Landmark XIV-J The joint characteristic map on the unchanged carrier Reconstruction descent, a supported-sector normal-jet reference, and the exact J2 stop before quantum admission

We enter the joint-map obligation left unresolved by XIV-I. The internal retarded amplitude module, including both $K_5$ placements, is a copied prerequisite. It is not identified with the required composition against the grouped quantum cross-lines.

Two additional interfaces are established. First, a joint map on reconstructed kernels must annihilate the kernel of internal reconstruction. An explicit normal-distribution identity shows that the highest polynomial degree of an amplitude presentation need not descend to an intrinsic symbol of its reconstructed kernel.

Second, the supported response sector $K_6+K_8+K_9$ is put into an exact finite normal-jet form on the codimension-five response incidence. Its grouped cross-line composition is constructed on the local complement of the codimension-eight characteristic locus. The construction retains every normal derivative and both terms of the scalar root, and has an explicit wavefront bound. It is a reference theorem for that sector, not a composition theorem for the full six-branch sum.

A conditional normal-jet extension interface identifies the additional coefficient distributions that a supported realization would require. None is selected or constructed across the characteristic locus. No generalized joint composition for the full sum, complete joint section, transverse estimate, contact gluing, or output admission is supplied. The first unresolved gate remains J2. No ${R_1}^{(1)}$ is admitted in ${\mathcal{E}}_H$, and XV-B remains closed.

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

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

release{ Constructed: the reconstruction-descent criterion; the exact normal-jet representation of the supported response sector; its grouped reference composition away from $N$; and a conditional normal-jet extension interface.

First missing gate: J2. The required map for the complete reconstructed six-branch sum on tests meeting $N$ has not been supplied.

Not released: a joint principal-symbol section, a complete quantum remainder, extension obligations, or an admitted ${R_1}^{(1)}$.

XV-B remains closed. The free carrier recorded in XV-A cannot replace the missing analytic coefficient.}

The manuscript inputs below are used with their stated domains. Their inclusion is not an independent verification of every predecessor claim. No pole identity, incidence inventory, elementary phase, or transition geometry is re-proved.

A successor receives:

1. the unchanged XIV-H package;

2. the finite retarded-pole module and its exact normalization;

3. the pole identity, uniform estimates, and reconstruction map;

4. the two complete $K_5$ amplitude maps with $z$ retained;

5. their exact finite lower-degree remainder and internal wavefront bounds;

6. the common precomposition sum and its canonical-domain value;

7. the precise missing I3 joint operator map;

8. the still closed I4–I6 and J3–J5 gates.

The internal amplitude ending is Constructed at I2. The full joint characteristic map is Unresolved at I3. No transverse degree, cancellation theorem, or admitted ${R_1}^{(1)}$ is handed over XIVI.

In the Fourier convention

write

On a conic neighborhood

the fixed internal multiplier is

This is the Fourier transform of the existing retarded kernel, not a prescription for a new graph product.

The finite internal amplitude generators are

For smooth coefficient arrays the reconstruction is

For both $K_5$ placements, $j+|\beta |\le 4$. The module has uniform finite-seminorm estimates on normalized frequency annuli meeting the pole, uniform after finitely many coefficient derivatives on compact parameter sets. Its finite lower-degree remainder is an internal frequency-distribution estimate only.

The reconstruction is a joint distributional map with $z$ retained; it is not a claim that singular kernels have pointwise slices at every residual value XIVI.

Define constant fiber maps by

The indices $I,J,L,M$ range over the inherited symmetric metric components. All formulas below are componentwise with the frozen tensor pairings.

The untested source incidence is

It is defined and supported on $v=a=z$. Its smooth coefficients are those of the actual source.

Using an explicit output index, the response kernel is

The coordinate and scalar response legs are

and

The derivative and mass terms are both retained.

On the separated chronological patch, $K_8,K_9$ are finite sums of derivatives of total order at most three of smooth coefficients times

Their regular upper conormal order is $3/2$ on

The copied convolution is

It is an identity of existing internal kernels XIVH,XIVI.

