source incidence is defined before residual testing and retains its complete coefficients;

2. the two gravitational-response legs have the exact kernel maps (eq:K8reduced) and (eq:K9);

3. on the retained separated-endpoint patch these legs have regular conormal representations of upper order $\frac{3}{2}$;

4. the two $K_5$ placements have the exact tensor-product pullback formula (eq:K5);

5. their regular separated-cone and noncharacteristic-diagonal orders are those proved above;

6. their internal diagonal and limiting flow-out pieces intersect cleanly but cannot be identified as one smooth ordinary Lagrangian relation;

7. the uniform internal amplitude construction at that transition, and hence the common characteristic representation, is not supplied.

The ending is Unresolved at H2, within G2/F2/E1/D2/J2.

No universal representation obstruction, joint-section conclusion, characteristic prescription, or admitted ${R_1}^{(1)}$ is asserted.

Proof. Proposition prop:sourceincidence proves (i). Proposition prop:K8exact and Proposition prop:K9 prove (ii). Corollary cor:K8order and Proposition prop:K9 prove (iii).

Proposition prop:K5exact proves (iv). Propositions prop:K5flow and prop:K5diag prove (v). Proposition prop:branching proves (vi). Theorem thm:H2 identifies the missing amplitude theorem and the correct first stop. Proposition prop:nocancellation separates these results from a cancellation verdict.

The internal source vertex is no longer hidden inside a smooth tested operator. Its two-field incidence is written explicitly.

Both gravitational-response legs have exact separate kernel formulas. The coordinate leg's intermediate retarded propagation is evaluated without touching the quantum cross-lines. The scalar leg retains its derivative and mass terms.

The remaining internal issue is also more precise. For $K_5$, the exact tree kernel exists, but its regular diagonal and flow-out symbols do not form one ordinary amplitude chart at the internal characteristic diagonal. Their clean intersection is a relation-interface problem, not the excess identification rejected in XIV-F.

These results do not solve that interface problem or the complete characteristic contraction.

A successor receives:

1. the unchanged XIV-G package;

2. the untested source incidence and its regular relation;

3. the exact coordinate-leg convolution map and its regular conormal representation;

4. the separate complete scalar-leg root map;

5. the exact two-placement $K_5$ tensor pullbacks;

6. their regular relation charts and orders;

7. the clean diagonal/flow-out intersection and the exclusion of a single ordinary Lagrangian identification;

8. the precise missing uniform internal amplitude theorem and the H2 stop.

It receives no transverse degree of the target, no common characteristic representation, no cancellation theorem, and no permission to call the tangent defect an excess. It receives no ${R_1}^{(1)}$.

To advance the same route, a successor must first construct the missing internal amplitude map without replacing the frozen model or discarding a placement. It must then justify one common operator representation of the complete regrouped sum. Only thereafter may it enter the joint-section and remainder gates.

A defined tree kernel is not a uniform internal symbol theorem. A clean intersection is not one ordinary relation. An internal conormal order is not a transverse degree of the quantum target. An undefined quantum coefficient is not an anomaly.

scope. The new proofs concern the untested source incidence, the three internal kernel maps, an existing retarded convolution, regular internal symbol charts, and their characteristic-diagonal transition geometry. No cross-line characteristic product is constructed.

correction. The first missing amplitude theorem is an H2 requirement. The manuscript therefore reports Unresolved at H2 instead of asserting an unsupported H2 construction and moving the same gap to H3.

scope. The internal response relation and the two $K_5$ relation pieces are not $J$, the excluded matching, or $S_6$. Their dimensions and tangent calculations have separate meanings. No tangent defect is renamed an excess.

scope. All orders assigned here belong to specified internal regular conormal charts. They are upper filtered orders unless equality is separately established. No transverse scaling degree of the grouped target is assigned.

and internal-vertex scope. All source coefficients remain those of the actual density-weighted Euler source. Internal equalities arise only from proved local incidences. No hidden vertex is assigned a retained configuration point.

scope. The analytic inputs are exactly those stated in H0. XIV-A and XIV-B are not hypotheses of any product, amplitude, or normalization argument. The inherited relative classical ledger is not recomputed.

calculus. Individual conormal conventions are used only after the relevant regular submanifolds and relations are identified. Defined submersion pullbacks and proper internal convolutions are distinguished from the unconstructed uniform amplitude map. Extension frameworks are not applied to the missing target.

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

XIVG

N. Cheung, Landmark XIV-G: Operator and pullback composition of the regrouped retained germ. Supplied manuscript XIV-G.tex. Immediate predecessor: route, local source typing, residual-testing limitation, factorwise audit, and exact G2 stop.

XIVF

N. Cheung, Landmark XIV-F: Paired composition on a proposed excess bundle. Supplied manuscript XIV-F.tex. Excluded clean-excess identification and geometric boundaries only.

XIVE

N. Cheung, Landmark XIV-E: Joint symbol of the six-branch retained germ, or one named characteristic prescription. Supplied manuscript XIV-E.tex. Exact regrouping, complete root, tested-kernel bound, and characteristic lifts.

XIVD

N. Cheung, Landmark XIV-D: Cancellation or prescription on the retained noncoincident characteristic stratum. Supplied manuscript XIV-D.tex. Six-branch germ, actual local incidences, and branch-limited numerator.

XIVC

N. Cheung, Landmark XIV-C: The joint two-line extension of one selected binary pair. Supplied manuscript XIV-C.tex. Retained stratum, positive relation, and contact rigidity.

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. Four successor obligations and primitive collision extensions.

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, retarded 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. Response formulas, source-slot sign, roots, and existing 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 local source.

Hormander

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

Wunsch

J. Wunsch, Microlocal analysis and evolution equations: Lecture notes from the 2008 CMI/ETH Summer School, arXiv:0812.3181v3, 2023. Sections 8–9: individual conormal orders, symbols, canonical relations, and clean composition conventions.

BGP

C. Bär, N. Ginoux, and F. Pfäffle, Wave Equations on Lorentzian Manifolds and Quantization, ESI Lectures in Mathematics and Physics, European Mathematical Society, 2007, arXiv:0806.1036. Causal Green operators and support conventions. The internal convolution used here is computed explicitly.

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. Recorded extension framework only; not applied to the missing characteristic 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 only.