The output index in (eq:T) removes a predecessor index suppression. It does not change the response map.

Put

Then

Its two seed products are

Their actual coordinate maps are tensor-product diffeomorphisms, so the kernels are defined at $a=z$.

On regular separated-cone charts the upper conormal order is one. On the stated comparable-frequency, noncharacteristic diagonal charts it is $1/2$.

The internal diagonal and limiting flow-out pieces intersect cleanly in dimension fifteen and have tangent sum seventeen. They do not form one ordinary smooth Lagrangian. This is not XIV-F's non-clean matching XIVH.

Let ${\mathsf{D}}^{\alpha \mu ,I,J}$ be the constant fiber map defined by

The regrouped $K_4+K_7$ kernel is

The local density piece is

The exact retained sum is

Its reconstructed value agrees with the canonical composition on the inherited proved domain $U_{\mathrm{can}}$.

No map composing this sum with the grouped cross-lines on tests meeting $N$ was constructed XIVI.

Smooth residual testing can hide attachment-diagonal wavefront directions. The local symbol of the tested source does not provide either composed response symbol.

The factorwise audit of XIV-G excludes automatic application of the ordinary final product theorem at the retained opposite covectors. It is limited to the diagnosed numerical incidence and is not a full-pair nonvanishing theorem XIVG.

XIV-F's selected matching has dimension fifteen and linearized dimension seventeen. Its rank-two quotient is a tangent defect, not an excess fiber. The covector-sum projection has rank fourteen, one-dimensional local fibers, and an isotropic, non-Lagrangian image. Its geometry is a boundary, not a relation used below XIVF.

For smooth directions,

where

For $F={E_m}^{\mathrm{A}}f$,

These are exact tree-kernel identities before characteristic composition.

On the retained chronological neighborhood,

The elementary common product phase was not clean XIVE.

In

the retained mixed germ is the complete six-branch sum $K_4+\cdots +K_9$. Other branches are zero only on their proved local germ or species channel.

The test locus is

with $\mathbf{n}\in S^2$. It has dimension eight and codimension eight in $W$.

For

the copied positive relation is

Defined interaction collision contacts vanish on the retained separated attachment neighborhood.

The identity

is a branch-limited numerator, not a joint section XIVD,XIVC.

The neighborhoods have pairwise separated closures for $A,B,Y$, with $\overline{Z}\subset A$, and lie in the open slab. They retain

No characteristic configuration is removed.

XIII-C retains four obligations:

1. a joint characteristic prescription or complete-pair cancellation;

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

3. compatible partial contacts and singular spectators;

4. output wavefront control after proper pushforward.

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

The retarded convention is

Strict full-support ordering gives zero or the scaled commutator. The selected tree datum is admitted. Properness controls a distributional pushforward after the distribution exists; it does not define a missing joint product XIIIB.

The normalization and filled slots remain

The roots are

The scalar-sourced slot is included once XIIE.

The actual source is

The unintegrated cubic representative is fixed before weighting, and the selected action weights only that cubic coefficient XIIC.

XV-A is recorded at the exit only. 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 frozen spacetime, inverse, and carrier are

All switches, observations, masses, and widths retain their predecessor values.

The pair is

with

The retained formal target remains

Let ${\mathcal{K}}_{\mathrm{int}}$ be the space of actual reconstructed internal kernels obtained from the finite presentations in the copied precomposition package, with their tensor indices retained.

The class considered here consists of linear operator maps

that:

1. on the named reconstructed sum give the specified contraction with the grouped cross-lines and mixed vertex Hessian;

2. retain both $K_5$ placements and the residual variable $z$;

3. agree with the canonical composition on its proved domain;

4. are defined on arbitrary compact tests meeting $N$;

5. have a stated and proved complete-graph wavefront bound appropriate to their composition.

The product data and ${\mathfrak{C}}_{\mathrm{part}}$ are unchanged.

The domain consists of reconstructed kernels, not arbitrary amplitude arrays. No representative-independent operation on all free-shell quotient classes is asserted.

A generalized composition belongs to this class only after its relation, amplitude, and remainder hypotheses are supplied for the present target. No common product phase is introduced.

The endings are attached to the object at the gate reached:

1. Constructed: that object has its domain, amplitude or degree, and proved wavefront control.

2. Proved obstructed: an incompatibility excludes every map in Definition def:mapclass.

3. Unresolved: the joint map, joint section, prescription datum, or extension estimate is missing.

The already-copied factorwise failure is not a universal obstruction to this class.

The continuation after a successful joint map is fixed as the joint-section and complete-remainder exit. The prescription alternative is not entered.

Only the eventual admission line $Q_H{R}_1^{(1)}\in {\mathcal{E}}_H$ could open XV-B.

Let ${\mathcal{P}}_{\mathrm{int}}$ denote the finite direct sum of the internal coefficient presentation spaces. Its reconstruction map, including summation into the common tensor-valued kernel, is

Thus

This is a space of already defined internal distributions.

On $U_{\mathrm{can}}$, canonical contraction gives a linear map

Every identity between reconstructed kernels is respected by this canonical map on its proved domain.

A linear candidate

defines a map on reconstructed kernels if and only if

When this holds, the induced map $\mathsf{M}(\mathsf{Rec}A)=\tilde{\mathsf{M}}(A)$ is unique.

Agreement of $\tilde{\mathsf{M}}$ with canonical contraction on $U_{\mathrm{can}}$ proves (eq:descent) there only. It does not prove that the same identity holds across $N$.

Proof. If $\tilde{\mathsf{M}}=\mathsf{M}\mathsf{Rec}$, a presentation in $\ker \mathsf{Rec}$ has image zero.

Conversely, if two presentations have the same reconstruction, their difference belongs to $\ker \mathsf{Rec}$. Condition (eq:descent) therefore makes their images equal, so the induced map is well defined and unique.

For a null presentation, canonical contraction of its zero reconstruction is zero on $U_{\mathrm{can}}$. That statement only determines the restriction of its proposed global image. A distribution supported outside $U_{\mathrm{can}}$ is not detected by this restriction.

The following identity concerns the copied internal module. It introduces no cross-line product.

In an adapted $K_5$ chart,

Nevertheless the two reconstructed summands are separately nonzero:

Consequently the polynomial-degree splitting of a presentation does not by itself define an intrinsic joint-symbol splitting on ${\mathcal{K}}_{\mathrm{int}}$.

Proof. For a one-dimensional test $\psi$,

Hence $q{\delta }^{\prime }=-\delta$. Tensoring with the existing nonzero distribution $R(r)$ and using (eq:Rec) gives all displayed identities.

The copied internal module retains its proved reconstruction and frequency estimates. Lemma lem:null does not invalidate them.

It does show that its polynomial degree is a degree of a presentation. It cannot be promoted to the principal symbol of the reconstructed quantum composition without a descent theorem and the missing joint calculus.

In particular, no leading term is selected from an amplitude array before its reconstruction identities are accounted for.

This is not an obstruction to every joint map. It is an explicit compatibility condition for any proposed coefficient-level realization of such a map.

Define

This is the supported response sector, not the full six-branch kernel.

Work locally in a coordinate neighborhood $U\subset W$ of a retained point. The response incidence is regular there by the copied separated-endpoint geometry. Choose coordinates

Here $s$ has eleven components and includes the four independent $y$ coordinates.

All local distributions are represented relative to the chosen coordinate density. The smooth density conversion from the original flat variables is included in their coefficients.

The characteristic set remains

It is a submanifold of $S_{\mathrm{resp}}$ of relative codimension three, since $\dim S_{\mathrm{resp}}=11$ and $\dim N=8$.

ambient problem is unchanged{ The flag $N_U\subset S_{\mathrm{resp}}\subset U$ has codimensions three and five, whose sum is eight.

This bookkeeping does not replace the target by a codimension-three or codimension-four numerical extension. Every reconstructed distribution still lives in the sixteen-dimensional common space with $z$ retained.

No transverse degree or extension threshold is assigned using this flag.}

For a distribution $b$ on $S_0$ and a multiindex $\alpha \in {\mathbb{N}}^5$, define

Thus ${\mathsf{E}}_{\alpha }b=b(s){{\partial }_n}^{\alpha }{\delta }^{(5)}(n)$ in the chosen coordinates.

For each $\alpha$, (eq:Ejet) defines a continuous map

For any finite $m$, the map

is injective.

If $b_{\alpha }$ is supported in a set $L\subset S_0$, its insertion is supported in $L\times \{0\}$.

Proof. The map $\phi \mapsto {{\partial }_n}^{\alpha }\phi (s,0)$ is continuous from compactly supported tests in $U$ to compactly supported tests in $S_0$. Transposition defines (eq:Ejet) continuously and gives its support.

For independence choose $\chi (n)$ compactly supported and equal to one near zero, and use the tests

For every $|\alpha |\le m$,

These tests recover each $b_{\gamma }$ separately.

In the chart above, the supported response sector has an exact componentwise representation

The coefficient array is unique in the fixed coordinates.

Every term of $K_6,K_8,K_9$ is retained, including derivatives of their smooth coefficients, the actual source coefficients, and the mass term of $K_9$. No equality or vanishing of these coefficients is assumed.

Proof. The copied response formulas express $K_8,K_9$ as finite differential expressions of order at most three in smooth coefficients times $\delta (q_R){\delta }^{(4)}(a-z)$. Formula (eq:K6) has the same seed and differential order at most two.

Change to the regular coordinates $(s,n)$. The chain rule expresses every derivative as a finite sum of smooth-coefficient tangential and normal derivatives. Tangential derivatives act on coefficients. The remaining normal derivatives of the seed have order at most three.

For a smooth coefficient $c(s,n),$ the finite identity

follows by applying the Leibniz rule to its test-function pairing. It brings every coefficient into the stated normal form without dropping any derivative.

Summing the three kernels gives (eq:normalform). Uniqueness is Lemma lem:jetinsert.

For $b\in {\mathcal{D}}^{\prime }(S_0)$ let

Define

For every $\alpha$,

For a finite normal-jet sum, the corresponding union is a wavefront upper bound.

Proof. The tensor product $b(s)\otimes {\delta }^{(5)}(n)$ is defined. Its tensor-product wavefront bound is exactly the inclusion in (eq:liftcone), with zero covectors adjoined on the supports. Normal differentiation does not enlarge this bound. Finite sums preserve the asserted upper bound Hormander (Section 8.2).

These are bounds for actual supported internal kernels. They do not yet involve the quantum cross-lines.

In independent relative variables define

It is a tensor-product difference of defined distributions. Its pullback by

is defined because this map is a submersion. Denote this grouped pullback by ${\mathcal{C}}_{\mathrm{cross}}$.

This construction does not multiply ${\mathcal{C}}_{\mathrm{cross}}$ by an internal kernel at $N$.

The mixed vertex Hessian is

where $T$ is the inherited polarized stress tensor. Its finite expansion gives:

1. constant fiber contractions;

2. derivatives on the crossed scalar slot;

3. a smooth factor $g(y)$;

4. an unused scalar field or its first derivative at $y$.

Derivatives on crossed slots are applied in the two independent relative variables before the submersion pullback. For each such finite differential operator $D$, write ${{\mathcal{C}}_{\mathrm{cross}}}_D$ for the resulting grouped cross-line distribution.

An eventual transfer of an unused scalar derivative to the output kernel is a $y$ derivative. Since the normal coordinates $n$ are independent of $y$, this is tangential to $S_{\mathrm{resp}}$. All assertions below therefore apply componentwise to the complete finite vertex expansion.

The proof uses the following property of the frozen lines: at separated endpoints their singularities lie on the null cone and their wavefront covectors are conormal to that cone. Finite differentiation preserves that inclusion. The massive time-ordered line may have smooth interior terms; these do not add singular support.

This is the line-wavefront input already used in the predecessor graph calculus. It is also the standard separated line description in the functional framework FR (Sections 6–7 and Appendix A). No replacement Hadamard matrix is chosen.

On $S_{\mathrm{resp}}\cap (U\setminus N_U)$, every finite normal trace

is defined.

At a singular point of such a trace its tangential wavefront is bounded by a nonzero multiple of $i^{\ast }\mathrm{d}{q}_1$ or $i^{\ast }\mathrm{d}{q}_2$. No point in this domain has both $q_1=q_2=0$.

Proof. On $S_{\mathrm{resp}}$, $q_R=0$ and $a=z$. By the copied zero-locus statement (eq:zerolocus), simultaneous vanishing of $q_1,q_2$ would put the point in $N_U$. Thus away from $N_U$ at most one of the two cross-line factors is singular.

If neither is singular, the grouped distribution is smooth locally, and all normal traces exist.

If only $q_1=0$, the other line is smooth locally. The grouped distribution and its derivatives have wavefront bounded by nonzero multiples of $\mathrm{d}{q}_1$. The $y$ component of $\mathrm{d}{q}_1$ is nonzero because $y\ne a$. Every conormal of $S_{\mathrm{resp}}$ has zero $y$ component, since $q_R$ and $a-z$ do not depend on $y$. The restriction normal-set condition therefore holds.

The case with only $q_2=0$ is identical, using $y\ne b$. Differentiation does not enlarge the line-wavefront bound. The restriction theorem gives (eq:traces) and the tangential upper bound Hormander (Section 8.2).

This proof is confined to points away from $N_U$. It is not a retry of the ordinary product theorem at the retained opposite covectors.

For every vertex component $D$, the product

is defined on $U\setminus N_U$. In the fixed normal coordinates its exact normal form is

Smooth vertex weights, fiber contractions, and output $y$ derivatives may be applied to this distribution.

The reference agrees with the canonical grouped composition where that composition was already proved. It includes every normal derivative and both scalar-root terms.

Its support is contained in $S_{\mathrm{resp}}$, subject to the existing weights, and its wavefront set is bounded by

This is a local reference theorem for $K_{\mathrm{supp}}$, not for the full $K_{\mathrm{ret}}$.

Proof. At a point of $S_{\mathrm{resp}}\setminus N_U$, the proof of Lemma lem:traces bounds the cross-line wavefront by a conormal with nonzero $y$ component. The wavefront of $K_{\mathrm{supp}}$ is contained in $N^{\ast }{S}_{\mathrm{resp}}$, whose covectors have zero $y$ component. There is therefore no opposing pair at that point.

Off $S_{\mathrm{resp}}$ the internal supported distribution is locally zero, so the product is zero there. These local products glue on $U\setminus N_U$.

For smooth cross-line coefficients the exact formula is (eq:smoothjetproduct). For the present traceable distribution the same identity follows from the restriction and product calculus. Locally, one may regularize the single singular relative line in its admissible wavefront topology. Its normal traces converge by the restriction theorem; the products with the normal-jet distribution converge by the product theorem. The smooth identity therefore passes to the limit. This argument is entirely within the domain of Lemma lem:traces; no convergence at $N_U$ is claimed.

The normal traces have the tangential bounds in that lemma. Multiplication by $v_{\alpha }$ adds no wavefront, and Lemma lem:insertWF adds arbitrary response-normal covectors. This gives (eq:referenceWF). The finite vertex operations preserve the bound.

Every operation agrees with the canonical distributional operations on their common domain, proving agreement. The normal form was obtained from the complete response sector in Proposition prop:normalform, so no source or mass term has been removed.

Theorem thm:reference proves a reference on the local complement of $N_U$ for the supported sector only.

It does not prove

for the complete six-branch sum. In particular, it supplies no characteristic composition of $K_{47}$ or either $K_5$ placement with ${\mathcal{C}}_{\mathrm{cross}}$. Those kernels have different support geometry and remain in the target.

Collect the terms of (eq:reference) with the same normal derivative. For $|\gamma |\le 3$, put

Then

The ${b_{D,\gamma }}^{\circ }$ are distributions on $S_{\mathrm{resp}}\setminus N_U$. They need not be smooth. Their complete sum may have cancellations. No coefficient is assumed nonzero or zero.

The reference estimates on compact subsets away from $N_U$ do not provide uniform estimates as those compact subsets approach $N_U$.

The preceding reference is not a prescription across $N_U$. The following theorem names one exact interface for a possible supported realization. It does not impose this realization on every map in Definition def:mapclass.

For $m\ge 0$, let ${\mathcal{I}}_m$ be the space of compact tests $\phi$ whose normal jets satisfy

A distribution has normal order at most $m$ in this section when it annihilates ${\mathcal{I}}_m$.

This is a fixed-coordinate normal-jet condition. It is not a transverse scaling-degree assertion.

A distribution $t\in {\mathcal{D}}^{\prime }(U)$ annihilates ${\mathcal{I}}_m$ if and only if it has a unique expression

The coefficients are recovered continuously by

In particular such a distribution is supported on $S_{\mathrm{resp}}$.

Proof. An expression (eq:generalnormal) plainly annihilates ${\mathcal{I}}_m$. Its coefficients are unique by Lemma lem:jetinsert.

Conversely, choose the same cutoff $\chi$ as in (eq:Jjet) and subtract the finite normal Taylor jet of a test:

If $t$ annihilates this difference, its value is determined by the finite displayed jet. Defining the coefficients by (eq:extraction) therefore gives (eq:generalnormal). Continuity follows from continuity of the test-extension maps ${\mathsf{J}}_{\alpha }$.

A test supported off $S_{\mathrm{resp}}$ has every jet zero, so $t$ vanishes on that test.

The normal-jet description is the local supported-distribution structure in explicitly fixed coordinates. The general microlocal distinction between distributional tangential coefficients and smooth conormal coefficients is discussed in Dang (Section 8.2). Here the finite characterization and extraction formula are proved directly; no extension theorem is applied.

Fix a vertex component $D$. The following two assertions are equivalent:

1. There exists $t_D\in {\mathcal{D}}^{\prime }(U)$ of normal order at most three such that

   $$t_D{|}_{U\setminus N_U}={t_D}^{\circ }.$$

2. There exist distributions $b_{D,\gamma }\in {\mathcal{D}}^{\prime }(S_0)$, $|\gamma |\le 3$, whose restrictions to $S_{\mathrm{resp}}\setminus N_U$ are the coefficients (eq:bcoeff).

Given (ii), the corresponding supported realization is

Its wavefront satisfies

No existence, degree, or characteristic normalization of the coefficient extensions is asserted by this equivalence.

Proof. If (i) holds, Proposition prop:characterization extracts its unique coefficient distributions. Tests ${\mathsf{J}}_{\gamma }\psi$ with $\psi$ supported away from $N_U$ compare the extracted coefficients with (eq:collected); uniqueness shows that their restrictions equal (eq:bcoeff).

If (ii) holds, the insertion maps are defined for arbitrary coefficient distributions. Their finite sum has normal order at most three and has the required restriction. The wavefront statement is Lemma lem:insertWF.

A supported, normal-order-preserving realization of this sector would face exactly the coefficient interface above.

A generalized joint map in the full class might organize terms differently, or prove cancellations only after the entire six-branch sum is assembled. Theorem thm:conditionalextension does not claim to classify all such maps and is not a universal obstruction.

No coefficient extension is chosen. The prescription alternative remains unentered.

Locally choose further coordinates $(\tau ,\eta )$ on $S_{\mathrm{resp}}$ with $N_U=\{\eta =0,n=0\}$ and $\eta \in {\mathbb{R}}^3$. If a coefficient extension were conormal to $N_U$ inside $S_{\mathrm{resp}}$, its insertion would have covectors

These are precisely full ambient conormal covectors of $N_U$ in $U$.

Thus the normal-jet interface retains all eight ambient normal directions. It does not discard the attachment directions, and it assigns no degree from the relative three-dimensional notation.

No conormal landing condition or uniform estimate for the coefficient extensions is proved here.

If two distributions agree on an open domain $V\subset U$, their difference is supported in $U\setminus V$.

For the supported-sector reference of Theorem thm:reference, two extensions agreeing on $U\setminus N_U$ differ by a distribution supported in $N_U$.

For the full target, agreement on $U_{\mathrm{can}}$ alone gives support in $W\setminus U_{\mathrm{can}}$, not automatically in $N$.

Proof. The difference vanishes on every test compactly supported in $V$, which is exactly the asserted support statement. The two applications use their respective agreement domains.

This elementary distinction prevents an unproved identification of a canonical domain with an entire characteristic complement.

In a chart $(\tau ,\eta ,n)$ as above, a nonzero smooth compactly supported $a(\tau )$ defines

It is supported in $N_U$ and has wavefront contained in $N_U^{\ast }U\setminus 0$.

It is not a member of the inherited interaction collision-contact class on this retained separated patch.

Accordingly, agreement off $N_U$ and a wavefront allowance containing the full ambient conormal would not, by themselves, select a unique extension. This statement is conditional on such an allowance and does not assert an actual degree of the target.

Proof. The delta in the eight normal coordinates is a defined distribution. Its wavefront is conormal to their common zero set, and multiplication by the smooth tangential coefficient preserves that upper bound.

Every permitted interaction collision contact restricts to zero on the retained patch by the copied rigidity statement. The nonzero distribution (eq:AN) does not, so it is not such a contact.

Adding (eq:AN) to an existing extension would preserve agreement off $N_U$ and any wavefront upper bound that already permits its conormal. No extension is being constructed or modified here.

No degree of (eq:AN) is used to infer the degree of ${\mathcal{T}}_{\mathrm{form}}$, and no characteristic term is added to ${\mathfrak{C}}_{\mathrm{part}}$.

The supported-sector reference is not the required value of $\mathsf{M}$ on

It does not include:

1. a characteristic composition for $K_{47}$;

2. a characteristic composition for either $K_5$ placement;

3. coefficient extensions of the supported-sector reference;

4. a complete-graph wavefront theorem after summing all tensor and branch contributions;

5. a proof that a coefficient-level construction descends through every internal reconstruction identity.

The inherited internal pole estimates do not give any of these by themselves. They control a different operation before the new cross-line composition.

No attempt is made to multiply the complete sum by ${\mathcal{C}}_{\mathrm{cross}}$ at the retained opposite covectors using the ordinary product theorem.

For the class fixed in Definition def:mapclass:

1. reconstruction descent is a necessary condition, made explicit by Proposition prop:descent;

2. the supported sector has the exact normal form (eq:normalform);

3. its grouped composition on $U\setminus N_U$ is Constructed by Theorem thm:reference;

4. a supported normal-order-preserving extension has the conditional coefficient interface of Theorem thm:conditionalextension;

5. no full joint characteristic map on tests meeting $N$ is constructed.

The ending for the required full map is Unresolved at J2, within the inherited I3/G2/F2/E1/D2/J2 analytic boundary.

No incompatibility excluding every map in the fixed class has been proved.

Proof. The cited propositions and theorems establish (i)–(iv). Their domains stop short of a characteristic composition for the full six-branch kernel.

The conditional interface produces no extension coefficients, and the reference theorem treats only $K_{\mathrm{supp}}$ away from $N_U$. Neither gives a value for the full reconstructed sum on a test crossing $N$. No generalized composition theorem for that operation has been supplied, proving (v) and the unresolved ending.

The missing construction is not a converse failure theorem for all possible constructions, so the obstructed ending is not warranted.

first-missing-gate rule{ A reference for $K_6+K_8+K_9$ is not the joint map for $K_4+\cdots +K_9$.

A finite normal-jet formula away from $N$ is not a coefficient extension across $N$.

A conditional extension interface is not a prescription.

The first missing gate remains J2.}

The selected exit remains the joint-section exit. Because J2 is Unresolved, J3 is not entered as a construction.

A successful theorem would have to control:

1. lower terms of every numerical line;

2. internal vertices and their actual kernel maps;

3. attachment-diagonal derivatives;

4. the two defect directions as geometric constraints, not integration fibers;

5. every coefficient of the actual source;

6. both terms of the scalar root;

7. the grouped cross-line difference;

8. the complete tensor and branch summation;

9. independence from internal null presentations.

The copied internal frequency remainder is not this theorem. The exact normal-jet reference formula is not this theorem.

Neither a zero section nor a nonzero section is assumed. The branch numerator $Q\lambda$ supplies no section of the complete sum.

XIV-J proves neither cancellation nor failure of cancellation of ${\mathcal{T}}_{\mathrm{form}}$.

Proof. The new theorems concern reconstruction identities and a proper sub-sector on a domain avoiding $N$. No joint characteristic distribution or principal-symbol section of the entire sum is constructed there.

The source-sector coefficients could cancel among themselves or with other reconstructed branches in a later valid composition. Conversely, their existence away from $N$ does not prove such cancellation. Neither conclusion follows without the missing joint map and its complete remainder.

The prescription alternative is not entered. Its seven data remain obligations of a separately stated construction. No finite part, new characteristic contact, resolved base space, or XIII-E limit is substituted.

A J3 distribution is required before the remaining obligations of XIII-C can be entered for that distribution.

No uniform transverse scaling theorem is proved for ${\mathcal{T}}_{\mathrm{form}}$, the three- and four-vertex collision strata, or their intersections.

The parallel-line value eight remains a collision degree. The internal orders $3/2$, one, and $1/2$ remain orders of their copied charts. The internal pole-module frequency degree remains a degree of an amplitude presentation.

The normal order at most three in the supported-sector reference is an algebraic jet bound before characteristic extension. It is not a transverse degree of the full quantum coefficient.

Finite-order control on each compact subset away from $N$ does not give uniform control as the subset approaches $N$. Extension frameworks require their additional growth and wavefront hypotheses BF,Dang; none is inferred here merely from the finite jet count.

No joint contact gluing is performed. A later theorem must define each spectator composition, retain the inherited subgraph normalizations, and prove agreement on overlaps.

The frozen bound on permitted collision derivatives does not supply those compositions. No contact is chosen after the present calculation.

After a defined joint distribution and actual proper pushforward, the required output is a compact degree-two residual-field kernel.

A sufficient target bound is

under the fixed Minkowski cotangent identifications.

This is an output bound on the residual field variables, not an intermediate attachment-kernel bound. Equation (eq:referenceWF) is not that output theorem: its attachments have not undergone the missing joint construction and pushforward.

Properness of the inherited tree datum supplies support control only. It cannot establish this wavefront theorem for an absent coefficient.

The branches and species channels outside the retained germ also remain part of the global pair. A successful local characteristic construction would still require the appropriate global compatibility before admission of the full coefficient.

An admission line would have to identify ${R_1}^{(1)}$ as the pushed-forward coefficient of the constructed joint distribution, with the normalization (eq:Rconvention), and prove

No such line is asserted. The admitted tree datum is not substituted for ${R_1}^{(1)}$.

XIV-J releases no admitted ${R_1}^{(1)}$ and does not open XV-B.

Proof. The full joint map is Unresolved at J2. The joint section, remaining extension obligations, and output theorem have consequently not been constructed for the required coefficient.

Their absence prevents the stated admission conclusion. The exit contract for XV-B requires precisely that conclusion, not one of the partial reference results.

No differential is applied to the absent coefficient: neither ${\hat{s}}_0$, $s_{\mathrm{ext}}$, $s_{\Psi }$, nor $s_0$ appears in a calculation of ${R_1}^{(1)}$ here. No binary Ward defect is formed.

XV-A's stated free-carrier release is recorded only as an exit prerequisite XVA. It is not an analytic input to any theorem above.

If a later paper supplied the missing admission line, XV-B could attempt one relative interacting differential through one explicitly stated order in $\kappa$ and $\hslash$ on that free carrier.

The relative classical complex would remain a comparison object, not an identified quantum subalgebra. The recorded carrier uses the odd-sector antifield coupling